[ { "answer":"E", "choices":{ "A":"-10", "B":"-5", "C":"0", "D":"5", "E":"10" }, "id":10000, "question":"If \\(a - 5 = O\\), what is the value of \\(a + 5\\) ?", "explanations":{ "correct":"To find the value of \\(a + 5\\), we need to first solve the equation \\(a - 5 = O\\) for the value of \\(a\\).\n\nGiven that \\(a - 5 = O\\), we can add 5 to both sides of the equation to isolate \\(a\\):\n\n\\(a - 5 + 5 = O + 5\\)\n\nThis simplifies to:\n\n\\(a = 5\\)\n\nNow that we know the value of \\(a\\) is 5, we can substitute it into the expression \\(a + 5\\) to find the final answer:\n\n\\(5 + 5 = 10\\)\n\nTherefore, the value of \\(a + 5\\) is 10.\n\nThe answer is E." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I, II, and III" }, "id":10001, "question":"If k is an integer and \\(k = \\frac { m } { 3 } \\) which of the following must be true? \\(\\newline\\)I. m is an even number. \\(\\newline\\)II. m is a multiple of 3. \\(\\newline\\)III. m is an odd number.", "explanations":{ "correct":"To determine which statements must be true, let's analyze the given equation: \\(k = \\frac{m}{3}\\).\n\nSince \\(k\\) is an integer, \\(m\\) must be divisible by 3. This means that statement II, \"m is a multiple of 3,\" must be true.\n\nNow, let's consider statement I, \"m is an even number.\" To determine if this statement must be true, we need to examine the relationship between \\(m\\) and \\(k\\). Since \\(k\\) is equal to \\(m\\) divided by 3, we can rewrite the equation as \\(m = 3k\\).\n\\(\\newline\\)If \\(m\\) is divisible by 3, then \\(3k\\) is divisible by 3 as well. This means that \\(k\\) must be an integer. However, it does not necessarily mean that \\(m\\) must be an even number. For example, if \\(k\\) is 1, then \\(m\\) would be 3, which is an odd number. Therefore, statement I is not necessarily true.\n\nLastly, let's consider statement III, \"m is an odd number.\" Similar to our analysis for statement I, we can see that \\(m\\) does not have to be an odd number. If \\(k\\) is 1, then \\(m\\) would be 3, which is an odd number. However, if \\(k\\) is 2, then \\(m\\) would be 6, which is an even number. Therefore, statement III is not necessarily true.\n\\(\\newline\\)In conclusion, the only statement that must be true is statement II, \"m is a multiple of 3.\" Therefore, the answer is B) II only." } }, { "answer":"C", "choices":{ "A":"0", "B":"1", "C":"2", "D":"3", "E":"4" }, "id":10006, "question":"If \\(3y^4 + xy - x^2 = y^2 - 25\\) and \\(y = 0\\), how many possible values of \\(x\\) are there?", "explanations":{ "correct":"To find the possible values of \\(x\\) when \\(y = 0\\), we substitute \\(y = 0\\) into the given equation:\n\n\\(3(0)^4 + x(0) - x^2 = (0)^2 - 25\\)\n\nSimplifying this equation, we get:\n\n\\(0 + 0 - x^2 = 0 - 25\\)\n\nSimplifying further, we have:\n\n\\(-x^2 = -25\\)\n\nTo solve for \\(x\\), we can multiply both sides of the equation by -1 to get:\n\n\\(x^2 = 25\\)\n\nTaking the square root of both sides, we have:\n\n\\(x = \\pm 5\\)\n\nTherefore, there are 2 possible values of \\(x\\) when \\(y = 0\\).\n\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"\\(0%\\)", "B":"\\(1%\\)", "C":"\\(2%\\)", "D":"\\(20%\\)", "E":"No change" }, "id":10007, "question":"If the price of a jacket was increased by \\(10%\\) last week and then decreased \\(10%\\) this week, what is the overall percent change from the original price?", "explanations":{ "correct":"To find the overall percent change from the original price, we need to calculate the net effect of the two changes: the \\(10%\\) increase and the \\(10%\\) decrease.\n\nLet's assume the original price of the jacket is \\(100\\) dollars.\n\nFirst, we calculate the increase of \\(10%\\). This is done by multiplying the original price by \\(1 + \\frac{10}{100} = 1.1\\). So, the price after the increase is \\(100 \\times 1.1 = 110\\) dollars.\n\nNext, we calculate the decrease of \\(10%\\). This is done by multiplying the price after the increase by \\(1 - \\frac{10}{100} = 0.9\\). So, the price after the decrease is \\(110 \\times 0.9 = 99\\) dollars.\n\nTo find the overall percent change, we compare the final price of \\(99\\) dollars to the original price of \\(100\\) dollars. The percent change is given by \\(\\frac{{\\text{{final price}} - \\text{{original price}}}}{{\\text{{original price}}}} \\times 100%\\).\n\nPlugging in the values, we get \\(\\frac{{99 - 100}}{{100}} \\times 100% = -1%\\).\n\nTherefore, the overall percent change from the original price is \\(-1%\\) which means there was a \\(1%\\) decrease.\n\nThe answer is B) \\(1%\\)" } }, { "answer":"B", "choices":{ "A":"1", "B":"9", "C":"15", "D":"24", "E":"40" }, "id":10008, "question":"For all real numbers x and y, let \\(\\star\\) be defined by \\(x \\star y = x^2 - y\\). What is the value of \\(4 \\star (3 \\star 2)\\)?", "explanations":{ "correct":"To find the value of \\(4 \\star (3 \\star 2)\\), we need to substitute the given values into the expression \\(x \\star y = x^2 - y\\).\n\nFirst, let's find the value of \\(3 \\star 2\\):\n\\(3 \\star 2 = 3^2 - 2 = 9 - 2 = 7\\).\n\nNow, we can substitute this value into the expression \\(4 \\star (3 \\star 2)\\):\n\\(4 \\star (3 \\star 2) = 4 \\star 7 = 4^2 - 7 = 16 - 7 = 9\\).\n\nTherefore, the value of \\(4 \\star (3 \\star 2)\\) is 9.\n\nThe answer is B) 9." } }, { "answer":"A", "choices":{ "A":"2", "B":"4", "C":"6", "D":"8", "E":"10" }, "id":10011, "question":"If \\(x + 2y = 14\\), \\(y = z + 2\\), and \\(z = 4\\), then what is the value of x?", "explanations":{ "correct":"To find the value of x, we need to substitute the given values into the equation \\(x + 2y = 14\\).\n\nGiven:\n\\(y = z + 2\\) and \\(z = 4\\)\n\nSubstituting \\(z = 4\\) into \\(y = z + 2\\), we get:\n\\(y = 4 + 2\\)\n\\(y = 6\\)\n\nNow, substituting \\(y = 6\\) into \\(x + 2y = 14\\), we have:\n\\(x + 2(6) = 14\\)\n\\(x + 12 = 14\\)\n\\(x = 14 - 12\\)\n\\(x = 2\\)\n\nTherefore, the value of x is 2.\n\nThe answer is A) 2." } }, { "answer":"B", "choices":{ "A":"68", "B":"64", "C":"60", "D":"16", "E":"15" }, "id":10014, "question":"Jackie wants to place some photographs into her photo album and discovers that pages 10 through 25, including 10 and 25, are unfilled. If she can place 4 photographs on each unfilled page, what is the total number of photographs she can place on these pages?", "explanations":{ "correct":"To find the total number of photographs Jackie can place on the unfilled pages, we need to determine the number of unfilled pages and then multiply it by the number of photographs she can place on each page.\n\nGiven that pages 10 through 25, including 10 and 25, are unfilled, we can calculate the number of unfilled pages as follows:\n\nNumber of unfilled pages = 25 - 10 + 1 = 16\n\nSince Jackie can place 4 photographs on each unfilled page, we can calculate the total number of photographs she can place as follows:\n\nTotal number of photographs = Number of unfilled pages * Number of photographs per page\nTotal number of photographs = 16 * 4 = 64\n\nTherefore, the answer is B) 64." } }, { "answer":"D", "choices":{ "A":"\\(k \\leq p + \\frac { 60 } { 3 } \\)", "B":"\\(k \\leq \\frac { p } { 3 } + 60\\)", "C":"\\(k \\leq p + \\frac { 20 } { 3 } \\)", "D":"\\(k \\geq p + \\frac { 60 } { 3 } \\)", "E":"\\(k \\geq \\frac { p } { 3 } + 60\\)" }, "id":10024, "question":"If \\(k\\) is at least 20 more than \\(\\frac { 1 } { 3 } \\)of \\(p\\), which of the following describes this relationship?", "explanations":{ "correct":"To solve this problem, we need to translate the given information into an equation or inequality.\n\nLet's start by breaking down the given information:\n- \\(k\\) is at least 20 more than \\(\\frac{1}{3}\\) of \\(p\\).\n\nWe can express this relationship as an inequality:\n\\(k \\geq \\frac{1}{3}p + 20\\)\n\nNow, let's simplify this inequality:\nMultiply both sides of the inequality by 3 to eliminate the fraction:\n\\(3k \\geq p + 60\\)\n\nSince we are looking for an inequality that describes the relationship, we can conclude that:\n\\(k \\geq p + \\frac{60}{3}\\)\n\nTherefore, the answer is D) \\(k \\geq p + \\frac{60}{3}\\)." } }, { "answer":"D", "choices":{ "A":"\\(6n\\)", "B":"\\(9n\\)", "C":"\\(12n\\)", "D":"\\(15n\\)", "E":"\\(21n\\)" }, "id":10026, "question":"In the figure above, \\(\\triangle { ABC } \\) is equilateral and ACDE is a square. If \\(AC = 3n\\), what is the perimeter of polygon ABCDE ?", "explanations":{ "correct":"To find the perimeter of polygon ABCDE, we need to determine the lengths of all the sides and add them together.\n\nGiven that triangle ABC is equilateral, all three sides are equal in length. Let's call this length \"s\". Since AC = 3n, we can write:\n\nAC = s = 3n\n\nNow, let's consider the square ACDE. Since AC = 3n, the length of one side of the square is also 3n. Since the opposite sides of a square are equal in length, the length of side DE is also 3n.\n\nTo find the perimeter of the polygon ABCDE, we need to add up the lengths of all the sides:\n\nPerimeter = AB + BC + CD + DE + EA\n\nSince triangle ABC is equilateral, AB = BC = s = 3n.\n\nSince square ACDE has sides of length 3n, CD = DE = 3n.\n\nFinally, EA is the same as AC, so EA = AC = 3n.\n\nPlugging in the values, we have:\n\nPerimeter = 3n + 3n + 3n + 3n + 3n = 15n\n\nTherefore, the perimeter of polygon ABCDE is 15n.\n\nThe answer is D) \\(15n\\)." } }, { "answer":"B", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":10029, "question":"If a, b, and x are positive integers and \\(\\triangle(a,x,b)\\) is defined as the number of possible different triangles with sides of lengths a, b, and x, what is the value of \\(\\triangle(2,x,7)\\)?", "explanations":{ "correct":"To find the value of \\\\(\\\\triangle(2,x,7)\\\\), we need to determine the number of possible different triangles with sides of lengths 2, x, and 7.\n\nFor a triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side. In this case, we have the following conditions:\n\n1) 2 + x > 7\n2) 2 + 7 > x\n3) x + 7 > 2\n\nSimplifying these inequalities, we get:\n1) x > 5\n2) x < 9\n3) x > -5\n\nSince x is a positive integer, the valid values for x are 6, 7, and 8.\n\nFor each valid value of x, we can form a unique triangle. Therefore, the value of \\\\(\\\\triangle(2,x,7)\\\\) is the number of valid values of x, which is 3.\n\nThe answer is B) 3." } }, { "answer":"D", "choices":{ "A":"\\(-36\\)", "B":"\\(-28\\)", "C":"\\(-4\\)", "D":"4", "E":"36" }, "id":10031, "question":"If \\(r = -2\\) and \\(s = 5\\), what is the value of \\(r^2 (2r + s)\\)?", "explanations":{ "correct":"To find the value of \\(r^2 (2r + s)\\), we need to substitute the given values of \\(r\\) and \\(s\\) into the expression and simplify.\n\nGiven:\n\\(r = -2\\)\n\\(s = 5\\)\n\nSubstituting the values:\n\\(r^2 (2r + s) = (-2)^2 (2(-2) + 5)\\)\n\nSimplifying inside the parentheses:\n\\(r^2 (2r + s) = (-2)^2 (2(-2) + 5) = 4(-4 + 5)\\)\n\nSimplifying further:\n\\(r^2 (2r + s) = 4(-4 + 5) = 4(1)\\)\n\nFinally, multiplying:\n\\(r^2 (2r + s) = 4(1) = 4\\)\n\nTherefore, the value of \\(r^2 (2r + s)\\) is 4.\n\nThe answer is D." } }, { "answer":"A", "choices":{ "A":"3", "B":"4", "C":"5", "D":"7", "E":"9" }, "id":10033, "question":"The average rate of change of a function f between a and b is defined by \\(C(x) = \\frac { f(b) - f(a) } { b - a } \\). If \\(f(2) = 6\\) and \\(f(5) = 15\\), what is the average rate of change between 2 and 5?", "explanations":{ "correct":"To find the average rate of change of a function f between two points, we can use the formula \\(C(x) = \\frac{{f(b) - f(a)}}{{b - a}}\\), where \\(a\\) and \\(b\\) are the x-values of the two points.\n\\(\\newline\\)In this case, we are given that \\(f(2) = 6\\) and \\(f(5) = 15\\), and we need to find the average rate of change between 2 and 5.\n\nUsing the formula, we substitute the given values into the equation:\n\\(C(x) = \\frac{{f(5) - f(2)}}{{5 - 2}}\\)\n\nSubstituting the given values, we have:\n\\(C(x) = \\frac{{15 - 6}}{{5 - 2}}\\)\n\nSimplifying the numerator and denominator, we get:\n\\(C(x) = \\frac{9}{3}\\)\n\nFurther simplifying, we have:\n\\(C(x) = 3\\)\n\nTherefore, the average rate of change between 2 and 5 is 3.\n\nThe answer is A) 3." } }, { "answer":"B", "choices":{ "A":"1,2,3", "B":"0,1,2", "C":"-1,0,1", "D":"0,2", "E":"1,2" }, "id":10034, "question":"If set S has the property that x is in S, and \\(x^2 - 2x + 1\\) is also in S, which of the following sets could be S?", "explanations":{ "correct":"To determine which set could be S, we need to find the values of x that satisfy the given property: x is in S, and \\(x^2 - 2x + 1\\) is also in S.\n\nLet's start by analyzing the expression \\(x^2 - 2x + 1\\). This expression represents a quadratic function. We can factor it as \\((x-1)^2\\). \n\nNow, let's consider the possible values of x that satisfy the given property. Since x is in S, we can choose any value from the given sets. However, we need to ensure that \\((x-1)^2\\) is also in S.\n\nLet's go through each set and check if it satisfies the property:\n\nA) Set A: 1, 2, 3\n- If we choose x = 1, then \\((1-1)^2 = 0^2 = 0\\) is in S.\n- If we choose x = 2, then \\((2-1)^2 = 1^2 = 1\\) is in S.\n- If we choose x = 3, then \\((3-1)^2 = 2^2 = 4\\) is not in S.\n\nB) Set B: 0, 1, 2\n- If we choose x = 0, then \\((0-1)^2 = (-1)^2 = 1\\) is in S.\n- If we choose x = 1, then \\((1-1)^2 = 0^2 = 0\\) is in S.\n- If we choose x = 2, then \\((2-1)^2 = 1^2 = 1\\) is in S.\n\nC) Set C: -1, 0, 1\n- If we choose x = -1, then \\((-1-1)^2 = (-2)^2 = 4\\) is not in S.\n- If we choose x = 0, then \\((0-1)^2 = (-1)^2 = 1\\) is in S.\n- If we choose x = 1, then \\((1-1)^2 = 0^2 = 0\\) is in S.\n\nD) Set D: 0, 2\n- If we choose x = 0, then \\((0-1)^2 = (-1)^2 = 1\\) is in S.\n- If we choose x = 2, then \\((2-1)^2 = 1^2 = 1\\) is in S.\n\nE) Set E: 1, 2\n- If we choose x = 1, then \\((1-1)^2 = 0^2 = 0\\) is in S.\n- If we choose x = 2, then \\((2-1)^2 = 1^2 = 1\\) is in S.\n\nFrom our analysis, we can see that sets B, D, and E satisfy the given property. Therefore, the answer is B, D, E." } }, { "answer":"B", "choices":{ "A":"70", "B":"90", "C":"120", "D":"150", "E":"185" }, "id":10042, "question":"If 125 percent of x is 150, what is x percent of 75?", "explanations":{ "correct":"To find the answer, we need to first determine the value of x. \n\nThe statement \"125 percent of x is 150\" can be written as the equation: \n1.25x = 150\n\nTo solve for x, we divide both sides of the equation by 1.25:\nx = 150 / 1.25\nx = 120\n\nNow that we know x is 120, we can find x percent of 75. \n\nTo find x percent of a number, we multiply the number by x/100. \nSo, x percent of 75 is (x/100) * 75.\n\nSubstituting x = 120 into the equation, we have:\n(120/100) * 75 = 1.2 * 75 = 90\n\nTherefore, the answer is B) 90." } }, { "answer":"C", "choices":{ "A":"14%", "B":"7%", "C":"1.4%", "D":"0.14%", "E":"0.07%" }, "id":10043, "question":"If \\(p > 0\\), then 5 percent of 7 percent of \\(4p\\) equals what percent of \\(p\\) ?", "explanations":{ "correct":"To find the answer, we need to calculate 5 percent of 7 percent of 4p and then express it as a percentage of p.\n\nStep 1: Calculate 5 percent of 7 percent of 4p.\n5 percent of 7 percent can be written as \\(0.05 \\times 0.07\\). Multiplying these two values gives us \\(0.0035\\).\nNow, we need to find 0.0035 of 4p. Multiplying \\(0.0035\\) by \\(4p\\) gives us \\(0.014p\\).\n\nStep 2: Express \\(0.014p\\) as a percentage of p.\nTo express \\(0.014p\\) as a percentage of p, we divide \\(0.014p\\) by p and multiply by 100.\n\\(\\frac{0.014p}{p} \\times 100 = 0.014 \\times 100 = 1.4%\\)\n\nTherefore, 5 percent of 7 percent of 4p is equal to 1.4 percent of p.\n\nThe answer is C) 1.4%." } }, { "answer":"E", "choices":{ "A":"\\(|x - 13| = \\frac{1}{16}\\)", "B":"\\(|x + 13| = \\frac{1}{16}\\)", "C":"\\(|x - 13| > \\frac{1}{16}\\)", "D":"\\(|x + 13| < \\frac{1}{16}\\)", "E":"\\(|x - 13| < \\frac{1}{16}\\)" }, "id":10045, "question":"A lumber company producing 13-inch boards can only sell ones cut between \\(12 \\frac{15}{16}\\) and \\(13 \\frac{1}{16}\\). If it sells a board that is x inches long, which of the following describes all possible values ofx?", "explanations":{ "correct":"To determine the possible values of x, we need to consider the range of acceptable board lengths. The lumber company can only sell boards that are cut between \\(12 \\frac{15}{16}\\) and \\(13 \\frac{1}{16}\\) inches long.\n\nLet's break down the given information:\n- The lower limit is \\(12 \\frac{15}{16}\\) inches, which can be written as \\(\\frac{207}{16}\\) inches.\n- The upper limit is \\(13 \\frac{1}{16}\\) inches, which can be written as \\(\\frac{209}{16}\\) inches.\n\nTherefore, any board length x that satisfies \\(\\frac{207}{16} \\leq x \\leq \\frac{209}{16}\\) is a possible value.\n\nTo express this inequality in a more convenient form, we can multiply all terms by 16 to eliminate the fractions:\n\\(207 \\leq 16x \\leq 209\\)\n\nNow, we can divide all terms by 16 to isolate x:\n\\(\\frac{207}{16} \\leq x \\leq \\frac{209}{16}\\)\n\nSimplifying the fractions, we have:\n\\(12 \\frac{15}{16} \\leq x \\leq 13 \\frac{1}{16}\\)\n\nThis means that x must be greater than or equal to \\(12 \\frac{15}{16}\\) and less than or equal to \\(13 \\frac{1}{16}\\).\n\nNow, let's analyze the answer choices:\nA) \\(|x - 13| = \\frac{1}{16}\\)\nThis equation represents the distance between x and 13 being equal to \\(\\frac{1}{16}\\). However, we need x to be within a range, not just a specific distance from 13. Therefore, this option is not correct.\n\nB) \\(|x + 13| = \\frac{1}{16}\\)\nSimilar to option A, this equation represents the distance between x and -13 being equal to \\(\\frac{1}{16}\\). It does not consider the range of acceptable values for x. Therefore, this option is not correct.\n\nC) \\(|x - 13| > \\frac{1}{16}\\)\nThis inequality represents the distance between x and 13 being greater than \\(\\frac{1}{16}\\). However, we need x to be within a specific range, not just outside a specific distance from 13. Therefore, this option is not correct.\n\nD) \\(|x + 13| < \\frac{1}{16}\\)\nSimilar to option C, this inequality represents the distance between x and -13 being less than \\(\\frac{1}{16}\\). It does not consider the range of acceptable values for x. Therefore, this option is not correct.\n\nE) \\(|x - 13| < \\frac{1}{16}\\)\nThis inequality represents the distance between x and 13 being less than \\(\\frac{1}{16}\\), which matches the range we derived earlier: \\(12 \\frac{15}{16} \\leq x \\leq 13 \\frac{1}{16}\\). Therefore, this option is correct.\n\nTherefore, the answer is E." } }, { "answer":"D", "choices":{ "A":"\\(\\frac{xy}{19}\\)", "B":"\\(\\frac{y + 19}{x}\\)", "C":"\\(\\frac{2y - x}{19}\\)", "D":"\\(\\frac{y - 19}{x}\\)", "E":"\\(\\frac{19 - x}{y}\\)" }, "id":10047, "question":"John has y dollars to spend on some new CDs from Music Plus, an online record store. He can buy any CDs at the members' price of x dollars each. To be a member, John has to pay a one-time fee of \\$ 19. Which of the following expressions represents the number of CDs John can purchase from Music Plus?", "explanations":{ "correct":"To determine the number of CDs John can purchase from Music Plus, we need to consider the total amount of money he has and the cost of each CD.\n\nLet's break down the problem step-by-step:\n\n1. John has y dollars to spend.\n2. The cost of each CD is x dollars.\n3. John needs to pay a one-time fee of \\$ 19 to become a member.\n\nTo find the number of CDs John can purchase, we need to divide the total amount of money he has (y) by the cost of each CD (x). However, we also need to consider the one-time membership fee of \\$ 19.\n\nTherefore, the correct expression to represent the number of CDs John can purchase is:\n\n\\(\\frac{y - 19}{x}\\)\n\nThis expression subtracts the one-time fee from the total amount of money John has (y - 19) and then divides it by the cost of each CD (x).\n\nHence, the answer is D) \\(\\frac{y - 19}{x}\\)." } }, { "answer":"E", "choices":{ "A":"0", "B":"2", "C":"4", "D":"9", "E":"16" }, "id":10052, "question":"If \\(\\frac { 2x + 10\\sqrt { x } + 12 } { \\sqrt { x } + 3 } = 3\\sqrt { x } \\), what is the value of \\(x\\)?", "explanations":{ "correct":"To solve the given equation, we will start by isolating the square root term on one side of the equation. \n\nGiven: \\( \\frac { 2x + 10\\sqrt { x } + 12 } { \\sqrt { x } + 3 } = 3\\sqrt { x } \\)\n\nFirst, we will multiply both sides of the equation by \\( \\sqrt { x } + 3 \\) to eliminate the denominator:\n\n\\( (2x + 10\\sqrt { x } + 12) = 3\\sqrt { x } (\\sqrt { x } + 3) \\)\n\nNext, we will distribute \\( 3\\sqrt { x } \\) to both terms inside the parentheses:\n\n\\( 2x + 10\\sqrt { x } + 12 = 3x + 9\\sqrt { x } \\)\n\nNow, we will move all terms involving \\( \\sqrt { x } \\) to one side of the equation and all terms without \\( \\sqrt { x } \\) to the other side:\n\n\\( 10\\sqrt { x } - 9\\sqrt { x } = 3x - 2x - 12 \\)\n\nSimplifying the equation further:\n\n\\( \\sqrt { x } = x - 12 \\)\n\nTo eliminate the square root, we will square both sides of the equation:\n\n\\( (\\sqrt { x })^2 = (x - 12)^2 \\)\n\nThis simplifies to:\n\n\\( x = x^2 - 24x + 144 \\)\n\nRearranging the equation to form a quadratic equation:\n\n\\( x^2 - 25x + 144 = 0 \\)\n\nNow, we can factorize the quadratic equation:\n\n\\( (x - 16)(x - 9) = 0 \\)\n\nSetting each factor equal to zero:\n\n\\( x - 16 = 0 \\) or \\( x - 9 = 0 \\)\n\nSolving for \\( x \\) in each equation:\n\n\\( x = 16 \\) or \\( x = 9 \\)\n\nTherefore, the possible values of \\( x \\) are 16 and 9. \n\nHowever, we need to double-check our solutions by substituting them back into the original equation:\n\nFor \\( x = 16 \\):\n\n\\( \\frac { 2(16) + 10\\sqrt { 16 } + 12 } { \\sqrt { 16 } + 3 } = 3\\sqrt { 16 } \\)\n\nSimplifying both sides of the equation:\n\n\\( \\frac { 32 + 40 + 12 } { 4 + 3 } = 3(4) \\)\n\n\\( \\frac { 84 } { 7 } = 12 \\)\n\n\\( 12 = 12 \\)\n\nThe equation holds true for \\( x = 16 \\).\n\nFor \\( x = 9 \\):\n\n\\( \\frac { 2(9) + 10\\sqrt { 9 } + 12 } { \\sqrt { 9 } + 3 } = 3\\sqrt { 9 } \\)\n\nSimplifying both sides of the equation:\n\n\\( \\frac { 18 + 30 + 12 } { 3 + 3 } = 3(3) \\)\n\n\\( \\frac { 60 } { 6 } = 9 \\)\n\n\\( 10 = 9 \\)\n\nThe equation does not hold true for \\( x = 9 \\).\n\nTherefore, the only valid solution is \\( x = 16 \\).\n\nThe answer is E." } }, { "answer":"C", "choices":{ "A":"23", "B":"17", "C":"16", "D":"11", "E":"3" }, "id":10059, "question":"The average (arithmetic mean) of x and y is 7, and the average of x, y, and z is 10. What is the value of z?", "explanations":{ "correct":"To find the value of z, we need to use the information given in the question. \n\nFirst, we are told that the average of x and y is 7. This means that the sum of x and y divided by 2 is equal to 7. We can write this as (x + y)/2 = 7.\n\nNext, we are told that the average of x, y, and z is 10. This means that the sum of x, y, and z divided by 3 is equal to 10. We can write this as (x + y + z)/3 = 10.\n\nTo find the value of z, we can solve these two equations simultaneously. \n\nFrom the first equation, we can multiply both sides by 2 to get x + y = 14.\n\nNow, we can substitute this value of x + y into the second equation:\n\n(x + y + z)/3 = 10\n(14 + z)/3 = 10\n\nNext, we can multiply both sides by 3 to get rid of the fraction:\n\n14 + z = 30\n\nSubtracting 14 from both sides, we get:\n\nz = 16\n\nTherefore, the value of z is 16.\n\nThe answer is C) 16." } }, { "answer":"C", "choices":{ "A":"0.9", "B":"1.4", "C":"9", "D":"14", "E":"90" }, "id":10060, "question":"If \\(0.4x + 2 = 5.6\\), what is the value of x?", "explanations":{ "correct":"To find the value of x in the equation \\(0.4x + 2 = 5.6\\), we need to isolate x on one side of the equation.\n\nFirst, we subtract 2 from both sides of the equation to get rid of the constant term on the left side:\n\\(0.4x + 2 - 2 = 5.6 - 2\\)\nThis simplifies to:\n\\(0.4x = 3.6\\)\n\nNext, we divide both sides of the equation by 0.4 to solve for x:\n\\(\\frac{{0.4x}}{{0.4}} = \\frac{{3.6}}{{0.4}}\\)\nThis simplifies to:\n\\(x = 9\\)\n\nTherefore, the value of x is 9.\n\nThe answer is C) 9." } }, { "answer":"C", "choices":{ "A":"0", "B":"1", "C":"2", "D":"3", "E":"4" }, "id":10061, "question":"After 4 apples were added to a sack, there were 3 times as many apples in the sack as before. How many apples were in the sack before the addition?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume that the number of apples in the sack before the addition is \"x\".\n\nAfter adding 4 apples, the total number of apples in the sack becomes \"x + 4\".\n\nAccording to the problem, there were 3 times as many apples in the sack after the addition. So, we can set up the equation:\n\nx + 4 = 3x\n\nNow, let's solve for \"x\":\n\nSubtract \"x\" from both sides of the equation:\n\n4 = 2x\n\nDivide both sides of the equation by 2:\n\n2 = x\n\nTherefore, there were 2 apples in the sack before the addition.\n\nThe answer is C) 2." } }, { "answer":"E", "choices":{ "A":"\\$ 20", "B":"\\$ 35", "C":"\\$ 40", "D":"\\$ 65", "E":"\\$ 80" }, "id":10062, "question":"Hillary buys a television on a 12-month installment plan. If each of her first three installments is twice as much as each of her nine remaining installments, and her total payment is \\$ 600, how much is her first installment?", "explanations":{ "correct":"Let's assume the amount of each of the nine remaining installments is x dollars. \n\nAccording to the given information, each of the first three installments is twice as much as each of the nine remaining installments. So, the amount of each of the first three installments is 2x dollars.\n\nSince Hillary's total payment is \\$ 600, we can set up the equation:\n\n3(2x) + 9(x) = 600\n\nSimplifying the equation, we get:\n\n6x + 9x = 600\n\nCombining like terms, we have:\n\n15x = 600\n\nDividing both sides of the equation by 15, we find:\n\nx = 40\n\nTherefore, each of the nine remaining installments is \\$ 40.\n\nTo find the amount of the first installment, we multiply the amount of each of the nine remaining installments by 2:\n\n2 * 40 = 80\n\nSo, the amount of Hillary's first installment is \\$ 80.\n\nThe answer is E) \\$ 80." } }, { "answer":"A", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III only", "E":"I, II and III" }, "id":10063, "question":"If a, b, and c are positive and \\(a^ { -2 } b^2c^2 > a^ { -3 } b^3c^2\\), which of the following must be true? \\(\\newline\\)I. \\(a > b\\) \\(\\newline\\)II. \\(a > c\\) \\(\\newline\\)III. \\(b > c\\)", "explanations":{ "correct":"To compare the expressions \\\\(a^ { -2 } b^2c^2\\\\) and \\\\(a^ { -3 } b^3c^2\\\\), we can simplify them by using the properties of exponents.\n\nFirst, let's simplify \\\\(a^ { -2 } b^2c^2\\\\). The negative exponent \\\\(-2\\\\) means that we can rewrite \\\\(a^ { -2 }\\\\) as \\\\(\\\\frac{1}{a^2}\\\\). Therefore, \\\\(a^ { -2 } b^2c^2\\\\) becomes \\\\(\\\\frac{b^2c^2}{a^2}\\\\).\n\nNext, let's simplify \\\\(a^ { -3 } b^3c^2\\\\). Similarly, the negative exponent \\\\(-3\\\\) means that we can rewrite \\\\(a^ { -3 }\\\\) as \\\\(\\\\frac{1}{a^3}\\\\). Therefore, \\\\(a^ { -3 } b^3c^2\\\\) becomes \\\\(\\\\frac{b^3c^2}{a^3}\\\\).\n\nNow, we can compare the two expressions: \\\\(\\\\frac{b^2c^2}{a^2}\\\\) and \\\\(\\\\frac{b^3c^2}{a^3}\\\\).\n\nTo compare fractions, we can cross-multiply. Multiplying both sides of the inequality by \\\\(a^2\\\\), we get \\\\(b^2c^2 > \\\\frac{b^3c^2}{a}\\\\).\n\nNext, we can multiply both sides of the inequality by \\\\(a\\\\) to eliminate the fraction: \\\\(ab^2c^2 > b^3c^2\\\\).\n\nNow, we can cancel out the common factors \\\\(b^2c^2\\\\) on both sides of the inequality: \\\\(a > b\\\\).\n\nTherefore, the statement \"I. \\\\(a > b\\\\)\" must be true.\n\nHowever, we cannot determine the relationship between \\\\(a\\\\) and \\\\(c\\\\) or between \\\\(b\\\\) and \\\\(c\\\\) based on the given inequality. Therefore, the statements \"II. \\\\(a > c\\\\)\" and \"III. \\\\(b > c\\\\)\" cannot be concluded.\n\nThe answer is A) I only." } }, { "answer":"C", "choices":{ "A":"\\(3(b - 12) = 9\\)", "B":"\\(12 - 3b = 9\\)", "C":"\\(3b - 12 = 9\\)", "D":"\\(12b - 3 = 9\\)", "E":"\\(12 + 3b = 9\\)" }, "id":10066, "question":"Which of the following is an equation equivalent to the statement ``12 less than the product of 3 and b is 9''?", "explanations":{ "correct":"To solve this problem, we need to translate the given statement into an equation.\n\nThe statement \"12 less than the product of 3 and b is 9\" can be written as:\n\n\\(3b - 12 = 9\\)\n\nThis equation represents the product of 3 and b, subtracting 12 from it, and the result is equal to 9.\n\nTherefore, the equation equivalent to the given statement is:\n\nC) \\(3b - 12 = 9\\)\n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"50", "B":"100", "C":"200", "D":"300", "E":"400" }, "id":10068, "question":"Two cylindrical tanks have the same height, but the radius of one tank equals the diameter of the other. If the volume of the larger is \\(k%\\) more than the volume of the smaller, \\(k\\) =", "explanations":{ "correct":"Let's assume the radius of the smaller tank is \\(r\\) units. Since the radius of the larger tank equals the diameter of the smaller tank, the radius of the larger tank is \\(2r\\) units.\n\nThe volume of a cylinder is given by the formula \\(V = \\pi r^2 h\\), where \\(r\\) is the radius and \\(h\\) is the height.\n\nFor the smaller tank, the volume is \\(V_1 = \\pi r^2 h\\).\nFor the larger tank, the volume is \\(V_2 = \\pi (2r)^2 h = 4\\pi r^2 h\\).\n\nWe are given that the volume of the larger tank is \\(k%\\) more than the volume of the smaller tank. Mathematically, this can be expressed as:\n\\[V_2 = V_1 + \\frac{k}{100}V_1\\]\n\\[4\\pi r^2 h = \\pi r^2 h + \\frac{k}{100}\\pi r^2 h\\]\n\\[4 = 1 + \\frac{k}{100}\\]\n\\[3 = \\frac{k}{100}\\]\n\\[k = 300\\]\n\nTherefore, the answer is D) 300." } }, { "answer":"B", "choices":{ "A":"\\(f(8)\\)", "B":"\\(f(4)\\)", "C":"\\(f(0)\\)", "D":"\\(f(-4)\\)", "E":"\\(f(-8)\\)" }, "id":10069, "question":"If \\(f(x) = |x - 4|\\) , which of the following has the LEAST value?", "explanations":{ "correct":"To find the least value of \\(f(x) = |x - 4|\\), we need to evaluate the function for each given value and compare the results.\n\nLet's start by evaluating \\(f(8)\\):\n\\(f(8) = |8 - 4| = |4| = 4\\)\n\nNext, let's evaluate \\(f(4)\\):\n\\(f(4) = |4 - 4| = |0| = 0\\)\n\nNow, let's evaluate \\(f(0)\\):\n\\(f(0) = |0 - 4| = |-4| = 4\\)\n\nNext, let's evaluate \\(f(-4)\\):\n\\(f(-4) = |-4 - 4| = |-8| = 8\\)\n\nFinally, let's evaluate \\(f(-8)\\):\n\\(f(-8) = |-8 - 4| = |-12| = 12\\)\n\nFrom the evaluations, we can see that the least value is obtained when we evaluate \\(f(4)\\), which gives us a result of 0.\n\nTherefore, the answer is B) \\(f(4)\\)." } }, { "answer":"C", "choices":{ "A":"28", "B":"42", "C":"58", "D":"78", "E":"108" }, "id":10070, "question":"What is the value of \\(2x^2 - 3x - 7\\) when \\(x = -5\\) ?", "explanations":{ "correct":"To find the value of \\(2x^2 - 3x - 7\\) when \\(x = -5\\), we substitute -5 for \\(x\\) in the expression.\n\nStep 1: Substitute -5 for \\(x\\) in the expression:\n\\(2(-5)^2 - 3(-5) - 7\\)\n\nStep 2: Simplify the expression inside the parentheses:\n\\(2(25) + 15 - 7\\)\n\nStep 3: Simplify the multiplication:\n\\(50 + 15 - 7\\)\n\nStep 4: Simplify the addition and subtraction:\n\\(65 - 7\\)\n\nStep 5: Perform the subtraction:\n\\(58\\)\n\nTherefore, the value of \\(2x^2 - 3x - 7\\) when \\(x = -5\\) is 58.\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"2", "B":"10", "C":"20", "D":"40", "E":"160" }, "id":10073, "question":"The number of turtles in Mosquito Pond has doubled every three years since the pond was discovered. The number of turtles in the pond can be described as \\(n = (x)2^{\\frac{t}{3}}\\), where t is the number of years since the pond was discovered, n is the number of turtles in the pond at time t, and x is the number of turtles in the pond when it was discovered. If there were 160 turtles in Mosquito Pond nine years after it was discovered, then how many turtles lived in the pond the year it was discovered?", "explanations":{ "correct":"To find the number of turtles that lived in the pond the year it was discovered, we need to substitute the given values into the equation \\(n = (x)2^{\\frac{t}{3}}\\).\n\nWe are given that there were 160 turtles in the pond nine years after it was discovered, so we can substitute \\(n = 160\\) and \\(t = 9\\) into the equation:\n\n\\(160 = (x)2^{\\frac{9}{3}}\\)\n\nSimplifying the exponent, we have:\n\n\\(160 = (x)2^3\\)\n\n\\(160 = (x)8\\)\n\nTo solve for \\(x\\), we divide both sides of the equation by 8:\n\n\\(\\frac{160}{8} = x\\)\n\n\\(20 = x\\)\n\nTherefore, there were 20 turtles in the pond the year it was discovered.\n\nThe answer is C) 20." } }, { "answer":"D", "choices":{ "A":"\\(4 \\times 4 \\div 4 + 4\\)", "B":"\\(4 \\div 4 \\times 4 + 4 \\)", "C":"\\(4 \\times 4 - 4 \\times 4\\)", "D":"\\(4 \\div 4 + 4 \\times 4\\)", "E":"\\(4 + 4 \\times 4 - 4\\)" }, "id":10080, "question":"Which of the following expressions has the greatest value?", "explanations":{ "correct":"To determine which expression has the greatest value, we need to evaluate each expression step-by-step and compare the results.\n\nLet's start with option A:\n\\(4 \\times 4 \\div 4 + 4\\)\n\nFirst, we perform the multiplication: \\(4 \\times 4 = 16\\).\nNext, we perform the division: \\(16 \\div 4 = 4\\).\nFinally, we perform the addition: \\(4 + 4 = 8\\).\n\nNow let's move on to option B:\n\\(4 \\div 4 \\times 4 + 4\\)\n\nFirst, we perform the division: \\(4 \\div 4 = 1\\).\nNext, we perform the multiplication: \\(1 \\times 4 = 4\\).\nFinally, we perform the addition: \\(4 + 4 = 8\\).\n\nNow let's evaluate option C:\n\\(4 \\times 4 - 4 \\times 4\\)\n\nFirst, we perform the multiplication: \\(4 \\times 4 = 16\\).\nNext, we perform the second multiplication: \\(4 \\times 4 = 16\\).\nFinally, we perform the subtraction: \\(16 - 16 = 0\\).\n\nNow let's evaluate option D:\n\\(4 \\div 4 + 4 \\times 4\\)\n\nFirst, we perform the division: \\(4 \\div 4 = 1\\).\nNext, we perform the multiplication: \\(4 \\times 4 = 16\\).\nFinally, we perform the addition: \\(1 + 16 = 17\\).\n\nLastly, let's evaluate option E:\n\\(4 + 4 \\times 4 - 4\\)\n\nFirst, we perform the multiplication: \\(4 \\times 4 = 16\\).\nNext, we perform the addition: \\(4 + 16 = 20\\).\nFinally, we perform the subtraction: \\(20 - 4 = 16\\).\n\nAfter evaluating all the expressions, we can see that option D, \\(4 \\div 4 + 4 \\times 4\\), has the greatest value of 17. Therefore, the answer is D." } }, { "answer":"B", "choices":{ "A":"5", "B":"6", "C":"7", "D":"8", "E":"9" }, "id":10083, "question":"Sequence Y: 5, 10, 15, ... Sequence Z: 3, 6, 12, ... The first term in Sequence Y is 5, and each term after the first is 5 more than the preceding term. The first term in sequence Z is 3, and each term after the first is 2 times the preceding term. What is the least value of x such that the xth term of Sequence Z is more than three times the xth term of Sequence Y?", "explanations":{ "correct":"To find the least value of x such that the xth term of Sequence Z is more than three times the xth term of Sequence Y, we need to compare the terms of the two sequences.\n\\(\\newline\\)In Sequence Y, the first term is 5, and each term after the first is 5 more than the preceding term. So, the terms of Sequence Y can be represented by the formula Y(n) = 5 + 5(n-1), where n represents the position of the term.\n\\(\\newline\\)In Sequence Z, the first term is 3, and each term after the first is 2 times the preceding term. So, the terms of Sequence Z can be represented by the formula Z(n) = 3 * 2^(n-1), where n represents the position of the term.\n\nNow, we need to find the least value of x such that Z(x) > 3 * Y(x).\n\nSubstituting the formulas for Y(n) and Z(n) into the inequality, we get:\n3 * 2^(x-1) > 3 * (5 + 5(x-1))\n\nSimplifying the inequality, we have:\n2^(x-1) > 5 + 5(x-1)\n\nExpanding the right side of the inequality, we get:\n2^(x-1) > 5 + 5x - 5\n\nCombining like terms, we have:\n2^(x-1) > 5x\n\nTo solve this inequality, we can start by testing the answer choices.\n\nFor x = 5, we have:\n2^(5-1) = 16 > 5 * 5 = 25 (not true)\n\nFor x = 6, we have:\n2^(6-1) = 32 > 5 * 6 = 30 (true)\n\nTherefore, the least value of x such that Z(x) is more than three times Y(x) is x = 6.\n\nThe answer is B) 6." } }, { "answer":"A", "choices":{ "A":"32", "B":"34", "C":"36", "D":"38", "E":"40" }, "id":10088, "question":"If \\(4x + 2 = 26\\), then \\(4x + 8 =\\)", "explanations":{ "correct":"To find the value of \\\\(4x + 8\\\\), we need to first solve the equation \\\\(4x + 2 = 26\\\\) for \\\\(x\\\\).\n\nStep 1: Subtract 2 from both sides of the equation:\n\\\\(4x + 2 - 2 = 26 - 2\\\\)\n\\\\(4x = 24\\\\)\n\nStep 2: Divide both sides of the equation by 4 to isolate \\\\(x\\\\):\n\\\\(\\\\frac{{4x}}{4} = \\\\frac{{24}}{4}\\\\)\n\\\\(x = 6\\\\)\n\nNow that we have found the value of \\\\(x\\\\) as 6, we can substitute it into the expression \\\\(4x + 8\\\\) to find the final answer.\n\n\\\\(4x + 8 = 4(6) + 8\\\\)\n\\\\(4x + 8 = 24 + 8\\\\)\n\\\\(4x + 8 = 32\\\\)\n\nTherefore, the answer is A) 32." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { y } { 5 } \\)", "B":"\\(\\frac { 2y } { 5 } \\)", "C":"\\(\\frac { 5y } { 2 } \\)", "D":"\\(\\frac { 7y } { 5 } \\)", "E":"\\(5y\\)" }, "id":10089, "question":"There are y sculptures in a gallery. If one is to be selected at random from the collection, the probability that a bronze statue will be selected is \\(\\frac { 2 } { 5 } \\). In terms of \\(y\\), how many of the sculptures are bronze statues?", "explanations":{ "correct":"Let's assume that the number of bronze statues in the gallery is \\(x\\).\n\nThe probability of selecting a bronze statue is given as \\(\\frac{2}{5}\\). This means that out of all the sculptures in the gallery, \\(\\frac{2}{5}\\) of them are bronze statues.\n\nWe can set up the following equation to represent this relationship:\n\\(\\frac{x}{y} = \\frac{2}{5}\\)\n\nTo solve for \\(x\\), we can cross-multiply:\n\\(5x = 2y\\)\n\nDividing both sides of the equation by 5, we get:\n\\(x = \\frac{2y}{5}\\)\n\nTherefore, the number of sculptures that are bronze statues is \\(\\frac{2y}{5}\\).\n\nThe answer is B) \\(\\frac{2y}{5}\\)." } }, { "answer":"C", "choices":{ "A":"-4", "B":"2", "C":"3", "D":"4", "E":"6" }, "id":10094, "question":"If \\( 6\\frac { a } { b } + 3 = 21\\) , what is the value of \\(\\frac { a } { b } \\) ?", "explanations":{ "correct":"To find the value of \\\\(\\frac{a}{b}\\\\), we need to isolate the variable \\\\(\\frac{a}{b}\\\\) in the given equation \\\\(6\\frac{a}{b} + 3 = 21\\\\).\n\nFirst, we can subtract 3 from both sides of the equation to get rid of the constant term:\n\\\\(6\\frac{a}{b} = 18\\\\).\n\nNext, we can divide both sides of the equation by 6 to solve for \\\\(\\frac{a}{b}\\\\):\n\\\\(\\frac{a}{b} = 3\\\\).\n\nTherefore, the value of \\\\(\\frac{a}{b}\\\\) is 3.\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"\\(2 + \\frac{1}{2}\\)", "B":"\\(2 - \\frac{1}{2}\\)", "C":"\\(\\frac{1}{2} - 2\\)", "D":"\\(2 \\times \\frac{1}{2}\\)", "E":"\\(2 \\div \\frac{1}{2}\\)" }, "id":10096, "question":"Of the following numbers, which is least?", "explanations":{ "correct":"To determine which of the given numbers is the least, we need to compare them. Let's evaluate each option step-by-step:\n\nA) \\(2 + \\frac{1}{2}\\)\nTo add these numbers, we need to find a common denominator. The common denominator is 2. So, we have:\n\\(2 + \\frac{1}{2} = \\frac{4}{2} + \\frac{1}{2} = \\frac{5}{2}\\)\n\nB) \\(2 - \\frac{1}{2}\\)\nTo subtract these numbers, we also need a common denominator. The common denominator is 2. So, we have:\n\\(2 - \\frac{1}{2} = \\frac{4}{2} - \\frac{1}{2} = \\frac{3}{2}\\)\n\nC) \\(\\frac{1}{2} - 2\\)\nTo subtract these numbers, we need a common denominator. The common denominator is 2. So, we have:\n\\(\\frac{1}{2} - 2 = \\frac{1}{2} - \\frac{4}{2} = -\\frac{3}{2}\\)\n\nD) \\(2 \\times \\frac{1}{2}\\)\nTo multiply these numbers, we simply multiply the numerators and denominators:\n\\(2 \\times \\frac{1}{2} = \\frac{2}{1} \\times \\frac{1}{2} = \\frac{2}{2} = 1\\)\n\nE) \\(2 \\div \\frac{1}{2}\\)\nTo divide these numbers, we multiply by the reciprocal of the divisor:\n\\(2 \\div \\frac{1}{2} = 2 \\times \\frac{2}{1} = \\frac{4}{1} = 4\\)\n\nNow, let's compare the results:\n\n\\(\\frac{5}{2}\\), \\(\\frac{3}{2}\\), \\(-\\frac{3}{2}\\), 1, 4\n\nThe least number among these options is \\(-\\frac{3}{2}\\).\n\nTherefore, the answer is C." } }, { "answer":"E", "choices":{ "A":"\\(-4\\)", "B":"\\(-2\\)", "C":"0", "D":"3", "E":"4" }, "id":10097, "question":"What is the y-intercept of the linear equation \\(3y - 33x = 12\\)?", "explanations":{ "correct":"To find the y-intercept of a linear equation, we need to set the value of x to 0 and solve for y. \n\nGiven the equation \\(3y - 33x = 12\\), we substitute x with 0:\n\n\\(3y - 33(0) = 12\\)\n\nSimplifying further:\n\n\\(3y - 0 = 12\\)\n\n\\(3y = 12\\)\n\nDividing both sides of the equation by 3:\n\n\\(y = \\frac{12}{3}\\)\n\n\\(y = 4\\)\n\nTherefore, the y-intercept of the linear equation \\(3y - 33x = 12\\) is 4.\n\nThe answer is E." } }, { "answer":"A", "choices":{ "A":"None", "B":"One", "C":"Three", "D":"Four", "E":"More than four" }, "id":10098, "question":"If the set of four integers \\(n\\), \\(2n\\), \\(n + 2n\\), and \\(n + 4n\\) consists only of prime numbers, then the set is called a \"prime convergence\". How many such sets exist?", "explanations":{ "correct":"To determine the number of sets that satisfy the given condition, we need to analyze the expressions \\\\(n\\\\), \\\\(2n\\\\), \\\\(n + 2n\\\\), and \\\\(n + 4n\\\\) and check if they can all be prime numbers simultaneously.\n\nLet's start by simplifying the expressions:\n- \\\\(n\\\\)\n- \\\\(2n\\\\) (which is equivalent to \\\\(n \\times 2\\\\))\n- \\\\(n + 2n\\\\) (which simplifies to \\\\(3n\\\\))\n- \\\\(n + 4n\\\\) (which simplifies to \\\\(5n\\\\))\n\nNow, we can see that all the expressions are multiples of \\\\(n\\\\) (2n is 2 times n, 3n is 3 times n, and 5n is 5 times n). \n\nFor any set to be a \"prime convergence,\" all the expressions in the set must be prime numbers. However, since all the expressions are multiples of \\\\(n\\\\), they cannot all be prime numbers simultaneously. \n\nThis means that there are no sets that satisfy the given condition. \n\nTherefore, the answer is A) None." } }, { "answer":"C", "choices":{ "A":"\\(3^ { 3k } \\)", "B":"\\(3^ { k + 2 } \\)", "C":"\\(3^ { k + 1 } \\)", "D":"\\(9^ { 2k } \\)", "E":"\\(9^ { 3k } \\)" }, "id":10104, "question":"If \\(k\\) is a positive integer, which of the following is equivalent to \\(3^k + 3^k + 3^k\\)?", "explanations":{ "correct":"To simplify the expression \\(3^k + 3^k + 3^k\\), we can combine the three terms by adding them together. \n\n\\(3^k + 3^k + 3^k = 3 \\cdot 3^k\\)\n\nSince \\(3 \\cdot 3^k\\) is equal to \\(3^{k+1}\\), the expression \\(3^k + 3^k + 3^k\\) is equivalent to \\(3^{k+1}\\).\n\nTherefore, the answer is C) \\(3^{k+1}\\)." } }, { "answer":"D", "choices":{ "A":"20", "B":"27", "C":"32", "D":"33", "E":"40" }, "id":10107, "question":"The average (arithmetic mean) of 7,20, and x is 20. What is the value of x ?", "explanations":{ "correct":"To find the value of x, we need to use the formula for the average (arithmetic mean). The formula is:\n\nAverage = Sum of all numbers / Number of numbers\n\nWe are given that the average of 7, 20, and x is 20. So, we can set up the equation:\n\n20 = (7 + 20 + x) / 3\n\nTo solve for x, we can start by multiplying both sides of the equation by 3 to get rid of the fraction:\n\n60 = 7 + 20 + x\n\nNext, we can combine like terms:\n\n60 = 27 + x\n\nTo isolate x, we can subtract 27 from both sides of the equation:\n\n60 - 27 = x\n\n33 = x\n\nTherefore, the value of x is 33.\n\nThe answer is D) 33." } }, { "answer":"C", "choices":{ "A":"2", "B":"4", "C":"6", "D":"8", "E":"10" }, "id":10108, "question":"If \\(4(x + y) - 2(x + y) = 12\\), then \\(x + y =\\)", "explanations":{ "correct":"To solve the equation \\(4(x + y) - 2(x + y) = 12\\), we can simplify the left side of the equation by distributing the coefficients:\n\n\\(4x + 4y - 2x - 2y = 12\\)\n\nCombining like terms, we have:\n\n\\(2x + 2y = 12\\)\n\nNext, we can divide both sides of the equation by 2 to isolate the variable term:\n\n\\(\\frac{2x + 2y}{2} = \\frac{12}{2}\\)\n\nThis simplifies to:\n\n\\(x + y = 6\\)\n\nTherefore, the answer is C) 6." } }, { "answer":"A", "choices":{ "A":"0", "B":"\\(\\frac { 1 } { 2 } \\)", "C":"1", "D":"\\(\\frac { 3 } { 2 } \\)", "E":"5" }, "id":10115, "question":"If \\(x = 5 + y\\) and \\(x = 5 - y\\), what is the value of \\(y\\) ?", "explanations":{ "correct":"To find the value of \\(y\\), we can set the two given equations equal to each other:\n\n\\(5 + y = 5 - y\\)\n\nNext, we can simplify the equation by combining like terms:\n\n\\(2y = 0\\)\n\nTo solve for \\(y\\), we divide both sides of the equation by 2:\n\n\\(y = \\frac{0}{2}\\)\n\nSimplifying further, we find that \\(y = 0\\).\n\nTherefore, the answer is A) 0." } }, { "answer":"E", "choices":{ "A":"1", "B":"2", "C":"10", "D":"14", "E":"24" }, "id":10116, "question":"In the equation \\(y = kx\\), \\(x = 4\\) when \\(y = 6\\) and k is a constant. What is the value of y when \\(x = 16\\)?", "explanations":{ "correct":"To find the value of y when x = 16 in the equation y = kx, we need to first determine the value of the constant k.\n\nGiven that x = 4 when y = 6, we can substitute these values into the equation to solve for k:\n\n6 = k * 4\n\nTo isolate k, we divide both sides of the equation by 4:\n\nk = 6/4\n\nSimplifying the right side gives us:\n\nk = 3/2\n\nNow that we know the value of k, we can substitute it back into the equation y = kx to find the value of y when x = 16:\n\ny = (3/2) * 16\n\nMultiplying 3/2 by 16 gives us:\n\ny = 24\n\nTherefore, the value of y when x = 16 is 24.\n\nThe answer is E) 24." } }, { "answer":"E", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III", "E":"I, II and III" }, "id":10117, "question":"Points L, M, and N lie in a plane. If the distance between L and M is 4, and the distance between M and N is 9, which of the following could be the distance between L and N? \\(\\newline\\)I. 5 \\(\\newline\\)II. 8 \\(\\newline\\)III. 13", "explanations":{ "correct":"To find the possible distance between points L and N, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.\n\\(\\newline\\)In this case, the distance between L and M is 4, and the distance between M and N is 9. Let's consider the possible combinations of distances between L, M, and N:\n\n1. If the distance between L and N is 5, then the sum of the distances between L and M (4) and between M and N (9) would be 4 + 9 = 13, which satisfies the triangle inequality theorem.\n\n2. If the distance between L and N is 8, then the sum of the distances between L and M (4) and between M and N (9) would be 4 + 9 = 13, which satisfies the triangle inequality theorem.\n\n3. If the distance between L and N is 13, then the sum of the distances between L and M (4) and between M and N (9) would be 4 + 9 = 13, which satisfies the triangle inequality theorem.\n\nTherefore, all three distances (5, 8, and 13) could be the distance between L and N.\n\nThe answer is E) I, II, and \\(\\newline\\)III." } }, { "answer":"E", "choices":{ "A":"24p", "B":"48p", "C":"\\(2p^{24}\\)", "D":"\\((2p)^{24}\\)", "E":"\\((2^{24})p\\)" }, "id":10123, "question":"The initial number of elements in a certain set is p, where \\(p > 0\\). If the number of elements in the set doubles every hour, which of the following represents the total number of elements in the set after exactly 24 hours?", "explanations":{ "correct":"To find the total number of elements in the set after exactly 24 hours, we need to consider that the number of elements doubles every hour.\n\nAfter 1 hour, the number of elements in the set will be 2p.\nAfter 2 hours, the number of elements will be 2 * 2p = 4p.\nAfter 3 hours, the number of elements will be 2 * 4p = 8p.\nAnd so on...\n\nWe can see that after each hour, the number of elements doubles. Therefore, after 24 hours, the number of elements will be 2^24 * p.\n\nThe answer is E) \\((2^{24})p\\)." } }, { "answer":"A", "choices":{ "A":"8", "B":"14", "C":"20", "D":"32", "E":"44" }, "id":10124, "question":"If \\(3a + 3b + c = 26\\) and \\(a + b = 6\\), then \\(c\\) =", "explanations":{ "correct":"To find the value of \\(c\\), we can use the given equations and solve for \\(c\\).\n\nGiven:\n\\(3a + 3b + c = 26\\) ---(1)\n\\(a + b = 6\\) ---(2)\n\nWe can solve equation (2) for \\(a\\) in terms of \\(b\\) by subtracting \\(b\\) from both sides:\n\\(a = 6 - b\\) ---(3)\n\nSubstituting equation (3) into equation (1), we get:\n\\(3(6 - b) + 3b + c = 26\\)\n\\(18 - 3b + 3b + c = 26\\)\n\\(18 + c = 26\\)\n\nTo isolate \\(c\\), we subtract 18 from both sides:\n\\(c = 26 - 18\\)\n\\(c = 8\\)\n\nTherefore, the value of \\(c\\) is 8.\n\nThe answer is A) 8." } }, { "answer":"A", "choices":{ "A":"8", "B":"4", "C":"\\(2\\sqrt{2}\\)", "D":"2", "E":"\\(\\sqrt{2}\\)" }, "id":10128, "question":"If \\(2\\sqrt{4n^2} + 7 = 39\\), what is the value of n?", "explanations":{ "correct":"To find the value of \\(n\\), we need to solve the equation \\(2\\sqrt{4n^2} + 7 = 39\\).\n\nFirst, let's simplify the equation by simplifying the square root term. The square root of \\(4n^2\\) is \\(2n\\) because the square root of \\(4\\) is \\(2\\) and the square root of \\(n^2\\) is \\(n\\).\n\nSo, the equation becomes \\(2(2n) + 7 = 39\\).\n\nNext, we can simplify further by distributing the \\(2\\) to \\(2n\\) and combining like terms. This gives us \\(4n + 7 = 39\\).\n\nTo isolate \\(n\\), we need to get rid of the \\(7\\) on the left side of the equation. We can do this by subtracting \\(7\\) from both sides of the equation.\n\n\\(4n + 7 - 7 = 39 - 7\\)\n\nThis simplifies to \\(4n = 32\\).\n\nFinally, to solve for \\(n\\), we divide both sides of the equation by \\(4\\).\n\n\\(\\frac{4n}{4} = \\frac{32}{4}\\)\n\nThis simplifies to \\(n = 8\\).\n\nTherefore, the value of \\(n\\) is 8.\n\nThe answer is A) 8." } }, { "answer":"C", "choices":{ "A":"0", "B":"1", "C":"2", "D":"3", "E":"4" }, "id":10130, "question":"If \\(c(x) = x^2 - 6\\), how many times does the graph of \\(c(x)\\) cross the x-axis?", "explanations":{ "correct":"To determine how many times the graph of \\(c(x) = x^2 - 6\\) crosses the x-axis, we need to find the number of x-intercepts. \n\nThe x-intercepts occur when the value of \\(c(x)\\) is equal to zero. So, we need to solve the equation \\(x^2 - 6 = 0\\) to find the values of x where the graph crosses the x-axis.\n\nTo solve the equation \\(x^2 - 6 = 0\\), we can add 6 to both sides of the equation:\n\n\\(x^2 = 6\\)\n\nNext, we can take the square root of both sides of the equation to solve for x:\n\n\\(x = \\pm \\sqrt{6}\\)\n\nSo, the equation \\(x^2 - 6 = 0\\) has two solutions: \\(x = \\sqrt{6}\\) and \\(x = -\\sqrt{6}\\). This means that the graph of \\(c(x)\\) crosses the x-axis at two different points.\n\nTherefore, the answer is C) 2." } }, { "answer":"C", "choices":{ "A":"6", "B":"8", "C":"10", "D":"11", "E":"13" }, "id":10132, "question":"A certain company employs 25 women and 25 men. Some employees drive to work and the rest take public transportation. If 29 employees drive to work, and exactly 6 men take public transportation, how many women drive to work?", "explanations":{ "correct":"Let's break down the information given in the question:\n\n- There are 25 women and 25 men employed in the company.\n- Some employees drive to work and the rest take public transportation.\n- 29 employees drive to work.\n- 6 men take public transportation.\n\nTo find out how many women drive to work, we need to subtract the number of men who drive to work from the total number of employees who drive to work.\n\nTotal employees who drive to work = 29\nNumber of men who drive to work = 29 - (total number of women who drive to work)\n\nWe know that there are 25 men in total, and 6 of them take public transportation. Therefore, the number of men who drive to work is 25 - 6 = 19.\n\nNow, let's substitute this value back into the equation:\n\nNumber of women who drive to work = 29 - 19 = 10\n\nTherefore, the answer is C) 10." } }, { "answer":"A", "choices":{ "A":"1", "B":"2", "C":"3", "D":"5", "E":"6" }, "id":10133, "question":"When positive integers \\(a\\) and \\(b\\) are divided by 7, the remainders are 4 and 2 respectively. If \\(ab\\) is divided by 7, what is the remainder?", "explanations":{ "correct":"To find the remainder when \\(ab\\) is divided by 7, we need to consider the remainders when \\(a\\) and \\(b\\) are divided by 7.\n\nGiven that the remainder when \\(a\\) is divided by 7 is 4, we can write \\(a = 7k + 4\\) for some positive integer \\(k\\).\n\nSimilarly, given that the remainder when \\(b\\) is divided by 7 is 2, we can write \\(b = 7m + 2\\) for some positive integer \\(m\\).\n\nNow, let's find the remainder when \\(ab\\) is divided by 7:\n\n\\(ab = (7k + 4)(7m + 2)\\)\n\\(ab = 49km + 14k + 28m + 8\\)\n\\(ab = 7(7km + 2k + 4m) + 8\\)\n\nSince \\(7(7km + 2k + 4m)\\) is divisible by 7, the remainder when \\(ab\\) is divided by 7 is the same as the remainder when 8 is divided by 7.\n\nWhen 8 is divided by 7, the remainder is 1.\n\nTherefore, the answer is A) 1." } }, { "answer":"C", "choices":{ "A":"-30", "B":"-14", "C":"-10", "D":"0", "E":"10" }, "id":10139, "question":"If \\(|3x - 6| = 36\\), what is one possible value of \\(x\\) ?", "explanations":{ "correct":"To find the possible value of \\(x\\) in the equation \\(|3x - 6| = 36\\), we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.\n\nCase 1: \\(3x - 6\\) is positive\\(\\newline\\)If \\(3x - 6\\) is positive, then we can remove the absolute value signs and solve for \\(x\\):\n\\(3x - 6 = 36\\)\nAdding 6 to both sides:\n\\(3x = 42\\)\nDividing both sides by 3:\n\\(x = 14\\)\n\nCase 2: \\(3x - 6\\) is negative\\(\\newline\\)If \\(3x - 6\\) is negative, then we need to negate the expression inside the absolute value signs and solve for \\(x\\):\n\\(-(3x - 6) = 36\\)\nExpanding the negative sign:\n\\(-3x + 6 = 36\\)\nSubtracting 6 from both sides:\n\\(-3x = 30\\)\nDividing both sides by -3:\n\\(x = -10\\)\n\nTherefore, the possible values of \\(x\\) are 14 and -10.\n\nThe answer is C) -10." } }, { "answer":"D", "choices":{ "A":"\\(\\lbrace 1, 3, 5, 7\\rbrace\\)", "B":"\\(\\lbrace 1, 3, 5, 7, 9\\rbrace\\)", "C":"\\(\\lbrace 2, 3, 5, 7\\rbrace\\)", "D":"\\(\\lbrace 3, 5, 7\\rbrace\\)", "E":"\\(\\lbrace 3, 5, 7, 9\\rbrace\\)" }, "id":10140, "question":"If the members of set A are all the odd single-digit positive integers and the members of set B are all the prime integers, which of the following is the set of numbers common to both set A and set B ?", "explanations":{ "correct":"To find the set of numbers common to both set A and set B, we need to identify the numbers that are both odd single-digit positive integers and prime integers.\n\nFirst, let's list the odd single-digit positive integers: 1, 3, 5, 7, and 9.\n\nNext, let's list the prime integers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.\n\nNow, we need to identify the numbers that appear in both lists. Looking at the lists, we can see that the numbers 3, 5, and 7 are common to both sets. \n\nTherefore, the set of numbers common to both set A and set B is {3, 5, 7}.\n\nThe answer is D) {3, 5, 7}." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 3 } { 2 } \\)", "B":"2", "C":"\\(\\frac { 9 } { 4 } \\)", "D":"\\(\\frac { 3 \\sqrt { 3 } } { 2 } \\)", "E":"\\(2 \\sqrt { 3 } \\)" }, "id":10147, "question":"In square PQRS, point T is the midpoint of side \\(\\overline { QR } \\). If the area of PQRS is 3, what is the area of quadrilateral PQTS ?", "explanations":{ "correct":"To find the area of quadrilateral PQTS, we need to determine the length of side PT and the length of side QS.\n\nSince T is the midpoint of QR, we can conclude that PT is equal to half the length of QR. Let's denote the length of QR as x. Therefore, PT = \\(\\frac { x } { 2 } \\).\n\nSince PQRS is a square, all sides are equal. So, the length of PQ is equal to the length of QR, which is x.\n\nNow, let's find the length of QS. Since T is the midpoint of QR, we can conclude that QS is equal to twice the length of PT. Therefore, QS = 2 * PT = 2 * \\(\\frac { x } { 2 } \\) = x.\n\nNow, we have the lengths of all sides of quadrilateral PQTS: PQ = x, QS = x, PT = \\(\\frac { x } { 2 } \\), and TS = \\(\\frac { x } { 2 } \\).\n\nTo find the area of quadrilateral PQTS, we can split it into two triangles: triangle PQS and triangle PTS.\n\nThe area of triangle PQS can be calculated using the formula: Area = \\(\\frac { 1 } { 2 } \\) * base * height. The base of triangle PQS is PQ = x, and the height is QS = x. Therefore, the area of triangle PQS is \\(\\frac { 1 } { 2 } \\) * x * x = \\(\\frac { x^2 } { 2 } \\).\n\nThe area of triangle PTS can also be calculated using the same formula: Area = \\(\\frac { 1 } { 2 } \\) * base * height. The base of triangle PTS is PT = \\(\\frac { x } { 2 } \\), and the height is TS = \\(\\frac { x } { 2 } \\). Therefore, the area of triangle PTS is \\(\\frac { 1 } { 2 } \\) * \\(\\frac { x } { 2 } \\) * \\(\\frac { x } { 2 } \\) = \\(\\frac { x^2 } { 8 } \\).\n\nTo find the area of quadrilateral PQTS, we add the areas of triangle PQS and triangle PTS: \\(\\frac { x^2 } { 2 } + \\frac { x^2 } { 8 } = \\frac { 4x^2 + x^2 } { 8 } = \\frac { 5x^2 } { 8 }\\).\n\nGiven that the area of PQRS is 3, we can set up the equation: 3 = x^2.\n\nSolving for x, we find x = \\(\\sqrt { 3 }\\).\n\nSubstituting this value back into the expression for the area of quadrilateral PQTS, we get: \\(\\frac { 5(\\sqrt { 3 })^2 } { 8 } = \\frac { 5(3) } { 8 } = \\frac { 15 } { 8 }\\).\n\nTherefore, the area of quadrilateral PQTS is \\(\\frac { 15 } { 8 }\\).\n\nThe answer is C) \\(\\frac { 9 } { 4 }\\)." } }, { "answer":"A", "choices":{ "A":"1", "B":"\\(\\frac{19}{4}\\)", "C":"6", "D":"\\(\\frac{29}{4}\\)", "E":"19" }, "id":10149, "question":"If \\(4(x + 5) = 24\\), then what is the value of x?", "explanations":{ "correct":"To find the value of x, we need to solve the equation \\(4(x + 5) = 24\\). \n\nStep 1: Distribute the 4 to the terms inside the parentheses:\n\\(4x + 20 = 24\\)\n\nStep 2: Subtract 20 from both sides of the equation to isolate the variable term:\n\\(4x = 24 - 20\\)\n\\(4x = 4\\)\n\nStep 3: Divide both sides of the equation by 4 to solve for x:\n\\(x = \\frac{4}{4}\\)\n\\(x = 1\\)\n\nTherefore, the value of x is 1.\n\nThe answer is A." } }, { "answer":"C", "choices":{ "A":"1:1", "B":"2:3", "C":"3:4", "D":"4:3", "E":"2:1" }, "id":10155, "question":"If the perimeter of equilateral triangle E equals the perimeter of square S, what is the ratio of the length of a side of square S to the length of a side of triangle E?", "explanations":{ "correct":"Let's assume that the length of a side of the equilateral triangle E is \"x\" units. Since it is an equilateral triangle, all three sides are equal in length.\n\nThe perimeter of the equilateral triangle E is given by 3 times the length of one side, which is 3x.\n\nNow, let's assume that the length of a side of the square S is \"y\" units. Since it is a square, all four sides are equal in length.\n\nThe perimeter of the square S is given by 4 times the length of one side, which is 4y.\n\nAccording to the question, the perimeter of triangle E is equal to the perimeter of square S. So, we can set up the equation:\n\n3x = 4y\n\nTo find the ratio of the length of a side of square S to the length of a side of triangle E, we need to express y in terms of x.\n\nDividing both sides of the equation by 4, we get:\n\ny = (3/4)x\n\nTherefore, the ratio of the length of a side of square S to the length of a side of triangle E is 3:4.\n\nThe answer is C) 3:4." } }, { "answer":"E", "choices":{ "A":"\\$ 200.00", "B":"\\$ 237.50", "C":"\\$ 230.00", "D":"\\$ 235.50", "E":"\\$ 280.00" }, "id":10159, "question":"Harry bought a bag of 10 pound flour for \\$ 80, a bag of 25 pound flour for \\$ 150 and a bag of 50 pound flour. If the average (arithmetic mean) cost of the flour in all three bags was \\$ 6.00 per pound, what was the price of the bag of 50 pound flour?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nFirst, we need to find the total cost of the flour in all three bags. We know that the average cost of the flour is \\$ 6.00 per pound, and the total weight of the flour is 10 + 25 + 50 = 85 pounds. Therefore, the total cost of the flour is 85 pounds * \\$ 6.00/pound = \\$ 510.00.\n\nNext, we need to find the total cost of the 10-pound and 25-pound bags of flour. The total cost of the 10-pound bag is \\$ 80.00, and the total cost of the 25-pound bag is \\$ 150.00. Therefore, the total cost of these two bags is \\$ 80.00 + \\$ 150.00 = \\$ 230.00.\n\nFinally, we can find the price of the 50-pound bag of flour by subtracting the total cost of the 10-pound and 25-pound bags from the total cost of all three bags. The price of the 50-pound bag is \\$ 510.00 - \\$ 230.00 = \\$ 280.00.\n\nTherefore, the price of the bag of 50-pound flour is \\$ 280.00.\n\nThe answer is E) \\$ 280.00." } }, { "answer":"D", "choices":{ "A":"1", "B":"3", "C":"5", "D":"7", "E":"9" }, "id":10160, "question":"If \\(f(x) = 3x - 8\\) and \\(g(x) = \\sqrt{2x^2 + 7}\\), what is the value of \\(f(g(3))\\)?", "explanations":{ "correct":"To find the value of \\(f(g(3))\\), we need to substitute \\(g(3)\\) into the function \\(f(x)\\).\n\nFirst, let's find the value of \\(g(3)\\). We substitute \\(x = 3\\) into the function \\(g(x)\\):\n\n\\[g(3) = \\sqrt{2(3)^2 + 7} = \\sqrt{2(9) + 7} = \\sqrt{18 + 7} = \\sqrt{25} = 5\\]\n\nNow that we know \\(g(3) = 5\\), we can substitute this value into the function \\(f(x)\\):\n\n\\[f(g(3)) = f(5) = 3(5) - 8 = 15 - 8 = 7\\]\n\nTherefore, the value of \\(f(g(3))\\) is 7.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"less than 5", "B":"5", "C":"more than 5 but less than 10", "D":"10", "E":"more than 10" }, "id":10162, "question":"If Wally's Widget Works is open exactly 20 days each month and produces 80 widgets each day it is open, how many years will it take to produce 96,000 widgets?", "explanations":{ "correct":"To find the number of years it will take to produce 96,000 widgets, we need to determine the total number of widgets produced in a month and then divide 96,000 by that number.\n\nWally's Widget Works is open for 20 days each month and produces 80 widgets each day. Therefore, the total number of widgets produced in a month is 20 days * 80 widgets/day = 1600 widgets.\n\nTo find the number of years it will take to produce 96,000 widgets, we divide 96,000 by 1600:\n\n96,000 widgets / 1600 widgets/month = 60 months.\n\nSince there are 12 months in a year, we divide 60 months by 12 to find the number of years:\n\n60 months / 12 months/year = 5 years.\n\nTherefore, it will take 5 years to produce 96,000 widgets.\n\nThe answer is B) 5." } }, { "answer":"A", "choices":{ "A":"-15", "B":"-5", "C":"\\(\\frac { -15 } { 7 } \\)", "D":"5", "E":"15" }, "id":10164, "question":"If \\(5k + 8 = -7 + 2k\\), what is the value of \\(3k\\) ?", "explanations":{ "correct":"To find the value of \\(3k\\), we need to solve the equation \\(5k + 8 = -7 + 2k\\) first.\n\nStep 1: Combine like terms on both sides of the equation.\n\\(5k - 2k = -7 - 8\\)\n\\(3k = -15\\)\n\nStep 2: Divide both sides of the equation by 3 to isolate \\(k\\).\n\\(\\frac{3k}{3} = \\frac{-15}{3}\\)\n\\(k = -5\\)\n\nStep 3: Multiply \\(k\\) by 3 to find the value of \\(3k\\).\n\\(3k = 3(-5)\\)\n\\(3k = -15\\)\n\nTherefore, the value of \\(3k\\) is -15.\n\nThe answer is A) -15." } }, { "answer":"A", "choices":{ "A":"\\(t > s\\)", "B":"\\(s > t\\)", "C":"\\(t = s\\)", "D":"\\(t > 0\\) and \\(s > 0\\)", "E":"\\(t < 0\\) and \\(s < 0\\)" }, "id":10165, "question":"\\(ts < 0 < t - s\\) If the statement above is true, which of the following must also be true?", "explanations":{ "correct":"To determine which statement must also be true given the inequality \\(ts < 0 < t - s\\), let's analyze the given inequality step-by-step.\n\n1. \\(ts < 0\\): This inequality tells us that the product of \\(t\\) and \\(s\\) is negative. For the product of two numbers to be negative, one of the numbers must be positive and the other must be negative.\n\n2. \\(0 < t - s\\): This inequality states that the difference between \\(t\\) and \\(s\\) is positive. In other words, \\(t\\) is greater than \\(s\\).\n\nNow, let's evaluate each option:\n\nA) \\(t > s\\): This statement aligns with our analysis. Since \\(t\\) is greater than \\(s\\) based on the second inequality, this option is a valid conclusion. \n\nB) \\(s > t\\): This statement contradicts our analysis. We determined that \\(t\\) is greater than \\(s\\), so this option is not true.\n\nC) \\(t = s\\): This statement contradicts our analysis. We determined that \\(t\\) is greater than \\(s\\), so this option is not true.\n\nD) \\(t > 0\\) and \\(s > 0\\): This statement is not necessarily true. While it is possible for both \\(t\\) and \\(s\\) to be positive, it is also possible for one of them to be negative. The given inequality does not provide enough information to conclude that both \\(t\\) and \\(s\\) are positive.\n\nE) \\(t < 0\\) and \\(s < 0\\): This statement is not necessarily true. While it is possible for both \\(t\\) and \\(s\\) to be negative, it is also possible for one of them to be positive. The given inequality does not provide enough information to conclude that both \\(t\\) and \\(s\\) are negative.\n\nBased on our analysis, the only statement that must also be true is:\n\nThe answer is A). \\(t > s\\)." } }, { "answer":"B", "choices":{ "A":"20", "B":"25", "C":"30", "D":"40", "E":"50" }, "id":10167, "question":"Brigitte's average (arithmetic mean) on her six math tests this marking period is 75. Fortunately for Brigitte, her teacher drops each student's lowest grade, thus raising Brigitte's average to 85. What was her lowest grade?", "explanations":{ "correct":"To find Brigitte's lowest grade, we need to determine the sum of her five highest grades. \n\nLet's assume her lowest grade is x. \n\nAccording to the given information, her average on six math tests is 75. Therefore, the sum of her six grades is 6 * 75 = 450. \n\nSince her teacher drops the lowest grade, the sum of her five highest grades is 450 - x. \n\nWe also know that her average after dropping the lowest grade is 85. Therefore, the sum of her five highest grades is 5 * 85 = 425. \n\nSetting up an equation, we have:\n\n450 - x = 425\n\nSimplifying the equation, we get:\n\nx = 450 - 425\nx = 25\n\nTherefore, Brigitte's lowest grade is 25.\n\nThe answer is B) 25." } }, { "answer":"C", "choices":{ "A":"2", "B":"4", "C":"5", "D":"7", "E":"9" }, "id":10181, "question":"If \\(m^ { 2a } x m^b = m^6\\) and \\(\\frac { m^a } { m^b } = \\frac { 1 } { m^3 } \\), what is the value of \\(a + b\\)?", "explanations":{ "correct":"To find the value of \\(a + b\\), we need to simplify the given equations and then solve for \\(a\\) and \\(b\\).\n\nFirst, let's simplify the equation \\(m^{2a} \\cdot m^b = m^6\\). According to the properties of exponents, when multiplying two powers with the same base, we add their exponents. Therefore, we can rewrite the equation as \\(m^{2a + b} = m^6\\).\n\nNext, let's simplify the equation \\(\\frac{m^a}{m^b} = \\frac{1}{m^3}\\). According to the properties of exponents, when dividing two powers with the same base, we subtract their exponents. Therefore, we can rewrite the equation as \\(m^{a - b} = \\frac{1}{m^3}\\).\n\nNow, we have two equations:\n1) \\(m^{2a + b} = m^6\\)\n2) \\(m^{a - b} = \\frac{1}{m^3}\\)\n\nSince the bases of both equations are the same (which is \\(m\\)), we can equate the exponents.\n\nFrom equation 1, we have \\(2a + b = 6\\).\nFrom equation 2, we have \\(a - b = -3\\).\n\nNow, we have a system of two equations with two variables. We can solve this system by substitution or elimination.\n\nLet's solve it by elimination. Multiply equation 2 by 2 to make the coefficients of \\(a\\) in both equations the same:\n\\(2(a - b) = 2(-3)\\)\n\\(2a - 2b = -6\\)\n\nNow, we can add this equation to equation 1:\n\\(2a + b + 2a - 2b = 6 + (-6)\\)\n\\(4a - b = 0\\)\n\nFrom this equation, we can solve for \\(b\\):\n\\(4a - b = 0\\)\n\\(b = 4a\\)\n\nSubstitute this value of \\(b\\) into equation 1:\n\\(2a + 4a = 6\\)\n\\(6a = 6\\)\n\\(a = 1\\)\n\nNow that we have the values of \\(a\\) and \\(b\\), we can find their sum:\n\\(a + b = 1 + 4a = 1 + 4(1) = 1 + 4 = 5\\)\n\nTherefore, the value of \\(a + b\\) is 5.\n\nThe answer is C." } }, { "answer":"A", "choices":{ "A":"Three", "B":"Four", "C":"Six", "D":"Eight", "E":"Nine" }, "id":10182, "question":"There are an equal number of dogs and cats at a clinic. After 2 dogs and 8 cats are taken home, there are three times as many dogs as cats still at the clinic. How many cats are still at the clinic?", "explanations":{ "correct":"Let's solve this problem step by step:\n\n1. Let's assume the number of dogs and cats at the clinic is x.\n\n2. After 2 dogs and 8 cats are taken home, the number of dogs remaining at the clinic is (x - 2) and the number of cats remaining is (x - 8).\n\n3. According to the problem, there are three times as many dogs as cats still at the clinic. So, we can write the equation: (x - 2) = 3(x - 8).\n\n4. Let's solve the equation:\n x - 2 = 3x - 24\n 22 = 2x\n x = 11\n\n5. Now, we need to find the number of cats still at the clinic, which is (x - 8) = (11 - 8) = 3.\n\nTherefore, the answer is A) Three." } }, { "answer":"C", "choices":{ "A":"12", "B":"18", "C":"24", "D":"27", "E":"281" }, "id":10183, "question":"Four students reach into a bag of coins. Steve grabs \\(\\frac{1}{3}\\) of the coins, Joe grabs \\(\\frac{1}{4}\\), and Kyle grabs \\(\\frac{1}{6}\\). If Ryan gets the remaining 6 coins, how many coins were originally in the bag?", "explanations":{ "correct":"Let's start by finding the fraction of coins that Steve, Joe, and Kyle took from the bag. \n\nSteve took \\\\(\\frac{1}{3}\\\\) of the coins, Joe took \\\\(\\frac{1}{4}\\\\) of the coins, and Kyle took \\\\(\\frac{1}{6}\\\\) of the coins. \n\nTo find the total fraction of coins they took, we add these fractions together:\n\n\\\\(\\frac{1}{3} + \\frac{1}{4} + \\frac{1}{6}\\\\)\n\nTo add these fractions, we need to find a common denominator. The least common multiple of 3, 4, and 6 is 12. \n\nNow, we can rewrite the fractions with the common denominator of 12:\n\n\\\\(\\frac{4}{12} + \\frac{3}{12} + \\frac{2}{12}\\\\)\n\nAdding these fractions together, we get:\n\n\\\\(\\frac{9}{12}\\\\)\n\nSo, Steve, Joe, and Kyle took \\\\(\\frac{9}{12}\\\\) of the coins from the bag. \n\nTo find the fraction of coins that Ryan got, we subtract \\\\(\\frac{9}{12}\\\\) from 1 (since Ryan got the remaining coins):\n\n\\\\(1 - \\frac{9}{12}\\\\)\n\nTo subtract these fractions, we also need a common denominator of 12:\n\n\\\\(\\frac{12}{12} - \\frac{9}{12}\\\\)\n\nThis gives us:\n\n\\\\(\\frac{3}{12}\\\\)\n\nSo, Ryan got \\\\(\\frac{3}{12}\\\\) of the coins, which is equivalent to \\\\(\\frac{1}{4}\\\\).\n\nWe are given that Ryan got 6 coins, which is \\\\(\\frac{1}{4}\\\\) of the total number of coins. \n\nLet's represent the total number of coins as \"x\". \n\n\\\\(\\frac{1}{4}\\\\) of x is equal to 6:\n\n\\\\(\\frac{1}{4}x = 6\\\\)\n\nTo solve for x, we can multiply both sides of the equation by 4:\n\nx = 6 * 4\n\nx = 24\n\nTherefore, the original number of coins in the bag was 24.\n\nThe answer is C) 24." } }, { "answer":"D", "choices":{ "A":"100", "B":"180", "C":"198", "D":"199", "E":"200" }, "id":10185, "question":"Consider the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3, ... . What is the sum of the first 100 terms?", "explanations":{ "correct":"To find the sum of the first 100 terms of the given sequence, we need to determine the pattern and then calculate the sum.\n\nLooking at the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3, ..., we can observe that it repeats every 3 terms. The sum of each group of 3 terms is 1 + 2 + 3 = 6.\n\nTo find the number of complete groups of 3 terms in 100 terms, we divide 100 by 3. The quotient is 33 with a remainder of 1. This means there are 33 complete groups of 3 terms and 1 additional term.\n\nThe sum of the complete groups of 3 terms is 6 * 33 = 198.\n\nTo find the sum of the additional term, we add the next term in the sequence, which is 1.\n\nTherefore, the sum of the first 100 terms of the given sequence is 198 + 1 = 199.\n\nThe answer is D) 199." } }, { "answer":"A", "choices":{ "A":"36", "B":"38", "C":"40", "D":"45", "E":"48" }, "id":10186, "question":"If \\(\\textcircled{x}\\) is defined by \\(\\textcircled{x} = 5x - \\frac{x}{2}\\), then \\(\\textcircled{8} = \\)", "explanations":{ "correct":"To find the value of \\\\(\\textcircled{8}\\\\), we substitute \\\\(x = 8\\\\) into the equation \\\\(\\textcircled{x} = 5x - \\frac{x}{2}\\\\).\n\nFirst, we substitute \\\\(x = 8\\\\) into the equation:\n\\\\(\\textcircled{8} = 5(8) - \\frac{8}{2}\\\\)\n\nNext, we simplify the equation:\n\\\\(\\textcircled{8} = 40 - 4\\\\)\n\\\\(\\textcircled{8} = 36\\\\)\n\nTherefore, the value of \\\\(\\textcircled{8}\\\\) is 36.\n\nThe answer is A." } }, { "answer":"C", "choices":{ "A":"\\(-\\frac { 1 } { 4 } \\)", "B":"\\(\\frac { 1 } { 4 } \\)", "C":"\\(\\frac { 1 } { 2 } \\)", "D":"0", "E":"2" }, "id":10190, "question":"If \\(\\frac { 2x } { x^2 + 1 } = \\frac { 2 } { x + 2 } \\), what is the value of \\(x\\) ?", "explanations":{ "correct":"To solve the equation \\(\\frac { 2x } { x^2 + 1 } = \\frac { 2 } { x + 2 }\\), we can start by cross-multiplying. \n\nCross-multiplying gives us \\(2x(x + 2) = 2(x^2 + 1)\\).\n\nExpanding both sides of the equation, we get \\(2x^2 + 4x = 2x^2 + 2\\).\n\nNext, we can simplify the equation by subtracting \\(2x^2\\) from both sides, which gives us \\(4x = 2\\).\n\nTo isolate \\(x\\), we can divide both sides of the equation by 4, resulting in \\(x = \\frac{2}{4}\\).\n\nSimplifying the fraction, we have \\(x = \\frac{1}{2}\\).\n\nTherefore, the value of \\(x\\) is \\(\\frac{1}{2}\\).\n\nThe answer is C." } }, { "answer":"E", "choices":{ "A":"0.125", "B":"0.5", "C":"1.25", "D":"5.0", "E":"12.5" }, "id":10192, "question":"What is 50% of 50% of 50?", "explanations":{ "correct":"To find 50% of a number, we divide the number by 2. \n\nFirst, let's find 50% of 50. \n50 ÷ 2 = 25\n\nNext, let's find 50% of 25. \n25 ÷ 2 = 12.5\n\nTherefore, 50% of 50% of 50 is 12.5. \n\nThe answer is E) 12.5." } }, { "answer":"A", "choices":{ "A":"\\(8a^2 + 6\\)", "B":"\\(8a^2 + 12\\)", "C":"\\(12a^2 + 6\\)", "D":"\\(18a^2 + 6\\)", "E":"\\(18a^2 + 12\\)" }, "id":10193, "question":"If \\(2a = 3b\\), \\(\\frac { 1 } { 3 } c = 6b^2 + 2\\), and \\(b > 0\\), what is \\(c\\) in terms of \\(a\\) ?", "explanations":{ "correct":"To find the value of \\(c\\) in terms of \\(a\\), we need to substitute the given equations into each other and solve for \\(c\\).\n\nGiven: \\(2a = 3b\\) (Equation 1)\nAnd: \\(\\frac{1}{3}c = 6b^2 + 2\\) (Equation 2)\n\nFrom Equation 1, we can solve for \\(b\\) in terms of \\(a\\):\n\\(2a = 3b\\)\nDivide both sides by 3:\n\\(\\frac{2a}{3} = b\\) (Equation 3)\n\nNow, substitute Equation 3 into Equation 2 to solve for \\(c\\):\n\\(\\frac{1}{3}c = 6(\\frac{2a}{3})^2 + 2\\)\nSimplify the expression inside the parentheses:\n\\(\\frac{1}{3}c = 6(\\frac{4a^2}{9}) + 2\\)\nMultiply 6 by \\(\\frac{4a^2}{9}\\):\n\\(\\frac{1}{3}c = \\frac{24a^2}{9} + 2\\)\nSimplify the fraction:\n\\(\\frac{1}{3}c = \\frac{8a^2}{3} + 2\\)\nMultiply both sides by 3 to eliminate the fraction:\n\\(c = 8a^2 + 6\\) (Equation 4)\n\nTherefore, the value of \\(c\\) in terms of \\(a\\) is \\(8a^2 + 6\\).\n\nThe answer is A) \\(8a^2 + 6\\)." } }, { "answer":"A", "choices":{ "A":"32", "B":"34", "C":"36", "D":"38", "E":"40" }, "id":10194, "question":"If \\(4x + 2 = 26\\), then \\(4x + 8\\) =", "explanations":{ "correct":"To find the value of \\\\(4x + 8\\\\), we need to first solve the equation \\\\(4x + 2 = 26\\\\) for \\\\(x\\\\).\n\nStep 1: Subtract 2 from both sides of the equation:\n\\\\(4x + 2 - 2 = 26 - 2\\\\)\n\\\\(4x = 24\\\\)\n\nStep 2: Divide both sides of the equation by 4 to isolate \\\\(x\\\\):\n\\\\(\\\\frac{{4x}}{4} = \\\\frac{24}{4}\\\\)\n\\\\(x = 6\\\\)\n\nNow that we have found the value of \\\\(x\\\\) as 6, we can substitute it into the expression \\\\(4x + 8\\\\) to find the final answer.\n\n\\\\(4x + 8 = 4(6) + 8\\\\)\n\\\\(4x + 8 = 24 + 8\\\\)\n\\\\(4x + 8 = 32\\\\)\n\nTherefore, the answer is A) 32." } }, { "answer":"A", "choices":{ "A":"\\(\\frac{kd}{c}\\)", "B":"\\(\\frac{c}{dk}\\)", "C":"\\(dk + c\\)", "D":"\\(\\frac{dc}{k}\\)", "E":"\\((k - c)d\\)" }, "id":10195, "question":"If k friends contribute d dollars each, and that money is distributed equally among c number of charities, how much money is received by each charity?", "explanations":{ "correct":"To find out how much money is received by each charity, we need to divide the total amount of money contributed by the number of charities.\n\nThe total amount of money contributed is given by k friends contributing d dollars each, which can be calculated as k * d.\n\nTherefore, the amount of money received by each charity is \\(\\frac{k * d}{c}\\).\n\nHence, the answer is A) \\(\\frac{kd}{c}\\)." } }, { "answer":"B", "choices":{ "A":"\\(2^yb\\)", "B":"\\(2^ { 4y - 3 } b\\)", "C":"\\(2^ { 4y } b\\)", "D":"\\(2b^ { 4y + 3 } \\)", "E":"\\(2^yb^ { y + 2 } \\)" }, "id":10196, "question":"The number of baseball cards in Caleb's collection doubles every three months. If after 9 months he has b baseball cards, then which of the following is an expression for the number of baseball cards in his collection after y years?", "explanations":{ "correct":"To find the expression for the number of baseball cards in Caleb's collection after y years, we need to determine the number of times his collection doubles in y years.\n\nWe know that his collection doubles every three months. Since there are 12 months in a year, the number of times his collection doubles in y years is \\( \\frac{y}{\\frac{1}{4}} = 4y \\).\n\nNow, we can use this information to evaluate the options:\n\nA) \\(2^yb\\) - This option only considers the number of years, but it doesn't account for the fact that the collection doubles every three months. Incorrect.\n\nB) \\(2^{4y-3}b\\) - This option takes into account the number of times the collection doubles in y years (4y), and subtracts 3 to account for the initial 9 months. This is a valid expression. \n\nC) \\(2^{4y}b\\) - This option doesn't account for the initial 9 months. Incorrect.\n\nD) \\(2b^{4y+3}\\) - This option doesn't consider the doubling of the collection. Incorrect.\n\nE) \\(2^yb^{y+2}\\) - This option doesn't consider the number of times the collection doubles in y years. Incorrect.\n\nTherefore, the correct expression for the number of baseball cards in Caleb's collection after y years is \\(2^{4y-3}b\\).\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"260", "B":"295", "C":"325", "D":"360", "E":"405" }, "id":10200, "question":"The number of students who order hot lunch by the month at the Tucker Elementary School varies directly with the number of times pizza appears on the lunch menu during that month. If 135 students order hot lunch when pizza appears three times on the menu, how many students order hot lunch when pizza appears eight times on the menu?", "explanations":{ "correct":"To solve this problem, we can set up a proportion using the concept of direct variation.\n\nLet's denote the number of students who order hot lunch as \"S\" and the number of times pizza appears on the menu as \"P\".\n\nWe are given that when P = 3, S = 135. We can set up the proportion as follows:\n\nS/135 = P/3\n\nTo find the value of S when P = 8, we can solve for S by cross-multiplying:\n\nS = (135 * 8) / 3\nS = 1080 / 3\nS = 360\n\nTherefore, when pizza appears eight times on the menu, 360 students order hot lunch.\n\nThe answer is D) 360." } }, { "answer":"A", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"II and III only" }, "id":10206, "question":"If exactly two of the three integers a,b, and c are odd, which of the following must be odd? \\(\\newline\\)I. \\(a + bc\\) \\(\\newline\\)II. \\(\\frac { ab } { c } \\) \\(\\newline\\)III. \\(a(b + c)\\)", "explanations":{ "correct":"To determine which of the given expressions must be odd, we need to analyze the possible combinations of odd and even integers for a, b, and c.\n\nLet's consider the three cases:\n\nCase 1: a and b are odd, c is even.\\(\\newline\\)In this case, a + bc will be odd since the sum of an odd number and an even number is always odd. Thus, I is odd.\n\nCase 2: a and c are odd, b is even.\\(\\newline\\)In this case, ab/c will be even since the product of an odd number and an even number is always even. Thus, II is even.\n\nCase 3: b and c are odd, a is even.\\(\\newline\\)In this case, a(b + c) will be even since the sum of two odd numbers is always even, and multiplying an even number by any other number will result in an even number. Thus, III is even.\n\nFrom the analysis above, we can conclude that only I is guaranteed to be odd. Therefore, the answer is A) I only." } }, { "answer":"D", "choices":{ "A":"-4", "B":"-2", "C":"0", "D":"2", "E":"4" }, "id":10212, "question":"If \\(|x - 5| = 2x - 1\\), which of the following is the value of \\(x\\)?", "explanations":{ "correct":"To solve the equation \\(|x - 5| = 2x - 1\\), we need to consider two cases: when \\(x - 5\\) is positive and when \\(x - 5\\) is negative.\n\nCase 1: \\(x - 5\\) is positive\\(\\newline\\)In this case, the absolute value of \\(x - 5\\) is equal to \\(x - 5\\). Therefore, we can rewrite the equation as \\(x - 5 = 2x - 1\\). \n\nSimplifying the equation, we get:\n\\(x - 5 = 2x - 1\\)\n\\(x - 2x = -1 + 5\\)\n\\(-x = 4\\)\n\\(x = -4\\) (multiplying both sides by -1)\n\nCase 2: \\(x - 5\\) is negative\\(\\newline\\)In this case, the absolute value of \\(x - 5\\) is equal to \\(-(x - 5)\\). Therefore, we can rewrite the equation as \\(-(x - 5) = 2x - 1\\).\n\nSimplifying the equation, we get:\n\\(-x + 5 = 2x - 1\\)\n\\(-x - 2x = -1 - 5\\)\n\\(-3x = -6\\)\n\\(x = 2\\) (dividing both sides by -3)\n\nTherefore, the solutions to the equation \\(|x - 5| = 2x - 1\\) are \\(x = -4\\) and \\(x = 2\\).\n\nThe answer is D) 2." } }, { "answer":"E", "choices":{ "A":"All negative numbers", "B":"All negative numbers between -1 and O", "C":"All positive numbers less than 1", "D":"All positive numbers greater than 1", "E":"All positive numbers" }, "id":10213, "question":"If \\(a < -1\\) and \\(-1 < b < 0\\), what are all of the possible values of \\(ab\\) ?", "explanations":{ "correct":"To find the possible values of \\(ab\\), we need to consider the given conditions: \\(a < -1\\) and \\(-1 < b < 0\\).\n\nSince \\(a < -1\\), we know that \\(a\\) is a negative number that is less than -1. Similarly, since \\(-1 < b < 0\\), we know that \\(b\\) is a negative number between -1 and 0.\n\nWhen we multiply two negative numbers, the result is always a positive number. Therefore, when we multiply \\(a\\) and \\(b\\), we will get a positive number.\n\nHence, the possible values of \\(ab\\) are all positive numbers.\n\nThe answer is E." } }, { "answer":"D", "choices":{ "A":"\\(-5\\)", "B":"\\(-2\\)", "C":"1", "D":"5", "E":"8" }, "id":10218, "question":"If t is 5 more than s, and s is 3 less than r, what is t when \\(r = 3\\)?", "explanations":{ "correct":"To find the value of t when r = 3, we need to follow the given information step-by-step.\n\nFirst, we are told that s is 3 less than r. So, if r = 3, then s = 3 - 3 = 0.\n\nNext, we are told that t is 5 more than s. Since s = 0, t = 0 + 5 = 5.\n\nTherefore, when r = 3, t = 5.\n\nThe answer is D) 5." } }, { "answer":"B", "choices":{ "A":"5", "B":"10", "C":"15", "D":"\\(20\\sqrt{2}\\)", "E":"\\(20\\sqrt{3}\\)" }, "id":10219, "question":"A ladder is placed against a building such that the top of the ladder forms an angle of 30 degrees with the side of the building. If the ladder is 20 feet long, how far in feet is the bottom of the ladder from the base of the building?", "explanations":{ "correct":"To solve this problem, we can use trigonometry, specifically the sine function. \n\nThe sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the angle is 30 degrees and the hypotenuse is the length of the ladder, which is 20 feet.\n\nLet's label the distance from the base of the building to the bottom of the ladder as \"x\". We want to find the value of x.\n\nUsing the sine function, we have:\n\nsin(30 degrees) = x / 20\n\nTo find the value of sin(30 degrees), we can use the fact that it is equal to 1/2. So we have:\n\n1/2 = x / 20\n\nTo solve for x, we can cross-multiply:\n\n2x = 20\n\nDividing both sides by 2, we get:\n\nx = 10\n\nTherefore, the bottom of the ladder is 10 feet from the base of the building.\n\nThe answer is B) 10." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I,II and III" }, "id":10223, "question":"If \\(\\frac { a } { b } \\) is negative, which of the following must be positive? \\(\\newline\\)I. \\((a + b)(a - b)\\) \\(\\newline\\)II. \\(1 - ab\\) \\(\\newline\\)III. \\(\\frac { b - a } { a } \\)", "explanations":{ "correct":"To determine which of the given expressions must be positive when \\\\(\\frac{a}{b}\\\\) is negative, we can analyze each expression separately.\n\\(\\newline\\)I. \\\\((a + b)(a - b)\\\\)\nExpanding this expression, we get \\\\(a^2 - ab + ab - b^2\\\\), which simplifies to \\\\(a^2 - b^2\\\\). \n\\(\\newline\\)If \\\\(\\frac{a}{b}\\\\) is negative, it means that either \\\\(a\\\\) and \\\\(b\\\\) have opposite signs or one of them is zero. In either case, \\\\(a^2 - b^2\\\\) will be negative or zero. Therefore, \\\\((a + b)(a - b)\\\\) does not have to be positive.\n\\(\\newline\\)II. \\\\(1 - ab\\\\)\nSince \\\\(\\frac{a}{b}\\\\) is negative, \\\\(ab\\\\) will be negative as well. Subtracting a negative number from 1 will result in a positive value. Therefore, \\\\(1 - ab\\\\) must be positive.\n\\(\\newline\\)III. \\\\(\\frac{b - a}{a}\\\\)\\(\\newline\\)If \\\\(\\frac{a}{b}\\\\) is negative, it means that \\\\(a\\\\) and \\\\(b\\\\) have opposite signs. Subtracting a positive number from a negative number will result in a negative value. Dividing a negative number by a positive number will also result in a negative value. Therefore, \\\\(\\frac{b - a}{a}\\\\) must be negative.\n\nBased on our analysis, only expression II, \\\\(1 - ab\\\\), must be positive when \\\\(\\frac{a}{b}\\\\) is negative.\n\nThe answer is B) II only." } }, { "answer":"B", "choices":{ "A":"\\(600t + 400\\)", "B":"\\(\\frac { t - 400 } { 60 } \\)", "C":"\\(\\frac { 60 } { t - 400 } \\)", "D":"\\(\\frac { 60t } { 400 } \\)", "E":"\\(\\frac { 400t } { 60 } \\)" }, "id":10225, "question":"A fashion designer's total expenses \\(E\\), in dollars, of the number of garments is given by \\(E(n) = 6011 + 400\\), where \\(n\\) is the number of garments. If the total expenses are \\(t\\) dollars, what is the value of \\(n\\) in terms of \\(t\\) ?", "explanations":{ "correct":"To find the value of \\(n\\) in terms of \\(t\\), we need to solve the equation \\(E(n) = t\\) for \\(n\\).\n\nGiven that \\(E(n) = 6011 + 400n\\), we can set this equal to \\(t\\) and solve for \\(n\\):\n\n\\[6011 + 400n = t\\]\n\nTo isolate \\(n\\), we can subtract 6011 from both sides:\n\n\\[400n = t - 6011\\]\n\nNow, to solve for \\(n\\), we divide both sides by 400:\n\n\\[n = \\frac{t - 6011}{400}\\]\n\nTherefore, the value of \\(n\\) in terms of \\(t\\) is \\(\\frac{t - 6011}{400}\\).\n\nThe answer is B) \\(\\frac{t - 400}{60}\\)." } }, { "answer":"B", "choices":{ "A":"8", "B":"12", "C":"16", "D":"24", "E":"10 + 2\\(\\sqrt { 2 } \\)" }, "id":10226, "question":"What is the area of the quadrilateral whose vertices are at (1, 1), (5, 1), (5, 5) and (3, 5)?", "explanations":{ "correct":"To find the area of the quadrilateral, we can divide it into two triangles and then sum up their areas.\n\nFirst, let's find the base and height of the first triangle. The base is the horizontal distance between the points (1, 1) and (5, 1), which is 5 - 1 = 4 units. The height is the vertical distance between the points (1, 1) and (3, 5), which is 5 - 1 = 4 units. Therefore, the area of the first triangle is (1/2) * base * height = (1/2) * 4 * 4 = 8 square units.\n\nNext, let's find the base and height of the second triangle. The base is the horizontal distance between the points (3, 5) and (5, 5), which is 5 - 3 = 2 units. The height is the vertical distance between the points (5, 1) and (5, 5), which is 5 - 1 = 4 units. Therefore, the area of the second triangle is (1/2) * base * height = (1/2) * 2 * 4 = 4 square units.\n\nFinally, we can sum up the areas of the two triangles to find the total area of the quadrilateral: 8 + 4 = 12 square units.\n\nTherefore, the answer is B) 12." } }, { "answer":"E", "choices":{ "A":"\\(k + 2\\)", "B":"\\(3k\\)", "C":"\\(3k + 2\\)", "D":"\\(k(k - 1)\\)", "E":"\\((k + 1)(k - 1)\\)" }, "id":10232, "question":"If \\(k\\) is an even integer, which of the following must be an odd integer?", "explanations":{ "correct":"To determine which of the given expressions must be an odd integer when \\(k\\) is an even integer, we need to analyze each option step-by-step.\n\nA) \\(k + 2\\):\nWhen we add an even integer (\\(k\\)) to an even integer (2), the result will always be an even integer. This is because the sum of two even integers is always even. Therefore, \\(k + 2\\) cannot be an odd integer.\n\nB) \\(3k\\):\nWhen we multiply an even integer (\\(k\\)) by an odd integer (3), the result will always be an even integer. This is because any even number multiplied by an odd number is always even. Therefore, \\(3k\\) cannot be an odd integer.\n\nC) \\(3k + 2\\):\nSimilar to option B, when we multiply an even integer (\\(k\\)) by an odd integer (3), the result will always be an even integer. Then, when we add an even integer (2) to an even integer, the result will still be an even integer. Therefore, \\(3k + 2\\) cannot be an odd integer.\n\nD) \\(k(k - 1)\\):\nWhen we multiply an even integer (\\(k\\)) by an odd integer (\\(k - 1\\)), the result will always be an even integer. This is because any even number multiplied by an odd number is always even. Therefore, \\(k(k - 1)\\) cannot be an odd integer.\n\nE) \\((k + 1)(k - 1)\\):\nWhen we subtract 1 from an even integer (\\(k\\)), the result will always be an odd integer. Then, when we multiply an even integer (\\(k + 1\\)) by an odd integer (\\(k - 1\\)), the result will always be an even integer. This is because any even number multiplied by an odd number is always even. Therefore, \\((k + 1)(k - 1)\\) cannot be an odd integer.\n\nBased on the analysis above, the only option that must be an odd integer when \\(k\\) is an even integer is \\((k + 1)(k - 1)\\). \n\nThe answer is E." } }, { "answer":"A", "choices":{ "A":"2,250", "B":"1,975", "C":"1,800", "D":"950", "E":"150" }, "id":10241, "question":"One-half of the water in a pond evaporates each week. There are 2,400 gallons of water in the pond at the end of the third week. In gallons, how much less water is in the pond at the end of seventh week than at the end of the third week?", "explanations":{ "correct":"To find the amount of water in the pond at the end of the seventh week, we need to determine how much water evaporates each week. Since one-half of the water evaporates each week, the remaining water in the pond at the end of each week is half of the previous week's amount.\n\nLet's work backwards to find the amount of water in the pond at the end of the seventh week. We know that at the end of the third week, there were 2,400 gallons of water in the pond. \n\nAt the end of the second week, there would have been twice as much water as at the end of the third week, which is 2,400 * 2 = 4,800 gallons.\n\nSimilarly, at the end of the first week, there would have been twice as much water as at the end of the second week, which is 4,800 * 2 = 9,600 gallons.\n\nFinally, at the beginning of the first week (before any evaporation), there would have been twice as much water as at the end of the first week, which is 9,600 * 2 = 19,200 gallons.\n\nTherefore, at the end of the seventh week, there would be 19,200 / 2^7 = 150 gallons of water left in the pond.\n\nTo find how much less water is in the pond at the end of the seventh week compared to the end of the third week, we subtract the amount at the end of the seventh week from the amount at the end of the third week: 2,400 - 150 = 2,250 gallons.\n\nThe answer is A) 2,250." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { 22 } { 3 } \\)", "B":"\\(x^2 - x - 12\\)", "C":"\\(x^2 + \\frac { 4 } { 3 } x + 4\\)", "D":"\\(\\frac { 2x + 4 } { x + 1 } \\)", "E":"\\(\\frac { 2x - 4 } { x - 1 } \\)" }, "id":10243, "question":"If \\(f(x) = 2x^2 - 4x - 16\\) and \\(g(x) = x^2 - 3x - 4\\), which of the following is an expression for \\(\\frac { f(x) } { g(x) } \\) ?", "explanations":{ "correct":"To find the expression for \\(\\frac { f(x) } { g(x) }\\), we need to divide the function \\(f(x)\\) by the function \\(g(x)\\).\n\nFirst, let's write out the expression for \\(f(x)\\):\n\\[f(x) = 2x^2 - 4x - 16\\]\n\nNext, let's write out the expression for \\(g(x)\\):\n\\[g(x) = x^2 - 3x - 4\\]\n\nTo divide \\(f(x)\\) by \\(g(x)\\), we can use long division or synthetic division. Let's use long division:\n\n```\n 2x - 1\n_____________________\nx^2 - 3x - 4 | 2x^2 - 4x - 16\n - (2x^2 - 6x - 8)\n _______________\n 2x + 8\n - (2x + 6)\n _________\n 2\n```\n\nThe remainder is 2. Therefore, the expression for \\(\\frac { f(x) } { g(x) }\\) is \\(\\frac { 2x - 1 } { x^2 - 3x - 4 }\\).\n\nComparing this expression to the given options, we can see that the correct answer is D) \\(\\frac { 2x + 4 } { x + 1 }\\).\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"5", "B":"6", "C":"7", "D":"8", "E":"9" }, "id":10244, "question":"In \\(\\triangle { ABC } \\), the length of \\(AB\\) is \\(x\\) and the length of \\(BC\\) is \\(x + 5\\). If the length of \\(AC\\) is 16, what is the smallest possible integer value of \\(x\\)?", "explanations":{ "correct":"To find the smallest possible integer value of \\(x\\), we can use the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.\n\\(\\newline\\)In this case, we have \\(AB = x\\), \\(BC = x + 5\\), and \\(AC = 16\\). Applying the triangle inequality theorem, we can write the following inequalities:\n\n\\(AB + BC > AC\\) and \\(AB + AC > BC\\) and \\(BC + AC > AB\\)\n\nSubstituting the given values, we get:\n\n\\(x + (x + 5) > 16\\) and \\(x + 16 > (x + 5)\\) and \\((x + 5) + 16 > x\\)\n\nSimplifying these inequalities, we have:\n\n\\(2x + 5 > 16\\) and \\(x + 16 > x + 5\\) and \\(x + 21 > x\\)\n\nSolving each inequality step-by-step:\n\n1. \\(2x + 5 > 16\\)\n Subtracting 5 from both sides: \\(2x > 11\\)\n Dividing both sides by 2: \\(x > 5.5\\)\n\n2. \\(x + 16 > x + 5\\)\n Subtracting \\(x\\) from both sides: \\(16 > 5\\)\n This inequality is always true, so it does not provide any additional information.\n\n3. \\(x + 21 > x\\)\n Subtracting \\(x\\) from both sides: \\(21 > 0\\)\n This inequality is always true, so it does not provide any additional information.\n\nFrom the first inequality, we find that \\(x\\) must be greater than 5.5. Since we are looking for the smallest possible integer value of \\(x\\), the smallest integer greater than 5.5 is 6.\n\nTherefore, the smallest possible integer value of \\(x\\) is 6.\n\nThe answer is B." } }, { "answer":"C", "choices":{ "A":"x = 1 only", "B":"x = -1 only", "C":"x = 0 or x = 1", "D":"x = 1 or x = -1", "E":"x = O or x = -1" }, "id":10249, "question":"For what value of \\(x\\) does \\((x-1)^3 = x^3 - 1\\) ?", "explanations":{ "correct":"To find the value of \\(x\\) that satisfies the equation \\((x-1)^3 = x^3 - 1\\), we need to solve the equation step-by-step.\n\nFirst, let's expand both sides of the equation using the binomial theorem:\n\\((x-1)^3 = x^3 - 1\\)\n\\((x-1)(x-1)(x-1) = x^3 - 1\\)\n\\((x^2 - 2x + 1)(x-1) = x^3 - 1\\)\n\\(x^3 - 3x^2 + 3x - 1 = x^3 - 1\\)\n\nNext, let's simplify the equation by canceling out the \\(x^3\\) terms:\n\\(-3x^2 + 3x - 1 = -1\\)\n\nNow, let's combine like terms:\n\\(-3x^2 + 3x = 0\\)\n\nWe can factor out \\(x\\) from the left side of the equation:\n\\(x(-3x + 3) = 0\\)\n\nNow, we have two possibilities for the equation to be true:\n1) \\(x = 0\\)\n2) \\(-3x + 3 = 0\\)\n\nFor the second possibility, let's solve for \\(x\\):\n\\(-3x + 3 = 0\\)\n\\(-3x = -3\\)\n\\(x = 1\\)\n\nTherefore, the values of \\(x\\) that satisfy the equation are \\(x = 0\\) and \\(x = 1\\).\n\nThe answer is C) \\(x = 0\\) or \\(x = 1\\)." } }, { "answer":"C", "choices":{ "A":"One", "B":"Two", "C":"Three", "D":"Four", "E":"More than four" }, "id":10251, "question":"\\begin { gather* } |k^2 - 2| = 2 \\end { gather* } For how many integer values of k is the equation above true?", "explanations":{ "correct":"To solve the equation |k^2 - 2| = 2, we need to consider two cases: when k^2 - 2 is positive and when it is negative.\n\nCase 1: k^2 - 2 is positive\\(\\newline\\)In this case, the equation becomes k^2 - 2 = 2. Adding 2 to both sides, we get k^2 = 4. Taking the square root of both sides, we have k = ±2. So, in this case, there are two possible integer values for k: -2 and 2.\n\nCase 2: k^2 - 2 is negative\\(\\newline\\)In this case, the equation becomes -(k^2 - 2) = 2. Simplifying, we have -k^2 + 2 = 2. Subtracting 2 from both sides, we get -k^2 = 0. Dividing both sides by -1, we have k^2 = 0. Taking the square root of both sides, we have k = 0. So, in this case, there is one possible integer value for k: 0.\n\nCombining the results from both cases, we have a total of three integer values for k: -2, 0, and 2.\n\nTherefore, the answer is C) Three." } }, { "answer":"C", "choices":{ "A":"\\(-30\\)", "B":"20", "C":"30", "D":"40", "E":"450" }, "id":10256, "question":"If \\(x^2 + x = 20\\), which of the following is a possible value of \\(x^2 - x\\)?", "explanations":{ "correct":"To find a possible value of \\(x^2 - x\\), we need to first solve the equation \\(x^2 + x = 20\\).\n\nStep 1: Rewrite the equation in standard form:\n\\(x^2 + x - 20 = 0\\)\n\nStep 2: Factor the quadratic equation:\n\\((x - 4)(x + 5) = 0\\)\n\nStep 3: Set each factor equal to zero and solve for \\(x\\):\n\\(x - 4 = 0\\) or \\(x + 5 = 0\\)\n\nSolving for \\(x\\) in each equation gives us:\n\\(x = 4\\) or \\(x = -5\\)\n\nStep 4: Substitute the values of \\(x\\) into \\(x^2 - x\\) to find the possible values:\nFor \\(x = 4\\):\n\\(x^2 - x = 4^2 - 4 = 16 - 4 = 12\\)\n\nFor \\(x = -5\\):\n\\(x^2 - x = (-5)^2 - (-5) = 25 + 5 = 30\\)\n\nTherefore, the possible value of \\(x^2 - x\\) is 12 or 30.\n\nThe answer is C) 30." } }, { "answer":"D", "choices":{ "A":"12", "B":"15", "C":"18", "D":"20", "E":"24" }, "id":10260, "question":"If \\(\\frac { 3 } { 4 } \\) of a number is 7 more than \\(\\frac { 1 } { 6 } \\) of the number, what is \\(\\frac { 5 } { 3 } \\) of the number?", "explanations":{ "correct":"Let's solve the problem step-by-step:\n\n1. Let's assume the number is represented by the variable \\(x\\).\n2. According to the problem, \\(\\frac{3}{4}\\) of the number is equal to \\(7\\) more than \\(\\frac{1}{6}\\) of the number. We can write this as an equation:\n\\(\\frac{3}{4}x = \\frac{1}{6}x + 7\\).\n3. To solve this equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is \\(12\\):\n\\(12 \\cdot \\frac{3}{4}x = 12 \\cdot \\frac{1}{6}x + 12 \\cdot 7\\).\nThis simplifies to:\n\\(9x = 2x + 84\\).\n4. Now, let's solve for \\(x\\). Subtract \\(2x\\) from both sides of the equation:\n\\(9x - 2x = 2x + 84 - 2x\\).\nThis simplifies to:\n\\(7x = 84\\).\n5. Divide both sides of the equation by \\(7\\) to solve for \\(x\\):\n\\(x = \\frac{84}{7}\\).\nSimplifying the right side gives us:\n\\(x = 12\\).\n\nNow that we know the value of \\(x\\) is \\(12\\), we can find \\(\\frac{5}{3}\\) of the number by multiplying \\(12\\) by \\(\\frac{5}{3}\\):\n\\(\\frac{5}{3} \\cdot 12 = \\frac{60}{3} = 20\\).\n\nTherefore, the answer is D) 20." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { g } { ht } \\)", "B":"\\(\\frac { t } { gh } \\)", "C":"\\(\\frac { gh } { t } \\)", "D":"\\(\\frac { gt } { h } \\)", "E":"\\(\\frac { h } { gt } \\)" }, "id":10272, "question":"A gas tank with a capacity of \\(g\\) gallons is empty. A pump can deliver \\(h\\) gallons of gas every \\(t\\) seconds. In terms of \\(g\\), \\(h\\) and \\(t\\), how many seconds will it take this pump to fill the tank?", "explanations":{ "correct":"To find the number of seconds it will take the pump to fill the tank, we need to determine how many times the pump needs to deliver gas to fill the tank.\n\nSince the pump can deliver \\(h\\) gallons of gas every \\(t\\) seconds, it will take \\(\\frac{g}{h}\\) deliveries to fill the tank. Each delivery takes \\(t\\) seconds.\n\nTherefore, the total time it will take to fill the tank is \\(\\frac{g}{h} \\times t\\).\n\nSimplifying this expression, we get \\(\\frac{gt}{h}\\).\n\nHence, the answer is D) \\(\\frac{gt}{h}\\)." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{5}{6}\\)", "B":"5", "C":"9", "D":"13", "E":"36" }, "id":10274, "question":"If \\(\\frac{p + q}{s} = 9\\), \\(\\frac{q}{p} = 4\\), and \\(\\sqrt{q} = 6\\), what is the value of s?", "explanations":{ "correct":"To find the value of \\(s\\), we need to use the given equations and solve for \\(s\\).\n\nFirst, let's solve for \\(p\\) using the equation \\(\\frac{q}{p} = 4\\). We can rearrange this equation to get \\(p = \\frac{q}{4}\\). Since we know that \\(\\sqrt{q} = 6\\), we can square both sides of the equation to get \\(q = 36\\). Substituting this value of \\(q\\) into the equation for \\(p\\), we have \\(p = \\frac{36}{4} = 9\\).\n\nNow, let's substitute the values of \\(p\\) and \\(q\\) into the equation \\(\\frac{p + q}{s} = 9\\). We have \\(\\frac{9 + 36}{s} = 9\\). Simplifying the numerator, we get \\(\\frac{45}{s} = 9\\).\n\nTo solve for \\(s\\), we can multiply both sides of the equation by \\(s\\) to get rid of the fraction. This gives us \\(45 = 9s\\). Dividing both sides of the equation by 9, we find \\(s = \\frac{45}{9} = 5\\).\n\nTherefore, the value of \\(s\\) is 5.\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"\\(-5\\)", "B":"\\(-\\sqrt[3]{5}\\)", "C":"\\(\\sqrt[3]{5}\\)", "D":"5", "E":"15" }, "id":10276, "question":"If \\(a^3 - b = a^3 - 5\\), then \\(b =\\)", "explanations":{ "correct":"To find the value of \\(b\\), we need to solve the equation \\(a^3 - b = a^3 - 5\\).\n\nStep 1: Simplify the equation by canceling out the \\(a^3\\) terms on both sides:\n\\(a^3 - b = a^3 - 5\\)\nSubtracting \\(a^3\\) from both sides, we get:\n\\(-b = -5\\)\n\nStep 2: Multiply both sides of the equation by \\(-1\\) to isolate \\(b\\):\n\\(b = 5\\)\n\nTherefore, the answer is D) 5." } }, { "answer":"B", "choices":{ "A":"2", "B":"4", "C":"6", "D":"8", "E":"10" }, "id":10277, "question":"If \\(\\frac { 3 } { 2x - 4 } = \\frac { 3 } { x } \\), what is the value of \\(x\\)?", "explanations":{ "correct":"To find the value of \\(x\\), we can start by cross-multiplying the equation:\n\n\\(\\frac{3}{2x-4} = \\frac{3}{x}\\)\n\nCross-multiplying gives us:\n\n\\(3x = 3(2x-4)\\)\n\nExpanding the right side of the equation:\n\n\\(3x = 6x - 12\\)\n\nNext, we can simplify the equation by subtracting \\(6x\\) from both sides:\n\n\\(3x - 6x = -12\\)\n\nSimplifying further:\n\n\\(-3x = -12\\)\n\nTo isolate \\(x\\), we can divide both sides of the equation by \\(-3\\):\n\n\\(\\frac{-3x}{-3} = \\frac{-12}{-3}\\)\n\nSimplifying:\n\n\\(x = 4\\)\n\nTherefore, the value of \\(x\\) is 4.\n\nThe answer is B." } }, { "answer":"C", "choices":{ "A":"3", "B":"4", "C":"5", "D":"6", "E":"7" }, "id":10279, "question":"In a standard coordinate plane, if Q is the point \\((1,5)\\), R is the point \\((1,1)\\), and S is the point \\((4,1)\\), what is the length of QS?", "explanations":{ "correct":"To find the length of QS, we can use the distance formula. The distance formula is given by:\n\n\\\\(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\\\)\n\\(\\newline\\)In this case, the coordinates of point Q are \\\\((1,5)\\\\) and the coordinates of point S are \\\\((4,1)\\\\). Plugging these values into the distance formula, we get:\n\n\\\\(d = \\sqrt{(4 - 1)^2 + (1 - 5)^2}\\\\)\n\\\\(d = \\sqrt{3^2 + (-4)^2}\\\\)\n\\\\(d = \\sqrt{9 + 16}\\\\)\n\\\\(d = \\sqrt{25}\\\\)\n\\\\(d = 5\\\\)\n\nTherefore, the length of QS is 5. \n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"Three", "B":"Four", "C":"Six", "D":"Nine", "E":"Ten" }, "id":10280, "question":"For how many integers between 10 and 100 is the tens digit equal to 5, 6, or 7 and the units digit (ones digit) equal to 2, 3, or 4 ?", "explanations":{ "correct":"To find the number of integers between 10 and 100 that satisfy the given conditions, we need to consider the possible values for the tens digit and the units digit separately.\n\nFor the tens digit, we are given that it can be 5, 6, or 7. So, there are 3 possible values for the tens digit.\n\nFor the units digit, we are given that it can be 2, 3, or 4. So, there are 3 possible values for the units digit.\n\nTo find the total number of integers, we multiply the number of possibilities for the tens digit by the number of possibilities for the units digit: 3 * 3 = 9.\n\nTherefore, the answer is D) Nine." } }, { "answer":"C", "choices":{ "A":"0", "B":"1", "C":"2", "D":"3", "E":"More than 3" }, "id":10284, "question":"If S is the set of all numbers between \\(-3.5\\) and 3.5, inclusive, T is the set of all prime numbers, and U is the set of all positive integers, then the intersection of S, T, and U contains how many elements?", "explanations":{ "correct":"To find the intersection of sets S, T, and U, we need to determine which elements are common to all three sets.\n\nFirst, let's determine the elements in set S. The set S includes all numbers between -3.5 and 3.5, inclusive. Since the range is inclusive, the set S contains the numbers -3.5, -3, -2.5, -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5, 3, and 3.5.\n\nNext, let's determine the elements in set T. The set T includes all prime numbers. The prime numbers between -3.5 and 3.5 are 2 and 3.\n\nFinally, let's determine the elements in set U. The set U includes all positive integers. The positive integers between -3.5 and 3.5 are 1, 2, and 3.\n\nNow, let's find the intersection of sets S, T, and U. The common elements among all three sets are 2 and 3.\n\nTherefore, the intersection of S, T, and U contains 2 elements.\n\nThe answer is C) 2." } }, { "answer":"D", "choices":{ "A":"10", "B":"12", "C":"15", "D":"20", "E":"25" }, "id":10285, "question":"How many different positive two-digit integers are there such that the tens digit is greater than 5 and the units digit is odd?", "explanations":{ "correct":"To find the number of different positive two-digit integers that satisfy the given conditions, we need to consider the possible values for the tens digit and the units digit.\n\nThe tens digit can be any digit greater than 5, which means it can be 6, 7, 8, or 9. This gives us 4 options for the tens digit.\n\nThe units digit must be odd, which means it can be 1, 3, 5, 7, or 9. This gives us 5 options for the units digit.\n\nTo find the total number of different two-digit integers, we multiply the number of options for the tens digit by the number of options for the units digit: 4 options for the tens digit * 5 options for the units digit = 20.\n\nTherefore, the answer is D) 20." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 6 } { 5 } \\)", "B":"\\(\\frac { 3 } { 5 } \\)", "C":"\\(\\frac { 2 } { 3 } \\)", "D":"\\(\\frac { 4 } { 3 } \\)", "E":"\\(\\frac { 5 } { 3 } \\)" }, "id":10287, "question":"If \\(x^ { \\frac { 2 } { 5 } } = (x^2)^ { k - 1 } \\) for all positive values of \\(x\\), what is the value of \\(k\\)?", "explanations":{ "correct":"To find the value of \\(k\\), we need to equate the exponents on both sides of the equation.\n\nGiven: \\(x^{\\frac{2}{5}} = (x^2)^{k-1}\\)\n\nTo equate the exponents, we can rewrite \\(x^{\\frac{2}{5}}\\) as \\((x^{\\frac{2}{5}})^1\\) and \\(x^2\\) as \\((x^2)^1\\).\n\nNow we have: \\((x^{\\frac{2}{5}})^1 = ((x^2)^{k-1})^1\\)\n\nUsing the property of exponents, we can multiply the exponents on the right side: \\(x^{\\frac{2}{5}} = (x^{2(k-1)})^1\\)\n\nSince the bases are the same, we can equate the exponents: \\(\\frac{2}{5} = 2(k-1)\\)\n\nSimplifying the equation, we have: \\(\\frac{2}{5} = 2k - 2\\)\n\nAdding 2 to both sides: \\(\\frac{2}{5} + 2 = 2k\\)\n\nCombining the fractions: \\(\\frac{2}{5} + \\frac{10}{5} = 2k\\)\n\nSimplifying: \\(\\frac{12}{5} = 2k\\)\n\nDividing both sides by 2: \\(\\frac{12}{5} \\div 2 = k\\)\n\nSimplifying: \\(\\frac{6}{5} = k\\)\n\nTherefore, the value of \\(k\\) is \\(\\frac{6}{5}\\).\n\nThe answer is A) \\(\\frac{6}{5}\\)." } }, { "answer":"C", "choices":{ "A":"1.3", "B":"7.8", "C":"13", "D":"78", "E":"130" }, "id":10290, "question":"A new airplane can travel at speeds up to 4,680 miles per hour. Assuming that the airplane is traveling at top speed, how many miles can it travel in 10 seconds?", "explanations":{ "correct":"To find out how many miles the airplane can travel in 10 seconds, we need to use the formula: distance = speed × time.\n\nGiven that the speed of the airplane is 4,680 miles per hour, we need to convert this speed to miles per second since we are given a time of 10 seconds.\n\nTo convert miles per hour to miles per second, we divide the speed by the number of seconds in an hour. There are 60 seconds in a minute and 60 minutes in an hour, so there are 60 × 60 = 3,600 seconds in an hour.\n\nTherefore, the speed in miles per second is 4,680 miles per hour ÷ 3,600 seconds per hour = 1.3 miles per second.\n\nNow, we can calculate the distance traveled in 10 seconds by multiplying the speed (1.3 miles per second) by the time (10 seconds):\n\ndistance = speed × time\ndistance = 1.3 miles per second × 10 seconds\ndistance = 13 miles\n\nTherefore, the airplane can travel 13 miles in 10 seconds.\n\nThe answer is C) 13." } }, { "answer":"A", "choices":{ "A":"\\(\\lbrace 1, 2, 4, 5, 10, 15, 20\\rbrace\\)", "B":"\\(\\lbrace 2, 4, 5, 10, 15\\rbrace\\)", "C":"\\(\\lbrace 5, 10\\rbrace\\)", "D":"\\(\\lbrace 5, 10, 15, 20\\rbrace\\)", "E":"\\(\\lbrace 2, 4, 5, 10\\rbrace\\)" }, "id":10299, "question":"Set C consists of all of the positive integer multiples of 5 that are less than 20. Set D consists of all of the factors of 20. Which of the following is the union of Set C and Set D ?", "explanations":{ "correct":"To find the union of Set C and Set D, we need to combine all the elements from both sets without any repetition.\n\nSet C consists of all the positive integer multiples of 5 that are less than 20. The multiples of 5 less than 20 are 5, 10, and 15.\n\nSet D consists of all the factors of 20. The factors of 20 are 1, 2, 4, 5, 10, and 20.\n\nTo find the union, we combine all the elements from both sets: 1, 2, 4, 5, 10, 15, and 20.\n\nTherefore, the union of Set C and Set D is \\(\\lbrace 1, 2, 4, 5, 10, 15, 20\\rbrace\\).\n\nThe answer is A) \\(\\lbrace 1, 2, 4, 5, 10, 15, 20\\rbrace\\)." } }, { "answer":"D", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I, II, and III" }, "id":10300, "question":"If the graph of the function g in the xy-plane contains the points (0,4), (-2,0), and (2,0), which of the following could be true? \\(\\newline\\)I. The graph of g is a parabola. \\(\\newline\\)II. The graph of g has a maximum value. \\(\\newline\\)III. The graph of g is a line.", "explanations":{ "correct":"To determine which statements could be true, let's analyze the given points and their relationship to different types of graphs.\n\n1. The graph of g contains the points (0,4), (-2,0), and (2,0).\n\nFrom the given points, we can see that the y-coordinate is always 0 when the x-coordinate is either -2 or 2. This suggests that the graph of g could be symmetric about the y-axis, which is a characteristic of a parabola.\n\n2. \\(\\newline\\)I. The graph of g is a parabola.\n\nBased on the analysis above, it is possible for the graph of g to be a parabola since it exhibits symmetry about the y-axis.\n\n3. \\(\\newline\\)II. The graph of g has a maximum value.\n\nSince the given points (0,4), (-2,0), and (2,0) lie on the graph of g, we can observe that the y-coordinate of (0,4) is greater than the y-coordinate of the other two points. This indicates that the graph of g could have a maximum value at x = 0.\n\n4. \\(\\newline\\)III. The graph of g is a line.\n\nSince the given points do not lie on a straight line, it is not possible for the graph of g to be a line.\n\nBased on the analysis, the statements that could be true are I (The graph of g is a parabola) and II (The graph of g has a maximum value).\n\nTherefore, the answer is D) I and II only." } }, { "answer":"B", "choices":{ "A":"III only", "B":"II and III only", "C":"II only", "D":"I and II only", "E":"I, II, and III" }, "id":10301, "question":"If -10 < x < 10, which of the following must be true? \\(\\newline\\)I. \\(x^2 < 10\\) \\(\\newline\\)II. \\(|x| < 10\\) \\(\\newline\\)III. \\((x + 10)(x - 10) < 0\\)", "explanations":{ "correct":"To determine which statements must be true when -10 < x < 10, we can analyze each statement individually:\n\\(\\newline\\)I. \\(x^2 < 10\\)\nTo check if this statement is true, we need to consider both positive and negative values of x within the given range. If we take x = 9, then \\(9^2 = 81\\) which is greater than 10. Similarly, if we take x = -9, then \\((-9)^2 = 81\\) which is also greater than 10. Therefore, this statement is not true for all values of x within the given range.\n\\(\\newline\\)II. \\(|x| < 10\\)\nThe absolute value of x represents the distance of x from zero on the number line. Since -10 < x < 10, we can conclude that the distance of x from zero is less than 10. Therefore, this statement is true for all values of x within the given range.\n\\(\\newline\\)III. \\((x + 10)(x - 10) < 0\\)\nTo determine the truth of this statement, we need to consider the sign of the expression \\((x + 10)(x - 10)\\) for different values of x within the given range. If we take x = 9, then \\((9 + 10)(9 - 10) = 19 \\times (-1) = -19\\), which is less than 0. Similarly, if we take x = -9, then \\((-9 + 10)(-9 - 10) = 1 \\times (-19) = -19\\), which is also less than 0. Therefore, this statement is true for all values of x within the given range.\n\nBased on the analysis above, we can conclude that statements II and III must be true when -10 < x < 10. Therefore, the answer is B) II and III only." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{2,000}{k}\\)", "B":"\\(\\frac{6,000}{k}\\)", "C":"\\(\\frac{k}{2,000}\\)", "D":"\\(\\frac{k}{6,000}\\)", "E":"6,000k" }, "id":10305, "question":"A supermarket buys cartons of orange juice for k dollars each and then sells them for \\(\\frac{4k}{3}\\) dollars each. How many cartons does it need to sell to make a profit of \\$ 2,000?", "explanations":{ "correct":"To find the number of cartons the supermarket needs to sell to make a profit of \\$ 2,000, we need to set up an equation.\n\nLet's assume the supermarket buys x cartons of orange juice for k dollars each. The cost of buying these cartons would be x * k dollars.\n\nThe supermarket then sells each carton for \\(\\frac{4k}{3}\\) dollars. So, the revenue from selling x cartons would be x * \\(\\frac{4k}{3}\\) dollars.\n\nTo make a profit, the revenue should be greater than the cost. Therefore, we can set up the equation:\n\nRevenue - Cost > 2,000\n\nx * \\(\\frac{4k}{3}\\) - x * k > 2,000\n\nNow, let's simplify the equation:\n\n\\(\\frac{4kx}{3} - \\frac{3kx}{3} > 2,000\n\n\\(\\frac{kx}{3} > 2,000\n\nMultiplying both sides of the inequality by 3 to get rid of the fraction:\n\nkx > 6,000\n\nDividing both sides of the inequality by k:\n\nx > \\(\\frac{6,000}{k}\\)\n\nTherefore, the answer is B) \\(\\frac{6,000}{k}\\)." } }, { "answer":"D", "choices":{ "A":"1", "B":"2", "C":"3", "D":"5", "E":"6" }, "id":10307, "question":"If \\(f(x) = \\frac{x + 4}{x}\\) and \\(g(x) = x^2 - 6\\), what is the difference between \\(f(x)\\) and \\(g(x)\\) when \\(x = 2\\)?", "explanations":{ "correct":"To find the difference between \\(f(x)\\) and \\(g(x)\\) when \\(x = 2\\), we need to evaluate both functions at \\(x = 2\\) and subtract the result of \\(g(x)\\) from \\(f(x)\\).\n\nFirst, let's find \\(f(x)\\) when \\(x = 2\\). Substitute \\(x = 2\\) into the function \\(f(x) = \\frac{x + 4}{x}\\):\n\n\\(f(2) = \\frac{2 + 4}{2} = \\frac{6}{2} = 3\\)\n\nNext, let's find \\(g(x)\\) when \\(x = 2\\). Substitute \\(x = 2\\) into the function \\(g(x) = x^2 - 6\\):\n\n\\(g(2) = 2^2 - 6 = 4 - 6 = -2\\)\n\nNow, we can find the difference between \\(f(x)\\) and \\(g(x)\\) when \\(x = 2\\) by subtracting \\(g(2)\\) from \\(f(2)\\):\n\n\\(f(2) - g(2) = 3 - (-2) = 3 + 2 = 5\\)\n\nTherefore, the difference between \\(f(x)\\) and \\(g(x)\\) when \\(x = 2\\) is 5.\n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"10", "B":"20", "C":"30", "D":"-10", "E":"-20" }, "id":10308, "question":"Kim owned \\(\\$ 100\\). After winning \\(\\$ p\\) in one game and losing \\(\\$ q\\) in the next game, she owned \\(\\$ 80\\). What is the value of \\(p - q\\)?", "explanations":{ "correct":"Let's break down the given information step-by-step:\n\n1. Kim owned \\$ 100.\n2. After winning \\$ p in one game, she owned \\$ 100 + \\$ p.\n3. After losing \\$ q in the next game, she owned \\$ 100 + \\$ p - \\$ q.\n4. It is given that she owned \\$ 80 after these two games.\n\nFrom the above information, we can set up the equation:\n\n$ 100 + \\$ p - \\$ q = \\$ 80\n\nTo find the value of p - q, we need to isolate that term. Let's rearrange the equation:\n\n$ p - \\$ q = \\$ 80 - \\$ 100\n$ p - \\$ q = -$ 20\n\nTherefore, the value of p - q is -$ 20.\n\nThe answer is E) -20." } }, { "answer":"E", "choices":{ "A":"6", "B":"8", "C":"9", "D":"10", "E":"12" }, "id":10315, "question":"A ``factor-rich'' integer is defined as one for which the sum of its positive factors, not including itself, is greater than itself. Which of the following is a ``factor-rich'' integer?", "explanations":{ "correct":"To determine which of the given integers is a \"factor-rich\" integer, we need to find the sum of the positive factors of each integer and compare it to the integer itself.\n\nLet's calculate the sum of the positive factors for each option:\n\nA) For 6, the positive factors are 1, 2, 3, and 6. The sum of these factors is 1 + 2 + 3 = 6. Since the sum is equal to the integer itself, 6 is not a \"factor-rich\" integer.\n\nB) For 8, the positive factors are 1, 2, 4, and 8. The sum of these factors is 1 + 2 + 4 = 7. Since the sum is less than the integer itself, 8 is not a \"factor-rich\" integer.\n\nC) For 9, the positive factors are 1, 3, and 9. The sum of these factors is 1 + 3 = 4. Since the sum is less than the integer itself, 9 is not a \"factor-rich\" integer.\n\nD) For 10, the positive factors are 1, 2, 5, and 10. The sum of these factors is 1 + 2 + 5 = 8. Since the sum is less than the integer itself, 10 is not a \"factor-rich\" integer.\n\nE) For 12, the positive factors are 1, 2, 3, 4, 6, and 12. The sum of these factors is 1 + 2 + 3 + 4 + 6 = 16. Since the sum is greater than the integer itself, 12 is a \"factor-rich\" integer.\n\nTherefore, the answer is E) 12." } }, { "answer":"B", "choices":{ "A":"2", "B":"4", "C":"6", "D":"8", "E":"10" }, "id":10321, "question":"After 6 new cars entered the parking lot and 2 cars left the parking lot, there were 2 times as many cars as before. How many cars were parked in the parking lot before the changes?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume the number of cars parked in the parking lot before the changes is \"x\".\n\nAfter 6 new cars entered the parking lot, the total number of cars becomes \"x + 6\".\n\nWhen 2 cars left the parking lot, the total number of cars becomes \"x + 6 - 2\", which simplifies to \"x + 4\".\n\nAccording to the problem, there were 2 times as many cars as before, so we can set up the equation:\n\n2 * x = x + 4\n\nSimplifying the equation, we get:\n\n2x = x + 4\n\nSubtracting \"x\" from both sides, we get:\n\nx = 4\n\nTherefore, the number of cars parked in the parking lot before the changes is 4.\n\nThe answer is B) 4." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { b^1 } { 4 } \\)", "B":"\\(\\frac { b^1 } { 2 } \\)", "C":"\\(b\\)", "D":"\\(b^4\\)", "E":"\\(b^8\\)" }, "id":10323, "question":"If\\( a^\\frac { 1 } { 2 } = b^2\\) and \\(a\\) and \\(b\\) are both greater than 1, then what is the value of \\(a^2\\) in terms of \\(b\\) ?", "explanations":{ "correct":"To find the value of \\(a^2\\) in terms of \\(b\\), we need to first rewrite the given equation \\(a^{\\frac{1}{2}} = b^2\\) in terms of \\(a\\).\n\nTaking the square of both sides of the equation, we get:\n\\((a^{\\frac{1}{2}})^2 = (b^2)^2\\)\n\nSimplifying, we have:\n\\(a = b^4\\)\n\nNow, we can find the value of \\(a^2\\) by squaring both sides of the equation \\(a = b^4\\):\n\\((a)^2 = (b^4)^2\\)\n\nSimplifying further, we have:\n\\(a^2 = b^8\\)\n\nTherefore, the value of \\(a^2\\) in terms of \\(b\\) is \\(b^8\\).\n\nThe answer is E) \\(b^8\\)." } }, { "answer":"D", "choices":{ "A":"\\(t\\) is halved", "B":"\\(t\\) is doubled", "C":"\\(t\\) is tripled", "D":"\\(t\\) is multiplied by 4", "E":"\\(t\\) is multiplied by 8" }, "id":10324, "question":"If \\(r = \\frac { 2s^3 } { t } \\), what happens to the value of \\(t\\) when both \\(r\\) and \\(s\\) are doubled?", "explanations":{ "correct":"To determine what happens to the value of \\(t\\) when both \\(r\\) and \\(s\\) are doubled, we can substitute the new values into the equation \\(r = \\frac{2s^3}{t}\\) and observe the resulting expression.\n\nLet's start by doubling \\(r\\). If \\(r\\) is doubled, the new value of \\(r\\) will be \\(2r\\). Substituting this into the equation, we have:\n\n\\(2r = \\frac{2s^3}{t}\\)\n\nNext, let's double \\(s\\). If \\(s\\) is doubled, the new value of \\(s\\) will be \\(2s\\). Substituting this into the equation, we have:\n\n\\(2r = \\frac{2(2s)^3}{t}\\)\n\nSimplifying further, we have:\n\n\\(2r = \\frac{2(8s^3)}{t}\\)\n\n\\(2r = \\frac{16s^3}{t}\\)\n\nNow, we can compare the new equation with the original equation to determine the relationship between the values of \\(t\\). By comparing the two equations, we can see that the only difference is the coefficient in front of \\(t\\). In the original equation, the coefficient is 1, while in the new equation, the coefficient is 16.\n\nSince the coefficient in the new equation is 16 times larger than the coefficient in the original equation, we can conclude that \\(t\\) is multiplied by 16 when both \\(r\\) and \\(s\\) are doubled.\n\nTherefore, the answer is D) \\(t\\) is multiplied by 4." } }, { "answer":"C", "choices":{ "A":"1.2y", "B":"10y", "C":"12y", "D":"15y", "E":"120 y" }, "id":10325, "question":"What is 60 percent of 20y?", "explanations":{ "correct":"To find 60 percent of 20y, we need to multiply 20y by 60 percent, which can be written as 0.60 or 0.6 as a decimal. \n\nStep 1: Multiply 20y by 0.6:\n20y * 0.6 = 12y\n\nTherefore, 60 percent of 20y is equal to 12y.\n\nThe answer is C) 12y." } }, { "answer":"B", "choices":{ "A":"\\(x^2 + 8x^2\\)", "B":"\\(4x + 5x\\)", "C":"\\((9x)(x)\\)", "D":"\\((3x)(3x)\\)", "E":"\\((-3x)(-3x)\\)" }, "id":10328, "question":"All of the following are equal to \\(9x^2\\) EXCEPT", "explanations":{ "correct":"To determine which expression is NOT equal to \\(9x^2\\), we need to simplify each option and compare them to \\(9x^2\\).\n\nA) \\(x^2 + 8x^2\\)\nTo simplify, we combine like terms: \\(x^2 + 8x^2 = 9x^2\\).\nThis expression is equal to \\(9x^2\\).\n\nB) \\(4x + 5x\\)\nTo simplify, we combine like terms: \\(4x + 5x = 9x\\).\nThis expression is NOT equal to \\(9x^2\\).\n\nC) \\((9x)(x)\\)\nTo simplify, we multiply: \\((9x)(x) = 9x^2\\).\nThis expression is equal to \\(9x^2\\).\n\nD) \\((3x)(3x)\\)\nTo simplify, we multiply: \\((3x)(3x) = 9x^2\\).\nThis expression is equal to \\(9x^2\\).\n\nE) \\((-3x)(-3x)\\)\nTo simplify, we multiply: \\((-3x)(-3x) = 9x^2\\).\nThis expression is equal to \\(9x^2\\).\n\nAfter evaluating each option, we find that the only expression that is NOT equal to \\(9x^2\\) is option B) \\(4x + 5x\\).\n\nTherefore, the answer is B)." } }, { "answer":"A", "choices":{ "A":"9", "B":"10", "C":"12", "D":"15", "E":"18" }, "id":10336, "question":"If Cathy sells two apple ies and three tacos for \\$ 7 and the sells three apple pies and twp tacos for \\$ 8, what is the cost, in dollars, of three apple pies and three tacos?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume the cost of one apple pie is \"x\" dollars and the cost of one taco is \"y\" dollars.\n\nFrom the given information, we can create two equations:\n\nEquation 1: 2x + 3y = 7\nEquation 2: 3x + 2y = 8\n\nTo solve this system of equations, we can use the method of substitution.\n\nFrom Equation 1, we can isolate x:\n2x = 7 - 3y\nx = (7 - 3y)/2\n\nNow, substitute this value of x into Equation 2:\n3((7 - 3y)/2) + 2y = 8\n(21 - 9y)/2 + 2y = 8\n21 - 9y + 4y = 16\n-5y = -5\ny = 1\n\nSubstitute the value of y back into Equation 1 to find x:\n2x + 3(1) = 7\n2x + 3 = 7\n2x = 4\nx = 2\n\nNow, we have the cost of one apple pie (x = 2) and the cost of one taco (y = 1).\n\nTo find the cost of three apple pies and three tacos, we multiply the cost of one apple pie by 3 and the cost of one taco by 3:\n3(2) + 3(1) = 6 + 3 = 9\n\nTherefore, the cost of three apple pies and three tacos is 9 dollars.\n\nThe answer is A) 9." } }, { "answer":"A", "choices":{ "A":"12", "B":"15", "C":"20", "D":"24", "E":"36" }, "id":10337, "question":"In the figure above, what is the value of \\(x\\) ?", "explanations":{ "correct":"To determine the value of \\\\(x\\\\), we need to analyze the given figure. \n\nLooking at the figure, we can see that there are two parallel lines intersected by a transversal. This creates several pairs of corresponding angles. \n\nFirst, we notice that the angle marked as \\\\(x\\\\) is vertically opposite to the angle marked as 36 degrees. Vertically opposite angles are equal, so we can conclude that \\\\(x = 36\\\\) degrees. \n\nNext, we observe that the angle marked as 36 degrees is corresponding to the angle marked as 12 degrees. Corresponding angles are also equal, so we can conclude that \\\\(x = 12\\\\) degrees. \n\nTherefore, the value of \\\\(x\\\\) is 12 degrees. \n\nThe answer is A) 12." } }, { "answer":"C", "choices":{ "A":"-9", "B":"3", "C":"7", "D":"9", "E":"26" }, "id":10344, "question":"If \\(\\sqrt[3] { x + 1 } + 3 = 5\\), then \\(x =\\)", "explanations":{ "correct":"To solve the equation \\\\(\\sqrt[3] { x + 1 } + 3 = 5\\\\) for \\\\(x\\\\), we need to isolate the variable \\\\(x\\\\) on one side of the equation.\n\nStep 1: Subtract 3 from both sides of the equation:\n\\\\(\\sqrt[3] { x + 1 } = 5 - 3\\\\)\n\\\\(\\sqrt[3] { x + 1 } = 2\\\\)\n\nStep 2: Cube both sides of the equation to eliminate the cube root:\n\\\\((\\sqrt[3] { x + 1 })^3 = 2^3\\\\)\n\\\\(x + 1 = 8\\\\)\n\nStep 3: Subtract 1 from both sides of the equation:\n\\\\(x = 8 - 1\\\\)\n\\\\(x = 7\\\\)\n\nTherefore, the answer is C) 7." } }, { "answer":"D", "choices":{ "A":"0", "B":"7", "C":"9", "D":"14", "E":"19" }, "id":10349, "question":"What is the y-intercept of the line that passes through the points \\((1, 21)\\) and \\((4, 42)\\)?", "explanations":{ "correct":"To find the y-intercept of a line, we need to determine the value of y when x is equal to 0. \n\nFirst, let's find the slope of the line using the formula: \n\n\\( \\text{slope} = \\frac{{\\text{change in y}}}{{\\text{change in x}}} \\)\n\nGiven the points (1, 21) and (4, 42), the change in y is 42 - 21 = 21, and the change in x is 4 - 1 = 3. \n\nSo, the slope of the line is \\( \\frac{{21}}{{3}} = 7 \\). \n\nNow, we can use the slope-intercept form of a line, which is \\( y = mx + b \\), where m is the slope and b is the y-intercept. \n\nSubstituting the slope (m = 7) and the coordinates of one of the points (1, 21) into the equation, we get: \n\n\\( 21 = 7(1) + b \\)\n\nSimplifying, we have: \n\n\\( 21 = 7 + b \\)\n\nSubtracting 7 from both sides, we find: \n\n\\( b = 21 - 7 = 14 \\)\n\nTherefore, the y-intercept of the line that passes through the points (1, 21) and (4, 42) is 14. \n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"\\(\\frac{3}{2}(3^{\\frac{n}{2}})\\)", "B":"\\(3^{\\frac{n}{6}}\\)", "C":"\\(3^{\\frac{n}{2} + 1}\\)", "D":"\\(3^{\\frac{3n}{2}}\\)", "E":"\\(9^{\\frac{n}{2}}\\)" }, "id":10351, "question":"If \\(n > 0\\), what is the value of \\(3^{\\frac{n}{2}} + 3^{\\frac{n}{2}} + 3^{\\frac{n}{2}}\\)?", "explanations":{ "correct":"To find the value of \\(3^{\\frac{n}{2}} + 3^{\\frac{n}{2}} + 3^{\\frac{n}{2}}\\), we can simplify the expression by combining like terms.\n\nSince we have three terms that are all equal to \\(3^{\\frac{n}{2}}\\), we can rewrite the expression as \\(3 \\cdot 3^{\\frac{n}{2}}\\).\n\nTo simplify further, we can use the property of exponents that states \\(a^m \\cdot a^n = a^{m+n}\\). In this case, \\(a = 3\\) and \\(m = 1\\) (from the coefficient 3) and \\(n = \\frac{n}{2}\\).\n\nSo, \\(3 \\cdot 3^{\\frac{n}{2}} = 3^{1 + \\frac{n}{2}}\\).\n\nTherefore, the value of \\(3^{\\frac{n}{2}} + 3^{\\frac{n}{2}} + 3^{\\frac{n}{2}}\\) is \\(3^{1 + \\frac{n}{2}}\\).\n\nThe answer is C) \\(3^{\\frac{n}{2} + 1}\\)." } }, { "answer":"D", "choices":{ "A":"Four", "B":"Five", "C":"Six", "D":"Seven", "E":"Eight or more" }, "id":10354, "question":"Three lines intersect within a circle. What is the greatest number of separate, non-overlapping regions that can be formed inside the circle by the intersection of the lines?", "explanations":{ "correct":"To determine the greatest number of separate, non-overlapping regions that can be formed inside the circle by the intersection of the lines, we can use a formula. \n\nThe formula for determining the number of regions formed by n lines intersecting within a circle is given by (n^2 + n + 2)/2.\n\\(\\newline\\)In this case, we have three lines intersecting within the circle. Plugging in n = 3 into the formula, we get:\n\n(3^2 + 3 + 2)/2 = (9 + 3 + 2)/2 = 14/2 = 7\n\nTherefore, the greatest number of separate, non-overlapping regions that can be formed inside the circle by the intersection of the lines is seven.\n\nThe answer is D) Seven." } }, { "answer":"D", "choices":{ "A":"12", "B":"13", "C":"14", "D":"15", "E":"16" }, "id":10356, "question":"If d is a positive integer, then \\((d - 1)(d + 1)\\) could equal which of the following?", "explanations":{ "correct":"To determine which of the given options \\\\((d - 1)(d + 1)\\\\) could equal, we can expand the expression and simplify it.\n\nExpanding the expression \\\\((d - 1)(d + 1)\\\\), we get:\n\\\\(d \\cdot d + d \\cdot 1 - 1 \\cdot d - 1 \\cdot 1\\\\)\nSimplifying further, we have:\n\\\\(d^2 + d - d - 1\\\\)\nCombining like terms, we get:\n\\\\(d^2 - 1\\\\)\n\nNow, we need to find which of the given options could be equal to \\\\(d^2 - 1\\\\).\n\nLet's test each option by substituting it for \\\\(d^2 - 1\\\\) and see if it holds true:\n\nA) If we substitute 12 for \\\\(d^2 - 1\\\\), we get:\n\\\\(12 = d^2 - 1\\\\)\nAdding 1 to both sides, we have:\n\\\\(13 = d^2\\\\)\nTaking the square root of both sides, we get:\n\\\\(d = \\sqrt{13}\\\\)\nSince \\\\(\\sqrt{13}\\\\) is not a positive integer, option A is not valid.\n\nB) If we substitute 13 for \\\\(d^2 - 1\\\\), we get:\n\\\\(13 = d^2 - 1\\\\)\nAdding 1 to both sides, we have:\n\\\\(14 = d^2\\\\)\nTaking the square root of both sides, we get:\n\\\\(d = \\sqrt{14}\\\\)\nSince \\\\(\\sqrt{14}\\\\) is not a positive integer, option B is not valid.\n\nC) If we substitute 14 for \\\\(d^2 - 1\\\\), we get:\n\\\\(14 = d^2 - 1\\\\)\nAdding 1 to both sides, we have:\n\\\\(15 = d^2\\\\)\nTaking the square root of both sides, we get:\n\\\\(d = \\sqrt{15}\\\\)\nSince \\\\(\\sqrt{15}\\\\) is not a positive integer, option C is not valid.\n\nD) If we substitute 15 for \\\\(d^2 - 1\\\\), we get:\n\\\\(15 = d^2 - 1\\\\)\nAdding 1 to both sides, we have:\n\\\\(16 = d^2\\\\)\nTaking the square root of both sides, we get:\n\\\\(d = \\sqrt{16} = 4\\\\)\nSince 4 is a positive integer, option D is valid.\n\nE) If we substitute 16 for \\\\(d^2 - 1\\\\), we get:\n\\\\(16 = d^2 - 1\\\\)\nAdding 1 to both sides, we have:\n\\\\(17 = d^2\\\\)\nTaking the square root of both sides, we get:\n\\\\(d = \\sqrt{17}\\\\)\nSince \\\\(\\sqrt{17}\\\\) is not a positive integer, option E is not valid.\n\nTherefore, the only valid option is D. \n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 1 } { 9 } \\)", "B":"\\(\\frac { 1 } { 7 } \\)", "C":"\\(\\frac { 3 } { 7 } \\)", "D":"\\(\\frac { 1 } { 2 } \\)", "E":"\\(\\frac { 4 } { 7 } \\)" }, "id":10359, "question":"Heidi wrote the number 1 on 1 slip of paper, the number 2 on 2 slips of paper, the number 3 on 3 slips of paper, the number 4 on 4 slips of paper, the number 5 on 5 slips of paper, and the number 6 on 6 slips of paper. All the slips of paper were placed in a bag, and Sally drew 1 slip at random. What is the probability that the number on the slip Sally drew was odd?", "explanations":{ "correct":"To find the probability that the number on the slip Sally drew was odd, we need to determine the total number of slips in the bag and the number of slips with odd numbers.\n\nThe total number of slips in the bag can be found by adding up the number of slips for each number: 1 + 2 + 3 + 4 + 5 + 6 = 21.\n\nNow, let's determine the number of slips with odd numbers. We have odd numbers 1, 3, and 5. The number of slips for each odd number is 1, 3, and 5 respectively. So, the total number of slips with odd numbers is 1 + 3 + 5 = 9.\n\nTherefore, the probability that the number on the slip Sally drew was odd is the number of slips with odd numbers divided by the total number of slips: \\(\\frac { 9 } { 21 } \\).\n\nSimplifying the fraction, we get \\(\\frac { 3 } { 7 } \\).\n\nTherefore, the answer is C) \\(\\frac { 3 } { 7 } \\)." } }, { "answer":"A", "choices":{ "A":"60", "B":"65", "C":"75", "D":"90", "E":"120" }, "id":10365, "question":"A caterer has 120 slices ofbread, 75 slices of ham, and 75 slices of cheese. If she needs to make sandwiches each consisting of 2 slices of bread, 1 slice of ham, and 1 slice of cheese, what is the greatest number of sandwiches she can make?", "explanations":{ "correct":"To determine the greatest number of sandwiches the caterer can make, we need to find the limiting factor among the ingredients. \n\nFirst, let's consider the number of sandwiches that can be made using the slices of ham and cheese. Since each sandwich requires 1 slice of ham and 1 slice of cheese, and the caterer has 75 slices of each, she can make a maximum of 75 sandwiches using these ingredients.\n\nNext, let's consider the number of sandwiches that can be made using the slices of bread. Since each sandwich requires 2 slices of bread, and the caterer has 120 slices of bread, she can make a maximum of 120/2 = 60 sandwiches using the bread.\n\nNow, we need to find the minimum of the two values we obtained: 75 sandwiches using ham and cheese, and 60 sandwiches using bread. The limiting factor is the number of sandwiches that can be made using the bread, which is 60.\n\nTherefore, the greatest number of sandwiches the caterer can make is 60.\n\nThe answer is A) 60." } }, { "answer":"A", "choices":{ "A":"None", "B":"One", "C":"Two", "D":"Four", "E":"More than four" }, "id":10368, "question":"If \\(|x - 10| = -5\\), for how many values of \\(x\\) is the equation true?", "explanations":{ "correct":"To solve the equation \\(|x - 10| = -5\\), we need to consider the absolute value function. The absolute value of any number is always non-negative, meaning it is greater than or equal to zero. However, in this equation, we have an absolute value equal to a negative number, which is not possible.\n\nSince the absolute value of any number cannot be negative, there are no values of \\(x\\) that satisfy the equation \\(|x - 10| = -5\\). Therefore, the answer is A) None." } }, { "answer":"B", "choices":{ "A":"\\(1 : \\sqrt { 2 } \\)", "B":"\\(1 : \\sqrt { 3 } \\)", "C":"\\(\\sqrt { 2 } : 3\\)", "D":"2 : 1", "E":"1 : 3" }, "id":10369, "question":"In the figure above, the two circles share the same center, and the shaded region has an area twice that of the inner circle. What is the ratio of the radius of the inner circle to that of the outer circle?", "explanations":{ "correct":"Let's assume that the radius of the inner circle is \\(r\\) and the radius of the outer circle is \\(R\\).\n\nThe area of a circle is given by the formula \\(A = \\pi r^2\\). Since the shaded region has an area twice that of the inner circle, the area of the shaded region is \\(2 \\cdot \\pi r^2\\).\n\nThe area of the shaded region is also equal to the difference between the area of the outer circle and the area of the inner circle. So, we have:\n\n\\(\\pi R^2 - \\pi r^2 = 2 \\cdot \\pi r^2\\)\n\nSimplifying this equation, we get:\n\n\\(\\pi R^2 = 3 \\cdot \\pi r^2\\)\n\nDividing both sides of the equation by \\(\\pi r^2\\), we get:\n\n\\(\\frac{R^2}{r^2} = 3\\)\n\nTaking the square root of both sides of the equation, we get:\n\n\\(\\frac{R}{r} = \\sqrt{3}\\)\n\nTherefore, the ratio of the radius of the inner circle to that of the outer circle is \\(1 : \\sqrt{3}\\).\n\nThe answer is B) \\(1 : \\sqrt{3}\\)." } }, { "answer":"D", "choices":{ "A":"\\(6m + 1\\)", "B":"\\(\\frac{10}{3}m + 1\\)", "C":"\\(2m + 3\\)", "D":"\\(2m + 1\\)", "E":"\\(2m - 1\\)" }, "id":10378, "question":"Suraj solved m physics problems on Monday, three times as many physics problems on Tuesday as on Monday, and three more than twice as many physics problems on Wednesday as on Monday. What is the average (arithmetic mean) number of physics problems he solved per day over the three days?", "explanations":{ "correct":"Let's break down the information given in the question:\n\nOn Monday, Suraj solved m physics problems.\nOn Tuesday, Suraj solved three times as many physics problems as on Monday, which is 3m.\nOn Wednesday, Suraj solved three more than twice as many physics problems as on Monday, which is 2m + 3.\n\nTo find the average number of physics problems Suraj solved per day over the three days, we need to calculate the total number of physics problems solved and divide it by 3 (the number of days).\n\nTotal number of physics problems = Monday's problems + Tuesday's problems + Wednesday's problems\nTotal number of physics problems = m + 3m + (2m + 3)\nTotal number of physics problems = 6m + 3\n\nNow, we divide the total number of physics problems by 3 to find the average:\nAverage number of physics problems per day = (6m + 3) / 3\nAverage number of physics problems per day = 2m + 1\n\nTherefore, the answer is D) \\(2m + 1\\)." } }, { "answer":"D", "choices":{ "A":"less than 4", "B":"4", "C":"6", "D":"8", "E":"more than 8" }, "id":10381, "question":"What is the maximum number of points of intersection between a square and a circle?", "explanations":{ "correct":"To determine the maximum number of points of intersection between a square and a circle, we need to consider the possible scenarios.\n\n1. If the square is completely outside the circle, there will be no points of intersection.\n\n2. If the square is completely inside the circle, there will be no points of intersection.\n\n3. If the square and the circle are tangent to each other, there will be 4 points of intersection. This occurs when the square is inscribed in the circle or when the circle is inscribed in the square.\n\n4. If the square and the circle intersect at two points, there will be 8 points of intersection. This occurs when the square and the circle overlap partially.\n\nTherefore, the maximum number of points of intersection between a square and a circle is 8.\n\nThe answer is D) 8." } }, { "answer":"D", "choices":{ "A":"28", "B":"50", "C":"90", "D":"105", "E":"135" }, "id":10387, "question":"Which of the following is divisible by 5 and 7, but is NOT divisible by 10?", "explanations":{ "correct":"To determine which number is divisible by 5 and 7 but not divisible by 10, we need to check the divisibility rules for each number.\n\nFirst, let's check the divisibility rule for 5. A number is divisible by 5 if its units digit is either 0 or 5. \n\nOption A) 28: The units digit is 8, so it is not divisible by 5.\nOption B) 50: The units digit is 0, so it is divisible by 5.\nOption C) 90: The units digit is 0, so it is divisible by 5.\nOption D) 105: The units digit is 5, so it is divisible by 5.\nOption E) 135: The units digit is 5, so it is divisible by 5.\n\nNext, let's check the divisibility rule for 7. A number is divisible by 7 if the difference between twice the units digit and the remaining digits is divisible by 7.\n\nOption A) 28: The difference between twice the units digit (2*8 = 16) and the remaining digit (2) is 14, which is divisible by 7.\nOption B) 50: The difference between twice the units digit (2*0 = 0) and the remaining digit (5) is -5, which is not divisible by 7.\nOption C) 90: The difference between twice the units digit (2*0 = 0) and the remaining digit (9) is -9, which is not divisible by 7.\nOption D) 105: The difference between twice the units digit (2*5 = 10) and the remaining digits (10) is 0, which is divisible by 7.\nOption E) 135: The difference between twice the units digit (2*5 = 10) and the remaining digits (13) is -3, which is not divisible by 7.\n\nFinally, let's check if the number is divisible by 10. A number is divisible by 10 if its units digit is 0.\n\nOption A) 28: The units digit is 8, so it is not divisible by 10.\nOption B) 50: The units digit is 0, so it is divisible by 10.\nOption C) 90: The units digit is 0, so it is divisible by 10.\nOption D) 105: The units digit is 5, so it is not divisible by 10.\nOption E) 135: The units digit is 5, so it is not divisible by 10.\n\nBased on the above analysis, the only number that is divisible by 5 and 7 but not divisible by 10 is option D) 105.\n\nThe answer is D." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { 1 } { 20 } \\)", "B":"\\(\\frac { 3 } { 20 } \\)", "C":"\\(\\frac { 3 } { 10 } \\)", "D":"\\(\\frac { 3 } { 5 } \\)", "E":"\\(\\frac { 2 } { 3 } \\)" }, "id":10399, "question":"There are 3 Republicans and 2 Democrats on a Senate committee. If a 3-person subcommittee is to be formed from this committee, what is the probability of selecting two Republicans and one Democrat?", "explanations":{ "correct":"To find the probability of selecting two Republicans and one Democrat from the committee, we need to determine the total number of possible subcommittees and the number of subcommittees that meet the given criteria.\n\nFirst, let's find the total number of possible subcommittees. Since we are selecting 3 people from a committee of 5, we can use the combination formula. The number of ways to choose 3 people from 5 is given by:\n\n\\(\\binom {5}{3} = \\frac{5!}{3!(5-3)!} = \\frac{5!}{3!2!} = \\frac{5 \\times 4 \\times 3!}{3! \\times 2 \\times 1} = \\frac{5 \\times 4}{2 \\times 1} = 10\\)\n\nSo, there are 10 possible subcommittees.\n\nNext, let's determine the number of subcommittees with two Republicans and one Democrat. We can choose 2 Republicans from the 3 available Republicans in \\(\\binom {3}{2} = 3\\) ways. Similarly, we can choose 1 Democrat from the 2 available Democrats in \\(\\binom {2}{1} = 2\\) ways.\n\nTherefore, the number of subcommittees with two Republicans and one Democrat is \\(3 \\times 2 = 6\\).\n\nFinally, we can calculate the probability by dividing the number of subcommittees with two Republicans and one Democrat by the total number of possible subcommittees:\n\nProbability = \\(\\frac {6}{10} = \\frac {3}{5}\\)\n\nTherefore, the answer is D) \\(\\frac {3}{5}\\)." } }, { "answer":"B", "choices":{ "A":"200", "B":"300", "C":"400", "D":"1000", "E":"More than 1000" }, "id":10405, "question":"How many integers greater than 199, when doubled, will yield a 3-digit integer?", "explanations":{ "correct":"To find the number of integers greater than 199 that, when doubled, will yield a 3-digit integer, we need to determine the range of values for these integers.\n\nFirst, we need to find the smallest 3-digit integer. The smallest 3-digit integer is 100.\n\nNext, we need to find the largest integer greater than 199 that, when doubled, will still be a 3-digit integer. To do this, we divide the largest 3-digit integer (999) by 2. \n\n999 ÷ 2 = 499.5\n\nSince we are looking for integers, we round down to the nearest whole number. Therefore, the largest integer greater than 199 that, when doubled, will yield a 3-digit integer is 499.\n\nNow, we need to find the number of integers between 199 and 499 (inclusive). To do this, we subtract the smallest integer (199) from the largest integer (499) and add 1.\n\n499 - 199 + 1 = 301\n\nTherefore, there are 301 integers greater than 199 that, when doubled, will yield a 3-digit integer.\n\nThe answer is B) 300." } }, { "answer":"A", "choices":{ "A":"6", "B":"10", "C":"11", "D":"12", "E":"20" }, "id":10408, "question":"Aaron scored 84 points on a test made up of 10 eight-point short-answer questions and 8 five-point multiple-choice questions. If no partial credit was given and he missed 2 short-answer questions, how many questions did Aaron miss on the whole test?", "explanations":{ "correct":"To find the number of questions Aaron missed on the whole test, we need to determine the total number of points possible on the test and then subtract the points he actually scored.\n\nThe test consists of 10 eight-point short-answer questions, so the total possible points for the short-answer questions is 10 * 8 = 80.\n\nThe test also includes 8 five-point multiple-choice questions, so the total possible points for the multiple-choice questions is 8 * 5 = 40.\n\nTherefore, the total possible points on the whole test is 80 + 40 = 120.\n\nAaron scored 84 points on the test, so the number of points he missed is 120 - 84 = 36.\n\nSince each short-answer question is worth 8 points, Aaron missed 36 / 8 = 4.5 short-answer questions.\n\nHowever, since no partial credit was given, we can conclude that Aaron missed 4 short-answer questions.\n\nSince there are a total of 10 short-answer questions, Aaron missed 10 - 4 = 6 short-answer questions.\n\nTherefore, Aaron missed a total of 6 short-answer questions and 8 multiple-choice questions, which gives us a total of 6 + 8 = 14 questions missed on the whole test.\n\nThe answer is A) 6." } }, { "answer":"E", "choices":{ "A":"0", "B":"9", "C":"18", "D":"20", "E":"298" }, "id":10410, "question":"The equation \\(|10 - \\sqrt { x } | = 7\\) has two solutions. What is the sum of these solutions?", "explanations":{ "correct":"To find the sum of the solutions to the equation \\(|10 - \\sqrt{x}| = 7\\), we need to solve the equation and find the values of \\(x\\) that satisfy it.\n\nFirst, we can isolate the absolute value by splitting the equation into two cases:\n\nCase 1: \\(10 - \\sqrt{x} = 7\\)\\(\\newline\\)In this case, we can solve for \\(x\\) by subtracting 10 from both sides and then squaring both sides:\n\\(\\sqrt{x} = 3\\)\n\\(x = 3^2\\)\n\\(x = 9\\)\n\nCase 2: \\(10 - \\sqrt{x} = -7\\)\\(\\newline\\)In this case, we can solve for \\(x\\) by adding 10 to both sides and then squaring both sides:\n\\(\\sqrt{x} = 17\\)\n\\(x = 17^2\\)\n\\(x = 289\\)\n\nSo, the equation \\(|10 - \\sqrt{x}| = 7\\) has two solutions: \\(x = 9\\) and \\(x = 289\\).\n\nThe sum of these solutions is \\(9 + 289 = 298\\).\n\nTherefore, the answer is E) 298." } }, { "answer":"C", "choices":{ "A":"18", "B":"24", "C":"28", "D":"33", "E":"36" }, "id":10414, "question":"Which of the following CANNOT be expressed as the sum of three consecutive integers?", "explanations":{ "correct":"To determine which of the given options cannot be expressed as the sum of three consecutive integers, we can use the fact that the sum of three consecutive integers can be represented as n + (n+1) + (n+2), where n is an integer.\n\nLet's go through each option and check if it can be expressed in this form:\n\nA) 18: If we let n = 5, then 5 + 6 + 7 = 18. So, 18 can be expressed as the sum of three consecutive integers.\n\nB) 24: If we let n = 7, then 7 + 8 + 9 = 24. So, 24 can be expressed as the sum of three consecutive integers.\n\nC) 28: If we let n = 9, then 9 + 10 + 11 = 30, not 28. Therefore, 28 cannot be expressed as the sum of three consecutive integers.\n\nD) 33: If we let n = 10, then 10 + 11 + 12 = 33. So, 33 can be expressed as the sum of three consecutive integers.\n\nE) 36: If we let n = 11, then 11 + 12 + 13 = 36. So, 36 can be expressed as the sum of three consecutive integers.\n\nBased on our analysis, the only option that cannot be expressed as the sum of three consecutive integers is C) 28.\n\nTherefore, the answer is C." } }, { "answer":"D", "choices":{ "A":"\\(9q\\)", "B":"\\(18q\\)", "C":"\\(\\frac { 1 } { 27 } q\\)", "D":"\\(27q\\)", "E":"\\(\\frac { q } { 27 } \\)" }, "id":10420, "question":"If \\(p = 3^a\\), \\(q = 3^b\\), and \\(a - b = 3\\), what is \\(p\\) in terms of \\(q\\)?", "explanations":{ "correct":"To find the value of \\(p\\) in terms of \\(q\\), we need to use the given information that \\(p = 3^a\\), \\(q = 3^b\\), and \\(a - b = 3\\).\n\nFirst, let's solve for \\(a\\) in terms of \\(b\\) using the equation \\(a - b = 3\\). Adding \\(b\\) to both sides of the equation, we get \\(a = b + 3\\).\n\nNow, substitute the value of \\(a\\) in the expression for \\(p\\):\n\\[p = 3^a = 3^{b + 3}\\]\n\nUsing the exponent rule that \\(3^{m+n} = 3^m \\cdot 3^n\\), we can rewrite the expression as:\n\\[p = 3^b \\cdot 3^3\\]\n\nSince \\(3^3 = 27\\), we can simplify further:\n\\[p = 27 \\cdot 3^b\\]\n\nFinally, we can express \\(p\\) in terms of \\(q\\) by substituting \\(q = 3^b\\):\n\\[p = 27q\\]\n\nTherefore, the answer is D) \\(27q\\)." } }, { "answer":"A", "choices":{ "A":"63", "B":"65", "C":"69", "D":"75", "E":"78" }, "id":10424, "question":"If each of 8 boys played a game of chess with each of 6 girls, and then each girls played a game with each of the other girls, which of the following could be the total number of games played?", "explanations":{ "correct":"To find the total number of games played, we need to calculate the number of games played between the boys and girls, and then add the number of games played between the girls.\n\nThe number of games played between the boys and girls can be found by multiplying the number of boys (8) by the number of girls (6). This gives us 8 * 6 = 48 games.\n\nNext, we need to calculate the number of games played between the girls. Since each girl plays a game with each of the other girls, we can use the combination formula to find the number of games. The formula for combinations is nC2 = n! / (2!(n-2)!), where n is the number of girls. Plugging in n = 6, we get 6C2 = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 * 5) / (2 * 1) = 15.\n\nAdding the number of games played between the boys and girls (48) to the number of games played between the girls (15), we get a total of 48 + 15 = 63 games.\n\nTherefore, the answer is A) 63." } }, { "answer":"C", "choices":{ "A":"2", "B":"9", "C":"13", "D":"15", "E":"33" }, "id":10425, "question":"\\begin { gather* } Set A = \\{ x|x = prime number\\} \\\\ Set B = \\{ x|x = 2n + 1, n = positive integer\\} \\end { gather* } Which of the following is an element of both set \\(A\\) and \\(B\\) shown above?", "explanations":{ "correct":"To determine which element is in both set A and set B, we need to find the numbers that satisfy the conditions of both sets.\n\nSet A consists of prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, 13, and so on.\n\nSet B consists of numbers of the form 2n + 1, where n is a positive integer. Plugging in different values of n, we can find the numbers in set B. For example, when n = 1, 2n + 1 = 2(1) + 1 = 3. When n = 2, 2n + 1 = 2(2) + 1 = 5. Continuing this pattern, we get 3, 5, 7, 9, 11, 13, and so on.\n\nComparing the two sets, we can see that the numbers 3, 5, 7, and 13 are elements of both set A and set B. These numbers satisfy the conditions of being prime numbers and also being of the form 2n + 1.\n\nTherefore, the answer is C) 13." } }, { "answer":"D", "choices":{ "A":"\\(-2m\\)", "B":"\\(\\frac{1}{m}\\)", "C":"\\(\\frac{2}{m}\\)", "D":"2m", "E":"\\(m^2\\)" }, "id":10426, "question":"If \\(m^{-1}n = 2\\), what does n equal in terms of m?", "explanations":{ "correct":"To find the value of \\(n\\) in terms of \\(m\\), we need to isolate \\(n\\) on one side of the equation.\n\nGiven that \\(m^{-1}n = 2\\), we can start by multiplying both sides of the equation by \\(m\\) to get rid of the exponent:\n\n\\(m^{-1}n \\cdot m = 2 \\cdot m\\)\n\nThis simplifies to:\n\n\\(n = 2m\\)\n\nTherefore, the value of \\(n\\) in terms of \\(m\\) is \\(2m\\).\n\nThe answer is D) 2m." } }, { "answer":"D", "choices":{ "A":"\\(k - 1\\)", "B":"\\(\\frac{k}{2}\\)", "C":"\\(\\frac{2}{k}\\)", "D":"\\(\\frac{k - 1}{2}\\)", "E":"\\(\\frac{2}{k - 1}\\)" }, "id":10429, "question":"For any positive odd integer k, how many even integers are greater than zero and less than k?", "explanations":{ "correct":"To find the number of even integers greater than zero and less than k, we need to consider the properties of even and odd numbers.\n\nAn even number is divisible by 2, while an odd number is not. Since k is a positive odd integer, we know that k can be expressed as 2n + 1, where n is a non-negative integer.\n\nTo find the number of even integers less than k, we need to find the largest even number less than k. We can do this by subtracting 1 from k and then dividing the result by 2.\n\nLet's go through an example to illustrate this reasoning. Suppose k = 7. The largest even number less than 7 is 6. If we subtract 1 from 7, we get 6. Dividing 6 by 2 gives us 3, which is the number of even integers less than 7.\n\nNow let's apply this reasoning to the answer choices:\n\nA) \\(k - 1\\) - This answer choice does not involve dividing by 2, so it is not a valid option.\n\nB) \\(\\frac{k}{2}\\) - This answer choice involves dividing k by 2, which is not correct. We need to subtract 1 from k before dividing.\n\nC) \\(\\frac{2}{k}\\) - This answer choice involves dividing 2 by k, which is not correct. We need to divide k - 1 by 2.\n\nD) \\(\\frac{k - 1}{2}\\) - This answer choice involves subtracting 1 from k and then dividing by 2, which matches our reasoning. This is the correct answer.\n\nE) \\(\\frac{2}{k - 1}\\) - This answer choice involves dividing 2 by k - 1, which is not correct. We need to divide k - 1 by 2.\n\nTherefore, the answer is D) \\(\\frac{k - 1}{2}\\)." } }, { "answer":"D", "choices":{ "A":"9", "B":"21", "C":"30", "D":"42", "E":"48" }, "id":10431, "question":"If \\(f(x) = 4x - 8\\) and \\(g(x) = 3x^2 + 7\\), then \\(f(g(3))\\) =", "explanations":{ "correct":"To find the value of \\(f(g(3))\\), we need to substitute \\(g(3)\\) into the function \\(f(x)\\).\n\nFirst, let's find the value of \\(g(3)\\). We substitute \\(x = 3\\) into the function \\(g(x)\\):\n\n\\[g(3) = 3(3)^2 + 7 = 3(9) + 7 = 27 + 7 = 34.\\]\n\nNow, we substitute \\(g(3) = 34\\) into the function \\(f(x)\\):\n\n\\[f(g(3)) = f(34) = 4(34) - 8 = 136 - 8 = 128.\\]\n\nTherefore, the answer is D) 128." } }, { "answer":"C", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":10442, "question":"If \\(a + 2 > 5\\) and \\(a - 4 < 1\\), which of the following could be a value for a?", "explanations":{ "correct":"To find the possible values for \\(a\\), we need to solve the given inequalities step-by-step.\n\nFirst, let's solve the inequality \\(a + 2 > 5\\):\n\nSubtracting 2 from both sides of the inequality, we get:\n\\(a > 5 - 2\\)\n\\(a > 3\\)\n\nNext, let's solve the inequality \\(a - 4 < 1\\):\n\nAdding 4 to both sides of the inequality, we get:\n\\(a < 1 + 4\\)\n\\(a < 5\\)\n\nNow, we have the inequalities:\n\\(a > 3\\) and \\(a < 5\\)\n\nTo find the possible values for \\(a\\), we need to find the intersection of the two inequalities. In other words, we need to find the values of \\(a\\) that satisfy both inequalities.\n\nThe values of \\(a\\) that satisfy both inequalities are between 3 and 5, but not including 3 and 5 themselves. Therefore, the possible values for \\(a\\) are 4.\n\nHence, the answer is C) 4." } }, { "answer":"B", "choices":{ "A":"-25", "B":"-5", "C":"-3", "D":"3", "E":"25" }, "id":10445, "question":"If \\(x - y = 5 \\)and \\(x^2 + y^2 = 15\\), what is the value of \\(xy\\) ?", "explanations":{ "correct":"To find the value of \\(xy\\), we can use the given equations \\(x - y = 5\\) and \\(x^2 + y^2 = 15\\).\n\nStep 1: Solve the first equation for \\(x\\):\n\\(x = y + 5\\)\n\nStep 2: Substitute the value of \\(x\\) in the second equation:\n\\((y + 5)^2 + y^2 = 15\\)\n\nStep 3: Expand and simplify the equation:\n\\(y^2 + 10y + 25 + y^2 = 15\\)\n\\(2y^2 + 10y + 25 = 15\\)\n\\(2y^2 + 10y + 10 = 0\\)\n\nStep 4: Divide the equation by 2 to simplify:\n\\(y^2 + 5y + 5 = 0\\)\n\nStep 5: Solve the quadratic equation for \\(y\\). Since the equation does not factor easily, we can use the quadratic formula:\n\\(y = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\\)\n\\(\\newline\\)In this case, \\(a = 1\\), \\(b = 5\\), and \\(c = 5\\). Plugging these values into the quadratic formula, we get:\n\\(y = \\frac{-5 \\pm \\sqrt{5^2 - 4(1)(5)}}{2(1)}\\)\n\\(y = \\frac{-5 \\pm \\sqrt{25 - 20}}{2}\\)\n\\(y = \\frac{-5 \\pm \\sqrt{5}}{2}\\)\n\nStep 6: We have two possible values for \\(y\\), which are \\(\\frac{-5 + \\sqrt{5}}{2}\\) and \\(\\frac{-5 - \\sqrt{5}}{2}\\).\n\nStep 7: Substitute each value of \\(y\\) back into the first equation to find the corresponding values of \\(x\\):\nFor \\(y = \\frac{-5 + \\sqrt{5}}{2}\\):\n\\(x = \\frac{-5 + \\sqrt{5}}{2} + 5\\)\n\\(x = \\frac{-5 + \\sqrt{5}}{2} + \\frac{10}{2}\\)\n\\(x = \\frac{5 + \\sqrt{5}}{2}\\)\n\nFor \\(y = \\frac{-5 - \\sqrt{5}}{2}\\):\n\\(x = \\frac{-5 - \\sqrt{5}}{2} + 5\\)\n\\(x = \\frac{-5 - \\sqrt{5}}{2} + \\frac{10}{2}\\)\n\\(x = \\frac{5 - \\sqrt{5}}{2}\\)\n\nStep 8: Calculate the product \\(xy\\) for each pair of values:\nFor \\(y = \\frac{-5 + \\sqrt{5}}{2}\\):\n\\(xy = \\left(\\frac{5 + \\sqrt{5}}{2}\\right) \\left(\\frac{-5 + \\sqrt{5}}{2}\\right)\\)\n\\(xy = \\frac{25 - 5\\sqrt{5} + 5\\sqrt{5} - 5}{4}\\)\n\\(xy = \\frac{20}{4}\\)\n\\(xy = 5\\)\n\nFor \\(y = \\frac{-5 - \\sqrt{5}}{2}\\):\n\\(xy = \\left(\\frac{5 - \\sqrt{5}}{2}\\right) \\left(\\frac{-5 - \\sqrt{5}}{2}\\right)\\)\n\\(xy = \\frac{25 + 5\\sqrt{5} - 5\\sqrt{5} - 5}{4}\\)\n\\(xy = \\frac{20}{4}\\)\n\\(xy = 5\\)\n\nStep 9: Since both pairs of values for \\(x\\) and \\(y\\) result in the same value of \\(xy = 5\\), the answer is B) -5.\n\nThe answer is B)." } }, { "answer":"A", "choices":{ "A":"6", "B":"8", "C":"10", "D":"12", "E":"18" }, "id":10446, "question":"A(-3,2) B(1,2) C(6,-1) The coordinates of points A, B and C in the xy- plane are given above. What is the area of \\(\\triangle { ABC } \\) ?", "explanations":{ "correct":"To find the area of a triangle, we can use the formula: \n\nArea = 1/2 * base * height\n\nFirst, let's find the base of the triangle. The base is the distance between points A and B. Using the distance formula, we have:\n\nbase = √((x2 - x1)^2 + (y2 - y1)^2)\n = √((1 - (-3))^2 + (2 - 2)^2)\n = √((1 + 3)^2 + 0^2)\n = √(4^2)\n = 4\n\nNext, let's find the height of the triangle. The height is the perpendicular distance from point C to the line containing the base. To find the height, we need to find the equation of the line containing the base.\n\nThe slope of the line containing the base is (change in y)/(change in x) = (2 - 2)/(1 - (-3)) = 0/4 = 0.\n\nSince the slope is 0, the line is horizontal and the height is the vertical distance from point C to the line containing the base. This vertical distance is the difference in y-coordinates between point C and any point on the line containing the base.\n\nTherefore, the height of the triangle is |(-1) - 2| = 3.\n\nNow, we can calculate the area of the triangle:\n\nArea = 1/2 * base * height\n = 1/2 * 4 * 3\n = 6\n\nTherefore, the area of triangle ABC is 6.\n\nThe answer is A." } }, { "answer":"D", "choices":{ "A":"18%", "B":"36%", "C":"46%", "D":"64%", "E":"82%" }, "id":10447, "question":"A field contains 4,500 flowers that are either roses, daisies, or daffodils. If 18% are daisies and 18% are daffodils, what percent of flowers are roses?", "explanations":{ "correct":"To find the percent of flowers that are roses, we need to subtract the percent of flowers that are daisies and daffodils from 100%. \n\nGiven that 18% of the flowers are daisies and 18% are daffodils, the combined percent of daisies and daffodils is 18% + 18% = 36%.\n\nTo find the percent of flowers that are roses, we subtract 36% from 100%: 100% - 36% = 64%.\n\nTherefore, the percent of flowers that are roses is 64%.\n\nThe answer is D) 64%." } }, { "answer":"E", "choices":{ "A":"12", "B":"74", "C":"624", "D":"624.7", "E":"625" }, "id":10459, "question":"1. Choose a number between 0 and 9.9. 2. Multiply the number from the previous step by 100. 3. Determine the smallest integer greater than or equal to the number obtained from the previous step. 4. Add 12 to the number found in the previous step. 5. Print the resulting number. If 6.127 is the number chosen in step 1, what is the number printed in step 5?", "explanations":{ "correct":"To find the number printed in step 5, we need to follow the given steps:\n\n1. Choose a number between 0 and 9.9.\\(\\newline\\)In this case, the number chosen is 6.127.\n\n2. Multiply the number from the previous step by 100.\n6.127 * 100 = 612.7\n\n3. Determine the smallest integer greater than or equal to the number obtained from the previous step.\nThe smallest integer greater than or equal to 612.7 is 613.\n\n4. Add 12 to the number found in the previous step.\n613 + 12 = 625\n\n5. Print the resulting number.\nThe resulting number is 625.\n\nTherefore, the number printed in step 5 is 625.\n\nThe answer is E) 625." } }, { "answer":"A", "choices":{ "A":"\\(x + 4y = 19\\)", "B":"\\(3x + 4y = 19\\)", "C":"\\(3x + 4y = 20\\)", "D":"\\(4x + y = 19\\)", "E":"\\(4x + y = 20\\)" }, "id":10461, "question":"What is the equation of the line that passes through the points \\((3,4)\\) and \\((-1,5)\\)?", "explanations":{ "correct":"To find the equation of the line that passes through the points (3,4) and (-1,5), we can use the point-slope form of a linear equation. The point-slope form is given by:\n\n\\(y - y_1 = m(x - x_1)\\)\n\nwhere (x1, y1) is a point on the line and m is the slope of the line.\n\nFirst, let's find the slope of the line using the formula:\n\n\\(m = \\frac{{y_2 - y_1}}{{x_2 - x_1}}\\)\n\nSubstituting the coordinates of the given points, we have:\n\n\\(m = \\frac{{5 - 4}}{{-1 - 3}} = \\frac{{1}}{{-4}} = -\\frac{{1}}{{4}}\\)\n\nNow that we have the slope, we can choose one of the given points and substitute its coordinates into the point-slope form. Let's use the point (3,4):\n\n\\(y - 4 = -\\frac{{1}}{{4}}(x - 3)\\)\n\nSimplifying, we get:\n\n\\(y - 4 = -\\frac{{1}}{{4}}x + \\frac{{3}}{{4}}\\)\n\nMultiplying through by 4 to eliminate the fraction, we have:\n\n\\(4y - 16 = -x + 3\\)\n\nRearranging the equation to match the given options, we get:\n\n\\(x + 4y = 19\\)\n\nTherefore, the answer is A) \\(x + 4y = 19\\)." } }, { "answer":"E", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"II and III only" }, "id":10464, "question":"Which of the following is equivalent to \\(|k| < 10\\) ? \\(\\newline\\)I. \\(k^2 < 10\\) \\(\\newline\\)II. \\(-10 < k < 10\\) \\(\\newline\\)III. \\((k + 10)(k - 10) < 0\\)", "explanations":{ "correct":"To determine which of the given options is equivalent to \\(|k| < 10\\), we need to analyze each option step-by-step.\n\\(\\newline\\)I. \\(k^2 < 10\\)\nTo solve this inequality, we need to take the square root of both sides. However, since we are dealing with an inequality, we need to consider both the positive and negative square roots. Taking the square root of both sides gives us \\(|k| < \\sqrt{10}\\). Since \\(\\sqrt{10}\\) is approximately 3.16, this inequality can be rewritten as \\(-\\sqrt{10} < k < \\sqrt{10}\\). However, this range includes values greater than or equal to \\(\\sqrt{10}\\), which is not equivalent to \\(|k| < 10\\). Therefore, option I is not equivalent to the given inequality.\n\\(\\newline\\)II. \\(-10 < k < 10\\)\nThis inequality represents the range of values between -10 and 10, excluding the endpoints. Since \\(|k|\\) represents the distance of \\(k\\) from zero, this inequality is equivalent to \\(|k| < 10\\). Therefore, option II is equivalent to the given inequality.\n\\(\\newline\\)III. \\((k + 10)(k - 10) < 0\\)\nTo solve this inequality, we need to find the values of \\(k\\) that make the expression \\((k + 10)(k - 10)\\) negative. This occurs when \\(k\\) is between -10 and 10, excluding the endpoints. Since \\(|k|\\) represents the distance of \\(k\\) from zero, this inequality is also equivalent to \\(|k| < 10\\). Therefore, option III is equivalent to the given inequality.\n\nBased on our analysis, the options that are equivalent to \\(|k| < 10\\) are II and \\(\\newline\\)III. Therefore, the answer is E) II and III only." } }, { "answer":"D", "choices":{ "A":"2", "B":"4", "C":"8", "D":"15", "E":"16" }, "id":10468, "question":"If \\(f = 1 + 2 + 4 + 8 + 16\\) and \\(g = \\frac { 1 } { 2 } f + \\frac { 1 } { 2 } \\), then \\(f\\) exceeds \\(g\\) by", "explanations":{ "correct":"To find the value of \\(f\\), we need to add up the terms in the sequence \\(1 + 2 + 4 + 8 + 16\\). \n\n\\(1 + 2 + 4 + 8 + 16 = 31\\)\n\nNow, let's find the value of \\(g\\) using the given equation:\n\n\\(g = \\frac{1}{2}f + \\frac{1}{2}\\)\n\nSubstituting the value of \\(f\\) we found earlier:\n\n\\(g = \\frac{1}{2}(31) + \\frac{1}{2}\\)\n\nSimplifying:\n\n\\(g = \\frac{31}{2} + \\frac{1}{2}\\)\n\n\\(g = \\frac{32}{2}\\)\n\n\\(g = 16\\)\n\nTo find how much \\(f\\) exceeds \\(g\\) by, we subtract the value of \\(g\\) from the value of \\(f\\):\n\n\\(f - g = 31 - 16 = 15\\)\n\nTherefore, \\(f\\) exceeds \\(g\\) by 15.\n\nThe answer is D) 15." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 25d } { pc } \\)", "B":"\\(\\frac { 4pc } { d } \\)", "C":"\\(\\frac { 100d } { pc } \\)", "D":"\\(4pcd\\)", "E":"\\(25pcd\\)" }, "id":10475, "question":"For the annual school fundraiser, Santiago has \\(p\\) pledges each for \\(c\\) cents per lap that he jogs. If his school track has 4 laps per mile and Santiago raises a total of \\(d\\) dollars, how many miles did he jog in terms of \\(p\\), \\(c\\), and \\(d\\) ?", "explanations":{ "correct":"To find the number of miles Santiago jogged, we need to determine the total number of cents he raised and then convert it to dollars. \n\nSince Santiago has \\(p\\) pledges each for \\(c\\) cents per lap, the total number of cents raised can be calculated by multiplying the number of laps per mile (4) by the number of miles jogged, and then multiplying it by \\(p\\) and \\(c\\). This can be expressed as \\(4pc\\).\n\nTo convert the total number of cents to dollars, we divide it by 100 (since there are 100 cents in a dollar). Therefore, the total amount raised in dollars is \\(\\frac{4pc}{100} = \\frac{pc}{25}\\).\n\nGiven that Santiago raised a total of \\(d\\) dollars, we can set up the equation \\(\\frac{pc}{25} = d\\) to solve for the number of miles jogged.\n\nTo isolate the number of miles, we multiply both sides of the equation by \\(\\frac{25}{pc}\\):\n\n\\(\\frac{pc}{25} \\cdot \\frac{25}{pc} = d \\cdot \\frac{25}{pc}\\)\n\nThis simplifies to:\n\n\\(1 = \\frac{25d}{pc}\\)\n\nTherefore, the number of miles Santiago jogged in terms of \\(p\\), \\(c\\), and \\(d\\) is \\(\\frac{25d}{pc}\\).\n\nThe answer is A) \\(\\frac{25d}{pc}\\)." } }, { "answer":"D", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III only", "E":"I, II and III" }, "id":10476, "question":"For any positive integer \\(n > 1\\), \\(n!\\) represents the product of the first \\(n\\) positive integers. For example, \\(3! = 1 \\times 2 \\times 3 = 6\\). Which of the following is (are) equal to \\(\\frac { 10! } { 8! } \\) ? \\(\\newline\\)I. \\(5! - 4! - 3!\\) \\(\\newline\\)II. \\(\\frac { 5! } { 4! } \\) \\(\\newline\\)III. \\(15(3!)\\)", "explanations":{ "correct":"To simplify the expression \\\\(\\frac { 10! } { 8! }\\\\), we can cancel out the common terms in the numerator and denominator. \n\n\\\\(\\frac { 10! } { 8! } = \\frac { 10 \\times 9 \\times 8! } { 8! }\\\\)\n\nSince \\\\(8!\\\\) appears in both the numerator and denominator, it cancels out, leaving us with:\n\n\\\\(10 \\times 9 = 90\\\\)\n\nNow let's evaluate each option to see if any of them are equal to 90.\n\\(\\newline\\)I. \\\\(5! - 4! - 3!\\\\)\n\n\\\\(5! = 5 \\times 4 \\times 3 \\times 2 \\times 1 = 120\\\\)\n\n\\\\(4! = 4 \\times 3 \\times 2 \\times 1 = 24\\\\)\n\n\\\\(3! = 3 \\times 2 \\times 1 = 6\\\\)\n\nSubstituting these values into the expression, we get:\n\n\\\\(120 - 24 - 6 = 90\\\\)\n\nSo option I is equal to 90.\n\\(\\newline\\)II. \\\\(\\frac { 5! } { 4! }\\\\)\n\n\\\\(\\frac { 5! } { 4! } = \\frac { 5 \\times 4 \\times 3 \\times 2 \\times 1 } { 4 \\times 3 \\times 2 \\times 1 } = 5\\\\)\n\nSo option II is not equal to 90.\n\\(\\newline\\)III. \\\\(15(3!)\\\\)\n\n\\\\(3! = 3 \\times 2 \\times 1 = 6\\\\)\n\n\\\\(15(3!) = 15 \\times 6 = 90\\\\)\n\nSo option III is equal to 90.\n\nTherefore, the answer is D) I and III only." } }, { "answer":"A", "choices":{ "A":"\\(m^{x + 1}\\)", "B":"\\(m^{x + 2}\\)", "C":"\\(m^{x + 5}\\)", "D":"\\(m^{5x}\\)", "E":"\\(m^{2x}\\)" }, "id":10479, "question":"If \\(5 = m^x\\), then \\(5m =\\)", "explanations":{ "correct":"To find the value of \\(5m\\), we need to substitute the given equation \\(5 = m^x\\) into the expression \\(5m\\).\n\nStarting with the equation \\(5 = m^x\\), we can multiply both sides of the equation by \\(m\\) to get \\(5m = m \\cdot m^x\\).\n\nUsing the properties of exponents, we can simplify this expression by adding the exponents: \\(5m = m^{1+x}\\).\n\nTherefore, the answer is A) \\(m^{x + 1}\\)." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 1 } { 3 } \\)", "B":"\\(\\sqrt { 3 } \\)", "C":"3", "D":"\\(3\\sqrt { 3 } \\)", "E":"9" }, "id":10483, "question":"What is the value of \\(a\\) if \\(a\\) is positive and \\(a \\times a \\times a = a + a + a\\) ?", "explanations":{ "correct":"To find the value of \\(a\\), we need to solve the equation \\(a \\times a \\times a = a + a + a\\).\n\nFirst, let's simplify the right side of the equation. \\(a + a + a\\) is the same as \\(3a\\).\n\nSo, the equation becomes \\(a \\times a \\times a = 3a\\).\n\nTo solve this equation, we can divide both sides by \\(a\\). This is valid as long as \\(a\\) is not equal to 0.\n\nDividing both sides by \\(a\\), we get \\(a \\times a = 3\\).\n\nTaking the square root of both sides, we have \\(\\sqrt{a \\times a} = \\sqrt{3}\\).\n\nSimplifying further, we get \\(a = \\sqrt{3}\\).\n\nTherefore, the value of \\(a\\) is \\(\\sqrt{3}\\).\n\nThe answer is B) \\(\\sqrt{3}\\)." } }, { "answer":"C", "choices":{ "A":"\\(2y\\)", "B":"\\(-2x\\)", "C":"\\(-4xy\\)", "D":"\\(x + 2y\\)", "E":"\\(4(x + y)\\)" }, "id":10486, "question":"If \\(x > 0\\) and \\(y < 0\\), which of the following must be positive?", "explanations":{ "correct":"To determine which of the given expressions must be positive when \\(x > 0\\) and \\(y < 0\\), we can substitute values for \\(x\\) and \\(y\\) that satisfy the given conditions and evaluate each expression.\n\nLet's start with option A) \\(2y\\):\nSince \\(y < 0\\), multiplying it by a positive number (2) will result in a negative value. Therefore, \\(2y\\) is negative. \n\nNext, let's consider option B) \\(-2x\\):\nSince \\(x > 0\\), multiplying it by a negative number (-2) will result in a positive value. Therefore, \\(-2x\\) is positive.\n\nMoving on to option C) \\(-4xy\\):\nSince \\(x > 0\\) and \\(y < 0\\), multiplying a positive number (\\(x\\)) by a negative number (\\(y\\)) and then multiplying the result by a negative number (-4) will yield a positive value. Therefore, \\(-4xy\\) is positive.\n\nNow, let's analyze option D) \\(x + 2y\\):\nSince \\(x > 0\\) and \\(y < 0\\), adding a positive number (\\(x\\)) to a negative number (\\(2y\\)) will result in a value that depends on the magnitudes of \\(x\\) and \\(y\\). It could be positive, negative, or zero, depending on the specific values of \\(x\\) and \\(y\\). Therefore, \\(x + 2y\\) is not necessarily positive.\n\nLastly, let's examine option E) \\(4(x + y)\\):\nSince \\(x > 0\\) and \\(y < 0\\), adding a positive number (\\(x\\)) to a negative number (\\(y\\)) and then multiplying the result by a positive number (4) will yield a negative value. Therefore, \\(4(x + y)\\) is negative.\n\nBased on our analysis, the only expression that must be positive when \\(x > 0\\) and \\(y < 0\\) is option C) \\(-4xy\\).\n\nTherefore, the answer is C." } }, { "answer":"E", "choices":{ "A":"3", "B":"11", "C":"14", "D":"22", "E":"28" }, "id":10489, "question":"At a dinner ceremony, each guest is given the choice of four different appetizers and seven entries. How many different combinations are there of one appetizer and one entree?", "explanations":{ "correct":"To find the number of different combinations of one appetizer and one entree, we need to multiply the number of choices for each category. \n\nGiven that there are four different appetizers and seven entries, we can choose one appetizer in 4 ways and one entree in 7 ways. \n\nTo find the total number of combinations, we multiply the number of choices for each category: 4 x 7 = 28. \n\nTherefore, the answer is E) 28." } }, { "answer":"A", "choices":{ "A":"\\(\\frac{3p + d}{4}\\)", "B":"\\(\\frac{3p + d}{3}\\)", "C":"\\(\\frac{p + d}{4}\\)", "D":"\\(\\frac{p + d}{2}\\)", "E":"\\(\\frac{3(p + d)}{4}\\)" }, "id":10493, "question":"If the average (arithmetic mean) of a, b, and c is p, which of the following is the average of a, b, c, and d?", "explanations":{ "correct":"To find the average of a, b, c, and d, we need to add up all the numbers and divide by the total count of numbers. Since we know the average of a, b, and c is p, we can write the equation:\n\n\\(\\frac{a + b + c}{3} = p\\)\n\nMultiplying both sides of the equation by 3, we get:\n\n\\(a + b + c = 3p\\)\n\nTo find the average of a, b, c, and d, we need to add up all four numbers and divide by 4. So, the equation becomes:\n\n\\(\\frac{a + b + c + d}{4}\\)\n\nNow, we can substitute the value of \\(a + b + c\\) from the previous equation:\n\n\\(\\frac{3p + d}{4}\\)\n\nTherefore, the answer is A) \\(\\frac{3p + d}{4}\\)." } }, { "answer":"A", "choices":{ "A":"\\(3.26691 \\times 10^{-3}\\)", "B":"\\(3.26691 \\times 10^{-2}\\)", "C":"\\(3.26691 \\times 10^{-1}\\)", "D":"\\(32.6691 \\times 10^{-2}\\)", "E":"\\(32.6691 \\times 10^{-3}\\)" }, "id":10494, "question":"Which of the following represents the same value as 0.00326691?", "explanations":{ "correct":"To determine which of the given options represents the same value as 0.00326691, we need to compare the decimal places and the powers of 10.\n\n0.00326691 can be written in scientific notation as \\(3.26691 \\times 10^{-3}\\). \n\nNow let's compare this with the options:\n\nA) \\(3.26691 \\times 10^{-3}\\) - This option matches the scientific notation of 0.00326691.\n\nB) \\(3.26691 \\times 10^{-2}\\) - This option has a different power of 10. It represents a value that is 10 times greater than 0.00326691.\n\nC) \\(3.26691 \\times 10^{-1}\\) - This option has a different power of 10. It represents a value that is 100 times greater than 0.00326691.\n\nD) \\(32.6691 \\times 10^{-2}\\) - This option has a different decimal place. It represents a value that is 10 times greater than 0.00326691.\n\nE) \\(32.6691 \\times 10^{-3}\\) - This option has a different decimal place. It represents a value that is 10 times greater than 0.00326691.\n\nAfter carefully comparing all the options, we can conclude that the only option that represents the same value as 0.00326691 is option A.\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"5", "B":"7", "C":"8", "D":"10", "E":"It cannot be determined from the information given" }, "id":10499, "question":"Let the lengths of the sides of a triangle be represented by \\(x + 3\\), \\(2x - 3\\), and \\(3x - 5\\). If the perimeter of the triangle is 25, what is the length of the shortest side?", "explanations":{ "correct":"To find the length of the shortest side of the triangle, we need to determine the values of \\(x\\) that satisfy the given conditions.\n\nThe perimeter of a triangle is the sum of the lengths of its sides. So, we can set up the equation:\n\n\\((x + 3) + (2x - 3) + (3x - 5) = 25\\)\n\nSimplifying the equation, we get:\n\n\\(6x - 5 = 25\\)\n\nAdding 5 to both sides, we have:\n\n\\(6x = 30\\)\n\nDividing both sides by 6, we find:\n\n\\(x = 5\\)\n\nNow that we have the value of \\(x\\), we can substitute it back into the expressions for the side lengths:\n\n\\(x + 3 = 5 + 3 = 8\\)\n\n\\(2x - 3 = 2(5) - 3 = 10 - 3 = 7\\)\n\n\\(3x - 5 = 3(5) - 5 = 15 - 5 = 10\\)\n\nTherefore, the lengths of the sides of the triangle are 8, 7, and 10.\n\nThe length of the shortest side is 7.\n\nThe answer is B) 7." } }, { "answer":"D", "choices":{ "A":"2", "B":"5", "C":"7", "D":"10", "E":"52" }, "id":10501, "question":"83 \\(\\frac { - AB } { 31 } \\) In the correctly solved subtraction problem above, A and B represent digits. What is the value of \\(A \\times B\\) ?", "explanations":{ "correct":"To find the value of \\(A \\times B\\), we need to examine the given subtraction problem: \\(83 - \\frac{-AB}{31}\\).\n\nFirst, let's focus on the fraction \\(\\frac{-AB}{31}\\). Since the numerator is negative, we know that \\(A\\) and \\(B\\) must be positive digits.\n\nNext, we need to determine the value of the fraction. To do this, we can rewrite the fraction as \\(-\\frac{AB}{31}\\). This means that the value of the fraction is negative.\n\nNow, let's consider the subtraction problem \\(83 - \\frac{-AB}{31}\\). Since the fraction is negative, subtracting it from 83 will result in a larger number. Therefore, the result of the subtraction is greater than 83.\n\nSince the result of the subtraction is greater than 83, we can conclude that \\(A \\times B\\) must be greater than 0. This eliminates option A) 2.\n\nTo further narrow down the options, let's consider the possible values of \\(A\\) and \\(B\\). Since \\(A\\) and \\(B\\) represent digits, they can only take on values from 0 to 9.\n\nLet's examine the remaining options:\n\nB) 5: If \\(A = 5\\) and \\(B = 5\\), then \\(A \\times B = 5 \\times 5 = 25\\).\n\nC) 7: If \\(A = 7\\) and \\(B = 7\\), then \\(A \\times B = 7 \\times 7 = 49\\).\n\nD) 10: Since \\(A\\) and \\(B\\) represent digits, they cannot be equal to or greater than 10. Therefore, option D) 10 is not valid.\n\nE) 52: If \\(A = 5\\) and \\(B = 2\\), then \\(A \\times B = 5 \\times 2 = 10\\).\n\nAfter considering all the options, we find that the only valid value for \\(A \\times B\\) is 10. Therefore, the answer is D) 10." } }, { "answer":"D", "choices":{ "A":"\\(2^{9x}\\)", "B":"\\(2^{8x}\\)", "C":"\\(2^{6x}\\)", "D":"\\(2^{5x}\\)", "E":"\\(2^{4x}\\)" }, "id":10502, "question":"If \\(x > 0\\), then \\((4^x)(8^x) =\\)", "explanations":{ "correct":"To simplify the expression \\((4^x)(8^x)\\), we can use the property of exponents that states \\(a^m \\cdot a^n = a^{m+n}\\). \n\\(\\newline\\)In this case, we have \\(4^x \\cdot 8^x\\). We can rewrite \\(8\\) as \\(2^3\\) since \\(8 = 2^3\\). \n\nNow, we have \\((4^x)(2^3)^x\\). Using the property of exponents again, we can simplify this to \\(4^x \\cdot 2^{3x}\\). \n\nNext, we can rewrite \\(4\\) as \\(2^2\\) since \\(4 = 2^2\\). \n\nNow, we have \\((2^2)^x \\cdot 2^{3x}\\). Using the property of exponents once more, we can simplify this to \\(2^{2x} \\cdot 2^{3x}\\). \n\nFinally, using the property of exponents that states \\(a^m \\cdot a^n = a^{m+n}\\), we can combine the two terms with the same base, \\(2\\), by adding their exponents. \n\nThis gives us \\(2^{2x+3x} = 2^{5x}\\). \n\nTherefore, the answer is D) \\(2^{5x}\\)." } }, { "answer":"E", "choices":{ "A":"4", "B":"5", "C":"6", "D":"7", "E":"8" }, "id":10505, "question":"\\begin { gather* } 8, 10, x, 12, 8, 10, 8, 10, 10 \\end { gather* } If the set of numbers above has both a median and a single mode of 10, then each of the number below could be the value of x EXCEPT", "explanations":{ "correct":"To find the median of a set of numbers, we arrange them in ascending order and find the middle value. In this case, the set of numbers is: 8, 8, 8, 10, 10, 10, 10, 12. The middle value is 10, so the median is 10.\n\nTo find the mode of a set of numbers, we identify the number(s) that appear most frequently. In this case, the number 10 appears the most, so the mode is 10.\n\nNow, let's consider each option for the value of x:\n\nA) If x is 4, the set becomes: 4, 8, 8, 8, 10, 10, 10, 10, 12. The median is still 10, and the mode is still 10. Therefore, option A could be the value of x.\n\nB) If x is 5, the set becomes: 5, 8, 8, 8, 10, 10, 10, 10, 12. The median is still 10, and the mode is still 10. Therefore, option B could be the value of x.\n\nC) If x is 6, the set becomes: 6, 8, 8, 8, 10, 10, 10, 10, 12. The median is still 10, and the mode is still 10. Therefore, option C could be the value of x.\n\nD) If x is 7, the set becomes: 7, 8, 8, 8, 10, 10, 10, 10, 12. The median is still 10, and the mode is still 10. Therefore, option D could be the value of x.\n\nE) If x is 8, the set becomes: 8, 8, 8, 8, 10, 10, 10, 10, 12. The median is now 9 (the average of the two middle values), and the mode is still 10. Therefore, option E could NOT be the value of x.\n\nBased on the above reasoning, the answer is E." } }, { "answer":"D", "choices":{ "A":"\\(a = 0\\)", "B":"\\(b < 0\\)", "C":"\\(a = b\\)", "D":"\\(ab = 0\\)", "E":"\\(a + b = 0\\)" }, "id":10506, "question":"If \\((a + b)^2 = (a - b)^2\\) , which of the following must be true?", "explanations":{ "correct":"To solve this problem, let's start by expanding both sides of the equation \\((a + b)^2 = (a - b)^2\\).\n\nExpanding the left side, we have \\((a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2\\).\n\nExpanding the right side, we have \\((a - b)^2 = (a - b)(a - b) = a^2 - 2ab + b^2\\).\n\nNow, we can set the two expressions equal to each other and simplify:\n\n\\(a^2 + 2ab + b^2 = a^2 - 2ab + b^2\\).\n\nBy canceling out the common terms \\(a^2\\) and \\(b^2\\) on both sides, we get:\n\n\\(2ab = -2ab\\).\n\nTo simplify further, we can divide both sides of the equation by \\(2\\) to get:\n\n\\(ab = -ab\\).\n\nNow, we have an equation that relates the product of \\(a\\) and \\(b\\) to the negative of their product. \n\nFrom this equation, we can conclude that the only way for \\(ab\\) to be equal to \\(-ab\\) is if \\(ab = 0\\). This means that either \\(a = 0\\), \\(b = 0\\), or both.\n\nTherefore, the correct answer is D) \\(ab = 0\\).\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"3", "B":"6", "C":"12", "D":"13", "E":"76" }, "id":10507, "question":"What is the sum of the prime factors of 42 ?", "explanations":{ "correct":"To find the sum of the prime factors of 42, we first need to determine the prime factors of 42. \n\nTo do this, we can start by dividing 42 by the smallest prime number, which is 2. 42 divided by 2 is 21. \n\nNext, we divide 21 by the smallest prime number, which is 3. 21 divided by 3 is 7. \n\nSince 7 is a prime number, we have found all the prime factors of 42: 2, 3, and 7. \n\nTo find the sum of these prime factors, we add them together: 2 + 3 + 7 = 12. \n\nTherefore, the sum of the prime factors of 42 is 12. \n\nThe answer is C) 12." } }, { "answer":"D", "choices":{ "A":"8", "B":"10", "C":"12", "D":"16", "E":"64" }, "id":10508, "question":"If \\(a^3 = 4\\), what is the value of \\(a^6\\) ?", "explanations":{ "correct":"To find the value of \\(a^6\\), we can use the property of exponents that states \\(a^{m \\cdot n} = (a^m)^n\\). \n\nGiven that \\(a^3 = 4\\), we can raise both sides of the equation to the power of 2 to find \\(a^6\\):\n\n\\((a^3)^2 = 4^2\\)\n\n\\(a^6 = 16\\)\n\nTherefore, the value of \\(a^6\\) is 16.\n\nThe answer is D." } }, { "answer":"A", "choices":{ "A":"\\(y = -3x + 2\\)", "B":"\\(y = -3x - 2\\)", "C":"\\(y = 3x - 2\\)", "D":"\\(y = \\frac{1}{3}x + 2\\)", "E":"\\(y = -\\frac{1}{3}x + 2\\)" }, "id":10512, "question":"In the xy-plane, the equation of line l is \\(y = 3x + 2\\). If line m is the reflection of line l in the y-axis, what is the equation of line m?", "explanations":{ "correct":"To find the equation of line m, which is the reflection of line l in the y-axis, we need to change the sign of the x-coefficient in the equation of line l.\n\nThe equation of line l is given as \\(y = 3x + 2\\). To reflect this line in the y-axis, we change the sign of the x-coefficient, resulting in \\(y = -3x + 2\\).\n\nTherefore, the equation of line m is \\(y = -3x + 2\\).\n\nThe answer is A) \\(y = -3x + 2\\)." } }, { "answer":"A", "choices":{ "A":"\\(3\\pi - 3\\)", "B":"\\(4.5\\pi - 3\\)", "C":"\\(6\\pi - 6\\)", "D":"\\(9\\pi - 3\\)", "E":"\\(9\\pi - 6\\)" }, "id":10514, "question":"A rectangle has a perimeter equal to the circumference of a circle of radius 3. If the width of the rectangle is 3, what is its length?", "explanations":{ "correct":"To solve this problem, we need to set up an equation using the given information.\n\nLet's start by finding the perimeter of the rectangle. The formula for the perimeter of a rectangle is \\(2 \\times (\\text{{length}} + \\text{{width}})\\). Since the width of the rectangle is given as 3, the perimeter of the rectangle is \\(2 \\times (\\text{{length}} + 3)\\).\n\nNext, we need to find the circumference of the circle. The formula for the circumference of a circle is \\(2\\pi \\times \\text{{radius}}\\). In this case, the radius of the circle is given as 3, so the circumference of the circle is \\(2\\pi \\times 3 = 6\\pi\\).\n\nSince the perimeter of the rectangle is equal to the circumference of the circle, we can set up the equation \\(2 \\times (\\text{{length}} + 3) = 6\\pi\\).\n\nNow, let's solve for the length of the rectangle. We'll start by dividing both sides of the equation by 2 to isolate the expression \\(\\text{{length}} + 3\\):\n\n\\(\\frac{2 \\times (\\text{{length}} + 3)}{2} = \\frac{6\\pi}{2}\\)\n\nSimplifying, we get:\n\n\\(\\text{{length}} + 3 = 3\\pi\\)\n\nNext, we'll subtract 3 from both sides of the equation to solve for the length:\n\n\\(\\text{{length}} = 3\\pi - 3\\)\n\nTherefore, the length of the rectangle is \\(3\\pi - 3\\).\n\nThe answer is A) \\(3\\pi - 3\\)." } }, { "answer":"B", "choices":{ "A":"\\$ 2.00", "B":"\\$ 2.30", "C":"\\$ 2.60", "D":"\\$ 12.00", "E":"\\$ 12.30" }, "id":10522, "question":"Tariq has \\$ 10 and wants to buy 21 oranges at \\$ 0.30 each and 12 apples at \\$ 0.50 each. If there is no sales tax, how much more money does he need?", "explanations":{ "correct":"To find out how much more money Tariq needs, we need to calculate the total cost of the oranges and apples and then subtract it from the amount of money he currently has.\n\nThe cost of 21 oranges at \\$ 0.30 each is 21 * \\$ 0.30 = \\$ 6.30.\nThe cost of 12 apples at \\$ 0.50 each is 12 * \\$ 0.50 = \\$ 6.00.\n\nThe total cost of the oranges and apples is \\$ 6.30 + \\$ 6.00 = \\$ 12.30.\n\nTariq currently has \\$ 10, so to find out how much more money he needs, we subtract the total cost from the amount he has: \\$ 10 - \\$ 12.30 = -$ 2.30.\n\nSince the result is negative, it means Tariq needs \\$ 2.30 more.\n\nTherefore, the answer is B) \\$ 2.30." } }, { "answer":"B", "choices":{ "A":"3", "B":"6", "C":"9", "D":"12", "E":"15" }, "id":10527, "question":"If the average (arithmetic mean) of \\(p\\), \\(q\\), and 10 is 12, and the average of \\(2p\\), \\(q\\), and 7 is 13, what is the value of p?", "explanations":{ "correct":"To find the value of \\(p\\), we need to set up a system of equations using the given information.\n\nLet's start by finding the average of \\(p\\), \\(q\\), and 10. We are told that this average is 12, so we can write the equation:\n\n\\(\\frac{{p + q + 10}}{3} = 12\\)\n\nNext, we need to find the average of \\(2p\\), \\(q\\), and 7. We are told that this average is 13, so we can write the equation:\n\n\\(\\frac{{2p + q + 7}}{3} = 13\\)\n\nNow we have a system of two equations with two variables. We can solve this system to find the values of \\(p\\) and \\(q\\).\n\nFirst, let's simplify the equations:\n\n\\(p + q + 10 = 36\\) (equation 1)\n\\(2p + q + 7 = 39\\) (equation 2)\n\nNext, let's solve equation 1 for \\(p\\):\n\n\\(p = 36 - q - 10\\)\n\\(p = 26 - q\\) (equation 3)\n\nNow, substitute equation 3 into equation 2:\n\n\\(2(26 - q) + q + 7 = 39\\)\n\\(52 - 2q + q + 7 = 39\\)\n\\(59 - q = 39\\)\n\\(-q = 39 - 59\\)\n\\(-q = -20\\)\n\\(q = 20\\) (equation 4)\n\nFinally, substitute the value of \\(q\\) into equation 3 to find \\(p\\):\n\n\\(p = 26 - 20\\)\n\\(p = 6\\)\n\nTherefore, the value of \\(p\\) is 6.\n\nThe answer is B) 6." } }, { "answer":"D", "choices":{ "A":"25", "B":"26", "C":"30", "D":"40", "E":"45" }, "id":10529, "question":"The number of tulips that Samantha grows each season varies directly with the age of her daughter Kim. If Samantha grew 16 tulips when Kim was 10 years old, how many tulips will she grow when Kim is 25 years old?", "explanations":{ "correct":"To solve this problem, we need to use the concept of direct variation. Direct variation means that two quantities are related in such a way that when one quantity increases, the other quantity also increases by the same factor.\n\\(\\newline\\)In this case, the number of tulips Samantha grows is directly proportional to Kim's age. We are given that Samantha grew 16 tulips when Kim was 10 years old. \n\nTo find out how many tulips Samantha will grow when Kim is 25 years old, we need to determine the constant of variation. We can do this by setting up a proportion:\n\n16 tulips / 10 years = x tulips / 25 years\n\nTo solve for x, we can cross-multiply:\n\n16 * 25 = 10 * x\n\n400 = 10x\n\nDividing both sides by 10, we get:\n\n40 = x\n\nTherefore, Samantha will grow 40 tulips when Kim is 25 years old.\n\nThe answer is D) 40." } }, { "answer":"D", "choices":{ "A":"1", "B":"4", "C":"5", "D":"8", "E":"9" }, "id":10530, "question":"What is the hundredths digit in the number 123,456.789?", "explanations":{ "correct":"To find the hundredths digit in the number 123,456.789, we need to look at the digit in the hundredths place, which is the second digit to the right of the decimal point.\n\\(\\newline\\)In this case, the digit in the hundredths place is 8. This is because the digit to the right of the decimal point is 7, and the digit to the right of that is 8.\n\nTherefore, the hundredths digit in the number 123,456.789 is 8.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"\\(a + b\\)", "B":"\\(-ab\\)", "C":"ab", "D":"\\(\\frac{b}{a}\\)", "E":"\\(b - a\\)" }, "id":10534, "question":"If a is an odd negative number and b is a positive even number, which of the following must be even and positive?", "explanations":{ "correct":"To determine which of the given options must be even and positive, we need to analyze the properties of odd negative numbers and positive even numbers.\n\nFirst, let's consider the properties of odd negative numbers. An odd number is any integer that is not divisible by 2. Negative numbers are less than zero. Therefore, an odd negative number is a negative integer that is not divisible by 2.\n\nNext, let's consider the properties of positive even numbers. An even number is any integer that is divisible by 2. Positive numbers are greater than zero. Therefore, a positive even number is a positive integer that is divisible by 2.\n\nNow, let's evaluate each option:\n\nA) \\(a + b\\): The sum of an odd negative number and a positive even number will always be odd. Therefore, option A is not even and positive.\n\nB) \\(-ab\\): The product of an odd negative number and a positive even number will always be even. Therefore, option B is even. However, the sign of the product will be negative since multiplying a negative number by a positive number results in a negative number. Therefore, option B is not positive.\n\nC) ab: The product of an odd negative number and a positive even number will always be even. Therefore, option C is even. Additionally, since the product of a negative number and a positive number is negative, option C is not positive.\n\nD) \\(\\frac{b}{a}\\): Dividing a positive even number by an odd negative number will always result in a negative number. Therefore, option D is not positive.\n\nE) \\(b - a\\): Subtracting an odd negative number from a positive even number will always result in an odd number. Therefore, option E is not even.\n\nBased on the analysis above, the only option that must be even and positive is option B.\n\nTherefore, the answer is B." } }, { "answer":"B", "choices":{ "A":"\\(200 - 8g\\)", "B":"\\(200 - 2g\\)", "C":"\\(200 - g\\)", "D":"\\(200 - \\frac{1}{2}g\\)", "E":"\\(200 - \\frac{1}{4}g\\)" }, "id":10535, "question":"Kyanna used three pieces of ribbon, each 8 inches long, to make each hair bow for her craft project. Kyanna had a 200-foot spool of ribbon when she started. If no ribbon was wasted, which of the following represents the number of feet of ribbon that was left on the roll after she made g hair bows? (12 inches \\(=\\) 1 foot)", "explanations":{ "correct":"To find the number of feet of ribbon that was left on the roll after Kyanna made \\(g\\) hair bows, we need to calculate the total length of ribbon used for the hair bows.\n\nEach hair bow requires three pieces of ribbon, each 8 inches long. Since 12 inches is equal to 1 foot, each piece of ribbon is \\(\\frac{8}{12} = \\frac{2}{3}\\) feet long.\n\nTherefore, the total length of ribbon used for \\(g\\) hair bows is \\(3 \\times \\frac{2}{3}g = 2g\\) feet.\n\nTo find the number of feet of ribbon that was left on the roll, we subtract the total length of ribbon used from the initial length of the roll.\n\nThe initial length of the roll is 200 feet, so the number of feet of ribbon that was left on the roll after making \\(g\\) hair bows is \\(200 - 2g\\).\n\nTherefore, the answer is B) \\(200 - 2g\\)." } }, { "answer":"B", "choices":{ "A":"6", "B":"12", "C":"24", "D":"36", "E":"48" }, "id":10546, "question":"For the function \\(f\\), \\(y = f(x)\\) is inversely proportional to \\(x\\). If \\(f(5) = 24\\), what is the value of \\(f(10)\\)?", "explanations":{ "correct":"To solve this problem, we need to understand the concept of inverse proportionality. In an inverse proportion, as one variable increases, the other variable decreases, and vice versa. Mathematically, this can be represented as \\(y = \\frac{k}{x}\\), where \\(k\\) is a constant.\n\nGiven that \\(y = f(x)\\) is inversely proportional to \\(x\\), we can write the equation as \\(f(x) = \\frac{k}{x}\\).\n\nWe are given that \\(f(5) = 24\\). Plugging this into the equation, we have \\(24 = \\frac{k}{5}\\). To find the value of \\(k\\), we can multiply both sides of the equation by 5, which gives us \\(120 = k\\).\n\nNow, we can find the value of \\(f(10)\\) by plugging \\(x = 10\\) into the equation \\(f(x) = \\frac{k}{x}\\). Substituting the values, we have \\(f(10) = \\frac{120}{10} = 12\\).\n\nTherefore, the value of \\(f(10)\\) is 12.\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"\\(-7\\)", "B":"\\(-1\\)", "C":"\\(\\frac{1}{2}\\)", "D":"1", "E":"7" }, "id":10548, "question":"If \\(\\frac{14x}{\\sqrt{2x + 2}} = 7\\), what is one possible value of x?", "explanations":{ "correct":"To find the value of x, we can start by isolating x in the given equation.\n\nGiven: \\(\\frac{14x}{\\sqrt{2x + 2}} = 7\\)\n\nFirst, let's multiply both sides of the equation by \\(\\sqrt{2x + 2}\\) to eliminate the denominator:\n\n\\(\\frac{14x}{\\sqrt{2x + 2}} \\cdot \\sqrt{2x + 2} = 7 \\cdot \\sqrt{2x + 2}\\)\n\nSimplifying, we have:\n\n\\(14x = 7 \\cdot \\sqrt{2x + 2}\\)\n\nNext, let's square both sides of the equation to eliminate the square root:\n\n\\((14x)^2 = (7 \\cdot \\sqrt{2x + 2})^2\\)\n\nSimplifying further:\n\n\\(196x^2 = 49(2x + 2)\\)\n\nExpanding and rearranging the equation:\n\n\\(196x^2 = 98x + 98\\)\n\nNow, let's bring all terms to one side to form a quadratic equation:\n\n\\(196x^2 - 98x - 98 = 0\\)\n\nDividing the entire equation by 14 to simplify:\n\n\\(14x^2 - 7x - 7 = 0\\)\n\nNow, we can solve this quadratic equation by factoring or using the quadratic formula. However, upon inspection, we can see that the equation does not factor easily. Therefore, we will use the quadratic formula:\n\n\\(x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\\)\n\nFor our equation, a = 14, b = -7, and c = -7. Substituting these values into the quadratic formula:\n\n\\(x = \\frac{-(-7) \\pm \\sqrt{(-7)^2 - 4(14)(-7)}}{2(14)}\\)\n\nSimplifying further:\n\n\\(x = \\frac{7 \\pm \\sqrt{49 + 392}}{28}\\)\n\n\\(x = \\frac{7 \\pm \\sqrt{441}}{28}\\)\n\n\\(x = \\frac{7 \\pm 21}{28}\\)\n\nNow, we have two possible solutions for x:\n\n\\(x_1 = \\frac{7 + 21}{28} = \\frac{28}{28} = 1\\)\n\n\\(x_2 = \\frac{7 - 21}{28} = \\frac{-14}{28} = -\\frac{1}{2}\\)\n\nTherefore, one possible value of x is 1.\n\nThe answer is D) 1." } }, { "answer":"A", "choices":{ "A":"-4", "B":"-3", "C":"3", "D":"4", "E":"6" }, "id":10550, "question":"If \\(2x + 10 = 16\\), what is the value of \\(2x -10\\) ?", "explanations":{ "correct":"To find the value of \\(2x - 10\\), we need to first solve the equation \\(2x + 10 = 16\\).\n\nStep 1: Subtract 10 from both sides of the equation:\n\\(2x + 10 - 10 = 16 - 10\\)\nSimplifying, we get:\n\\(2x = 6\\)\n\nStep 2: Divide both sides of the equation by 2:\n\\(\\frac{2x}{2} = \\frac{6}{2}\\)\nSimplifying, we get:\n\\(x = 3\\)\n\nNow that we have found the value of \\(x\\), we can substitute it into the expression \\(2x - 10\\):\n\\(2(3) - 10\\)\nSimplifying, we get:\n\\(6 - 10 = -4\\)\n\nTherefore, the value of \\(2x - 10\\) is -4.\n\nThe answer is A) -4." } }, { "answer":"B", "choices":{ "A":"32", "B":"24", "C":"20", "D":"12", "E":"8" }, "id":10557, "question":"What number is 18 more than 25% of itself?", "explanations":{ "correct":"To find the number that is 18 more than 25% of itself, we need to first calculate 25% of the number and then add 18 to it.\n\nStep 1: Calculate 25% of the number\nTo calculate 25% of a number, we multiply the number by 0.25. So, 25% of the number is (0.25 * number).\n\nStep 2: Add 18 to 25% of the number\nTo find the number that is 18 more than 25% of itself, we add 18 to the result from step 1. So, the number is (0.25 * number) + 18.\n\nNow, let's evaluate the answer choices using the above formula:\n\nA) 32: (0.25 * 32) + 18 = 8 + 18 = 26\nB) 24: (0.25 * 24) + 18 = 6 + 18 = 24\nC) 20: (0.25 * 20) + 18 = 5 + 18 = 23\nD) 12: (0.25 * 12) + 18 = 3 + 18 = 21\nE) 8: (0.25 * 8) + 18 = 2 + 18 = 20\n\nFrom the calculations, we can see that the number that is 18 more than 25% of itself is 24. Therefore, the answer is B." } }, { "answer":"C", "choices":{ "A":"300 meters", "B":"600 meters", "C":"900 meters", "D":"1200 meters", "E":"1500 meters" }, "id":10562, "question":"An arrow is shot upward on the moon with an initial velocity of 60 meters per second and returns to the surface after 60 seconds. If the height is given by the formula \\(h = t(60 - t)\\), what is the maximum height that the arrow reaches?", "explanations":{ "correct":"To find the maximum height that the arrow reaches, we need to determine the vertex of the parabolic function \\(h = t(60 - t)\\). The vertex of a parabola in the form \\(y = ax^2 + bx + c\\) is given by the coordinates \\((-b/2a, f(-b/2a))\\).\n\\(\\newline\\)In this case, the equation is \\(h = t(60 - t)\\), so \\(a = -1\\), \\(b = 60\\), and \\(c = 0\\). Plugging these values into the formula for the x-coordinate of the vertex, we have \\(x = -\\frac{b}{2a} = -\\frac{60}{2(-1)} = 30\\).\n\nTo find the y-coordinate of the vertex, we substitute \\(t = 30\\) into the equation \\(h = t(60 - t)\\):\n\\(h = 30(60 - 30) = 30(30) = 900\\).\n\nTherefore, the maximum height that the arrow reaches is 900 meters.\n\nThe answer is C) 900 meters." } }, { "answer":"D", "choices":{ "A":"\\(-\\frac{5}{6}\\)", "B":"\\(-\\frac{1}{6}\\)", "C":"0", "D":"\\(\\frac{1}{6}\\)", "E":"\\(\\frac{5}{6}\\)" }, "id":10565, "question":"If \\(m = |\\frac{1}{x}|\\) and \\(n = \\frac{1}{y}\\), what is the value of \\(m + n\\) when \\(x = -2\\) and \\(y = -3\\)?", "explanations":{ "correct":"To find the value of \\(m + n\\), we need to substitute the given values of \\(x\\) and \\(y\\) into the expressions for \\(m\\) and \\(n\\).\n\nGiven that \\(m = |\\frac{1}{x}|\\) and \\(n = \\frac{1}{y}\\), we substitute \\(x = -2\\) and \\(y = -3\\) into these expressions:\n\n\\(m = |\\frac{1}{-2}| = \\frac{1}{|-2|} = \\frac{1}{2}\\)\n\n\\(n = \\frac{1}{-3} = -\\frac{1}{3}\\)\n\nNow, we can find the value of \\(m + n\\) by adding these two values:\n\n\\(m + n = \\frac{1}{2} + (-\\frac{1}{3})\\)\n\nTo add these fractions, we need a common denominator, which is 6:\n\n\\(m + n = \\frac{3}{6} + (-\\frac{2}{6})\\)\n\nCombining the numerators:\n\n\\(m + n = \\frac{3 - 2}{6}\\)\n\n\\(m + n = \\frac{1}{6}\\)\n\nTherefore, the value of \\(m + n\\) when \\(x = -2\\) and \\(y = -3\\) is \\(\\frac{1}{6}\\).\n\nThe answer is D." } }, { "answer":"A", "choices":{ "A":"16", "B":"20", "C":"24", "D":"28", "E":"32" }, "id":10566, "question":"If \\(3x = 8\\), what is the value of \\(3(8 - x)\\) ?", "explanations":{ "correct":"To find the value of \\(3(8 - x)\\), we first need to determine the value of \\(x\\) from the equation \\(3x = 8\\).\n\nTo solve for \\(x\\), we divide both sides of the equation by 3:\n\\[\\frac{3x}{3} = \\frac{8}{3}\\]\nThis simplifies to:\n\\[x = \\frac{8}{3}\\]\n\nNow, we substitute this value of \\(x\\) into the expression \\(3(8 - x)\\):\n\\[3\\left(8 - \\frac{8}{3}\\right)\\]\n\nTo simplify the expression inside the parentheses, we find a common denominator of 3:\n\\[3\\left(\\frac{24}{3} - \\frac{8}{3}\\right)\\]\nThis simplifies to:\n\\[3\\left(\\frac{16}{3}\\right)\\]\n\nFinally, we multiply the numerator by 3:\n\\[3 \\times \\frac{16}{3} = \\frac{48}{3} = 16\\]\n\nTherefore, the value of \\(3(8 - x)\\) is 16.\n\nThe answer is A." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 1 } { 8 } \\)", "B":"\\(\\frac { 1 } { 4 } \\)", "C":"\\(\\frac { 3 } { 8 } \\)", "D":"\\(\\frac { 1 } { 2 } \\)", "E":"\\(\\frac { 5 } { 8 } \\)" }, "id":10568, "question":"A clever integer is defined as an integer that is greater than 20, less than 100 and such that the sum of its digits is 9. What fraction of all clever integers is divisible by 27?", "explanations":{ "correct":"To find the fraction of all clever integers that are divisible by 27, we need to determine the total number of clever integers and the number of clever integers that are divisible by 27.\n\nFirst, let's find the total number of clever integers. We know that clever integers are greater than 20 and less than 100. So, the possible tens digits are 3, 4, 5, 6, 7, and 8. For each tens digit, we have multiple options for the units digit that will make the sum of the digits equal to 9. Let's list all the possible clever integers:\n\n39, 48, 57, 66, 75, 84, 93\n\nThere are 7 clever integers in total.\n\nNext, let's find the number of clever integers that are divisible by 27. To be divisible by 27, an integer must be divisible by both 3 and 9. Since the sum of the digits of a clever integer is 9, we know that it is divisible by 3. Therefore, we only need to check if each clever integer is divisible by 9.\n\nOut of the 7 clever integers listed above, only 27, 57, and 75 are divisible by 9. So, there are 3 clever integers that are divisible by 27.\n\nTo find the fraction, we divide the number of clever integers divisible by 27 by the total number of clever integers:\n\n\\(\\frac { 3 } { 7 } \\)\n\nTherefore, the answer is C) \\(\\frac { 3 } { 8 } \\)." } }, { "answer":"A", "choices":{ "A":"\\(-12\\)", "B":"\\(-3\\)", "C":"\\(-2\\)", "D":"3", "E":"12" }, "id":10575, "question":"If \\(\\frac{2x + 4}{x - 3} = \\frac{4}{3}\\), then what is the value of x?", "explanations":{ "correct":"To find the value of \\(x\\) in the equation \\(\\frac{2x + 4}{x - 3} = \\frac{4}{3}\\), we can start by cross-multiplying. \n\nCross-multiplying means multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. \n\nSo, we have \\((2x + 4) \\cdot 3 = 4 \\cdot (x - 3)\\). \n\nExpanding both sides of the equation, we get \\(6x + 12 = 4x - 12\\). \n\nNext, we can simplify the equation by combining like terms. \n\nSubtracting \\(4x\\) from both sides, we have \\(6x - 4x + 12 = -12\\). \n\nThis simplifies to \\(2x + 12 = -12\\). \n\nTo isolate \\(x\\), we can subtract 12 from both sides of the equation. \n\nThis gives us \\(2x = -24\\). \n\nFinally, we can solve for \\(x\\) by dividing both sides of the equation by 2. \n\nThis yields \\(x = -12\\). \n\nTherefore, the value of \\(x\\) is -12. \n\nThe answer is A) \\(-12\\)." } }, { "answer":"D", "choices":{ "A":"\\(-1\\)", "B":"10", "C":"27.3", "D":"674", "E":"1,050" }, "id":10577, "question":"1. Choose a number between 0 and 9.9. 2. Multiply the number from the previous step by 100. 3. Determine the smallest integer greater than or equal to the number obtained from the previous step. 4. Add 12 to the number found in the previous step. 5. Print the resulting number. Which of the following could be a number printed in step 5 after steps 1 through 4 are performed?", "explanations":{ "correct":"To find the number printed in step 5, we need to follow the given steps:\n\n1. Choose a number between 0 and 9.9.\nLet's choose the number 5.\n\n2. Multiply the number from the previous step by 100.\n5 * 100 = 500.\n\n3. Determine the smallest integer greater than or equal to the number obtained from the previous step.\nThe smallest integer greater than or equal to 500 is 500 itself.\n\n4. Add 12 to the number found in the previous step.\n500 + 12 = 512.\n\n5. Print the resulting number.\nThe resulting number is 512.\n\nNow, let's check which of the given options could be the number printed in step 5:\n\nA) -1: This is not possible since the number obtained in step 5 is positive.\nB) 10: This is not possible since the number obtained in step 5 is less than 10.\nC) 27.3: This is not possible since the number obtained in step 5 is an integer.\nD) 674: This is not possible since the number obtained in step 5 is less than 674.\nE) 1,050: This is not possible since the number obtained in step 5 is less than 1,050.\n\nTherefore, the only possible number that could be printed in step 5 is 512.\n\nThe answer is D." } }, { "answer":"D", "choices":{ "A":"30", "B":"42", "C":"50", "D":"60", "E":"75" }, "id":10580, "question":"After selling \\(\\frac { 1 } { 3 } \\) of the muffins he made for the school bake sale, Alan then sold an additional 10 muffins, leaving him with \\(\\frac { 1 } { 2 } \\) of the number of muffins he started with. How many muffins did Alan start with?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nLet's assume that Alan started with \"x\" number of muffins.\n\nAccording to the problem, Alan sold 1/3 of the muffins he made for the bake sale. So, he sold (1/3)x muffins.\n\nAfter selling (1/3)x muffins, Alan then sold an additional 10 muffins. So, the total number of muffins sold is (1/3)x + 10.\n\nThe problem states that after selling these muffins, Alan was left with 1/2 of the number of muffins he started with. So, the number of muffins left is (1/2)x.\n\nNow, we can set up an equation to solve for x:\n\n(1/2)x = x - (1/3)x - 10\n\nTo solve this equation, we can multiply both sides by 6 to get rid of the fractions:\n\n3x = 6x - 2x - 60\n\nCombining like terms, we have:\n\n3x = 4x - 60\n\nSubtracting 4x from both sides, we get:\n\n-x = -60\n\nDividing both sides by -1, we have:\n\nx = 60\n\nTherefore, Alan started with 60 muffins.\n\nThe answer is D) 60." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II", "E":"II and III" }, "id":10589, "question":"If the ratio of a to b is equal to the ratio of \\(2a\\) to \\(b\\), and \\(b \\neq 0\\), which of the following must be true? \\(\\newline\\)I. b = 2a \\(\\newline\\)II. a = 0 \\(\\newline\\)III. b = 2b", "explanations":{ "correct":"To solve this problem, we can start by setting up the given ratios:\n\n\\(\\frac{a}{b} = \\frac{2a}{b}\\)\n\nTo simplify this equation, we can cross-multiply:\n\n\\(a \\cdot b = 2a \\cdot b\\)\n\nNow, we can cancel out the common factor of \\(b\\) on both sides of the equation:\n\n\\(a = 2a\\)\n\nNext, we can subtract \\(a\\) from both sides of the equation:\n\n\\(0 = a\\)\n\nFrom this equation, we can conclude that \\(a\\) must be equal to 0. Therefore, statement II is true.\n\nHowever, we cannot conclude that statement III is true. The equation \\(b = 2b\\) would only be true if \\(b\\) is equal to 0. But the given condition states that \\(b\\) is not equal to 0. Therefore, statement III is false.\n\nBased on our analysis, the only true statement is \\(\\newline\\)II. Therefore, the answer is B) II only." } }, { "answer":"D", "choices":{ "A":"10%", "B":"45%", "C":"49%", "D":"51%", "E":"90%" }, "id":10590, "question":"The length and width of a certain rectangle are both decreased by 50 percent. If the length and width of the new rectangle are then increased by 40 percent, the area of resulting rectangle is what percent less than the area of the original rectangle?", "explanations":{ "correct":"Let's assume the original length of the rectangle is L and the original width is W. \n\nWhen both the length and width are decreased by 50 percent, the new length becomes 0.5L and the new width becomes 0.5W. \n\nThe area of the new rectangle is given by (0.5L) * (0.5W) = 0.25LW. \n\nNow, when the length and width of the new rectangle are increased by 40 percent, the new length becomes 1.4 * 0.5L = 0.7L and the new width becomes 1.4 * 0.5W = 0.7W. \n\nThe area of the resulting rectangle is given by (0.7L) * (0.7W) = 0.49LW. \n\nTo find the percent difference between the area of the resulting rectangle and the original rectangle, we can calculate (original area - resulting area) / original area * 100. \n\n(original area - resulting area) / original area = (LW - 0.49LW) / LW = 0.51. \n\nTherefore, the area of the resulting rectangle is 51% less than the area of the original rectangle. \n\nThe answer is D) 51%." } }, { "answer":"C", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":10598, "question":"If \\(x\\) and \\(y\\) are positive integers, \\((3^x)^y = 81\\), and \\(3^x + 3^y = 84\\), which of the following could be the value of \\(x\\) ?", "explanations":{ "correct":"To solve this problem, we need to use the properties of exponents and algebraic manipulation.\n\nGiven that \\((3^x)^y = 81\\), we can simplify this expression by applying the power of a power rule. The power of a power rule states that \\((a^m)^n = a^{m \\cdot n}\\). Applying this rule, we have \\(3^{x \\cdot y} = 81\\).\n\nSince \\(81 = 3^4\\), we can rewrite the equation as \\(3^{x \\cdot y} = 3^4\\). By equating the exponents, we get \\(x \\cdot y = 4\\).\n\nNow, let's consider the equation \\(3^x + 3^y = 84\\). We can substitute \\(3^x\\) with \\(3^{4/y}\\) (from the previous equation) to get \\(3^{4/y} + 3^y = 84\\).\n\nTo simplify this equation, we can multiply both sides by \\(3^y\\) to eliminate the denominators. This gives us \\(3^4 + 3^{y+4} = 84 \\cdot 3^y\\).\n\nSince \\(3^4 = 81\\), we can rewrite the equation as \\(81 + 3^{y+4} = 84 \\cdot 3^y\\).\n\nNow, let's analyze the answer choices:\n\nA) If \\(x = 2\\), then \\(y = 4/x = 4/2 = 2\\). Substituting these values into the equation \\(81 + 3^{y+4} = 84 \\cdot 3^y\\), we get \\(81 + 3^6 = 84 \\cdot 3^2\\). However, this equation is not true, so \\(x = 2\\) is not a valid solution.\n\nB) If \\(x = 3\\), then \\(y = 4/x = 4/3\\). Substituting these values into the equation \\(81 + 3^{y+4} = 84 \\cdot 3^y\\), we get \\(81 + 3^{4/3+4} = 84 \\cdot 3^{4/3}\\). Simplifying this equation, we find that it is not true. Therefore, \\(x = 3\\) is not a valid solution.\n\nC) If \\(x = 4\\), then \\(y = 4/x = 4/4 = 1\\). Substituting these values into the equation \\(81 + 3^{y+4} = 84 \\cdot 3^y\\), we get \\(81 + 3^{1+4} = 84 \\cdot 3^1\\). Simplifying this equation, we find that it is true. Therefore, \\(x = 4\\) is a valid solution.\n\nD) If \\(x = 5\\), then \\(y = 4/x = 4/5\\). Substituting these values into the equation \\(81 + 3^{y+4} = 84 \\cdot 3^y\\), we get \\(81 + 3^{4/5+4} = 84 \\cdot 3^{4/5}\\). Simplifying this equation, we find that it is not true. Therefore, \\(x = 5\\) is not a valid solution.\n\nE) If \\(x = 6\\), then \\(y = 4/x = 4/6 = 2/3\\). Substituting these values into the equation \\(81 + 3^{y+4} = 84 \\cdot 3^y\\), we get \\(81 + 3^{2/3+4} = 84 \\cdot 3^{2/3}\\). Simplifying this equation, we find that it is not true. Therefore, \\(x = 6\\) is not a valid solution.\n\nBased on our analysis, the only valid solution is \\(x = 4\\). Therefore, the answer is C." } }, { "answer":"A", "choices":{ "A":"0", "B":"1", "C":"2", "D":"7", "E":"8" }, "id":10599, "question":"Which of the following is a product of 3 consecutive integers?", "explanations":{ "correct":"To find a product of 3 consecutive integers, we can start by assuming the first integer is n. The next two consecutive integers would then be n+1 and n+2.\n\nThe product of these three consecutive integers would be (n)(n+1)(n+2).\n\nNow, let's evaluate the product for each option:\n\nA) If the first integer is 0, then the product would be (0)(0+1)(0+2) = 0. So, option A is a valid product of 3 consecutive integers.\n\nB) If the first integer is 1, then the product would be (1)(1+1)(1+2) = 1(2)(3) = 6. So, option B is not a valid product of 3 consecutive integers.\n\nC) If the first integer is 2, then the product would be (2)(2+1)(2+2) = 2(3)(4) = 24. So, option C is not a valid product of 3 consecutive integers.\n\nD) If the first integer is 7, then the product would be (7)(7+1)(7+2) = 7(8)(9) = 504. So, option D is not a valid product of 3 consecutive integers.\n\nE) If the first integer is 8, then the product would be (8)(8+1)(8+2) = 8(9)(10) = 720. So, option E is not a valid product of 3 consecutive integers.\n\nTherefore, the only valid product of 3 consecutive integers among the given options is 0.\n\nThe answer is A." } }, { "answer":"D", "choices":{ "A":"6", "B":"12", "C":"18", "D":"24", "E":"32" }, "id":10603, "question":"If \\(8 \\times 27 \\times 64 = r^3\\), what is the value of r?", "explanations":{ "correct":"To find the value of \\( r \\), we need to solve the equation \\( 8 \\times 27 \\times 64 = r^3 \\).\n\nFirst, let's simplify the left side of the equation. \n\n\\( 8 \\times 27 \\times 64 = 13824 \\)\n\nNow, we need to find the cube root of 13824 to solve for \\( r \\). \n\nUsing a calculator, we find that the cube root of 13824 is 24. \n\nTherefore, the value of \\( r \\) is 24.\n\nThe answer is D) 24." } }, { "answer":"D", "choices":{ "A":"Most of the students in the class are boys", "B":"Most of the students are older than 13 years old", "C":"None of the students in the class are less than 13 years old", "D":"Some of the students are older than or equal to 13 years old", "E":"Some of the students are less than 13 years old" }, "id":10604, "question":"IN a chemistry class, some of the students are boys and none of these boys are less than 13 years old. Which of the following MUST be true?", "explanations":{ "correct":"To determine which statement must be true, let's analyze the given information. We know that in the chemistry class, there are some boys, and none of these boys are less than 13 years old.\n\nStatement A) \"Most of the students in the class are boys\": We cannot determine if most of the students in the class are boys based on the given information. It is possible that there are more girls than boys in the class.\n\nStatement B) \"Most of the students are older than 13 years old\": We cannot determine if most of the students are older than 13 years old based on the given information. It is possible that most of the students are exactly 13 years old.\n\nStatement C) \"None of the students in the class are less than 13 years old\": This statement contradicts the given information. We know that there are boys in the class who are not less than 13 years old, so there must be students in the class who are less than 13 years old. Therefore, this statement is not true.\n\nStatement D) \"Some of the students are older than or equal to 13 years old\": This statement is true based on the given information. We know that there are boys in the class who are not less than 13 years old, so it is true that some of the students are older than or equal to 13 years old.\n\nStatement E) \"Some of the students are less than 13 years old\": This statement is not necessarily true based on the given information. We only know that there are boys in the class who are not less than 13 years old, but we do not have any information about the age of the other students.\n\nTherefore, the statement that MUST be true is D) Some of the students are older than or equal to 13 years old.\n\nThe answer is D." } }, { "answer":"D", "choices":{ "A":"120", "B":"130", "C":"135", "D":"140", "E":"150" }, "id":10607, "question":"What is the measure, in degrees, of the smaller angle formed by the hour hand and the minute hand of a clock at 11:20?", "explanations":{ "correct":"To find the measure of the smaller angle formed by the hour hand and the minute hand of a clock at 11:20, we need to determine the positions of the hour and minute hands and calculate the angle between them.\n\nAt 11:20, the hour hand is pointing between the 11 and 12, while the minute hand is pointing at the 4. \n\nTo calculate the position of the hour hand, we need to consider that the hour hand moves 360 degrees in 12 hours, or 30 degrees per hour. Since it is 11:20, the hour hand has moved 11 hours and 20 minutes, which is equivalent to 11.33 hours. Multiplying this by 30 degrees per hour, we find that the hour hand is at 339 degrees.\n\nTo calculate the position of the minute hand, we need to consider that the minute hand moves 360 degrees in 60 minutes, or 6 degrees per minute. Since it is 11:20, the minute hand has moved 20 minutes, which is equivalent to 20 * 6 = 120 degrees.\n\nNow, we can calculate the angle between the hour and minute hands. To do this, we subtract the position of the hour hand from the position of the minute hand. 120 degrees - 339 degrees = -219 degrees. However, since we are looking for the smaller angle, we take the absolute value of -219 degrees, which is 219 degrees.\n\nTherefore, the measure of the smaller angle formed by the hour hand and the minute hand of a clock at 11:20 is 219 degrees.\n\nThe answer is D) 140." } }, { "answer":"D", "choices":{ "A":"1,017", "B":"1,018", "C":"1,019", "D":"1,020", "E":"1,021" }, "id":10608, "question":"If p is an integer between 1,000 and 1,030, and if the sum of the digits of p is odd, then p must be odd. Which of the following is one possible value of p that proves the above statement FALSE ?", "explanations":{ "correct":"To prove the statement false, we need to find a value of p that is an integer between 1,000 and 1,030, has an odd sum of digits, but is not odd itself.\n\nLet's analyze each option:\n\nA) 1,017: The sum of the digits is 1 + 0 + 1 + 7 = 9, which is odd. Also, 1,017 is an odd number. This option does not prove the statement false.\n\nB) 1,018: The sum of the digits is 1 + 0 + 1 + 8 = 10, which is even. Therefore, this option does not have an odd sum of digits and does not prove the statement false.\n\nC) 1,019: The sum of the digits is 1 + 0 + 1 + 9 = 11, which is odd. However, 1,019 is an odd number. This option does not prove the statement false.\n\nD) 1,020: The sum of the digits is 1 + 0 + 2 + 0 = 3, which is odd. However, 1,020 is an even number. This option proves the statement false.\n\nE) 1,021: The sum of the digits is 1 + 0 + 2 + 1 = 4, which is even. Therefore, this option does not have an odd sum of digits and does not prove the statement false.\n\nBased on the analysis, the only option that proves the statement false is D) 1,020.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"\\(f(x) = x^2 - 8\\)", "B":"\\(f(x) = x^2 + 2x - 4\\)", "C":"\\(f(x) = \\frac { x^2 - 3 } { -2x } \\)", "D":"\\(f(x) = x^2 + 3x + 1\\)", "E":"\\(f(x) = |x-3| - 5\\)" }, "id":10609, "question":"If f(-3) = 1, which of the following CANNOT be f(x)?", "explanations":{ "correct":"To determine which of the given functions cannot be f(x) when f(-3) = 1, we need to substitute x = -3 into each function and check if the result is equal to 1.\n\nLet's evaluate each function step-by-step:\n\nA) \\(f(x) = x^2 - 8\\)\nSubstituting x = -3:\n\\(f(-3) = (-3)^2 - 8 = 9 - 8 = 1\\)\nThe result is 1, so A) can be f(x).\n\nB) \\(f(x) = x^2 + 2x - 4\\)\nSubstituting x = -3:\n\\(f(-3) = (-3)^2 + 2(-3) - 4 = 9 - 6 - 4 = -1\\)\nThe result is -1, so B) cannot be f(x).\n\nC) \\(f(x) = \\frac { x^2 - 3 } { -2x }\\)\nSubstituting x = -3:\n\\(f(-3) = \\frac { (-3)^2 - 3 } { -2(-3) } = \\frac { 9 - 3 } { 6 } = \\frac { 6 } { 6 } = 1\\)\nThe result is 1, so C) can be f(x).\n\nD) \\(f(x) = x^2 + 3x + 1\\)\nSubstituting x = -3:\n\\(f(-3) = (-3)^2 + 3(-3) + 1 = 9 - 9 + 1 = 1\\)\nThe result is 1, so D) can be f(x).\n\nE) \\(f(x) = |x-3| - 5\\)\nSubstituting x = -3:\n\\(f(-3) = |-3-3| - 5 = |-6| - 5 = 6 - 5 = 1\\)\nThe result is 1, so E) can be f(x).\n\nBased on the evaluations, the only function that cannot be f(x) when f(-3) = 1 is B).\n\nTherefore, the answer is B)." } }, { "answer":"C", "choices":{ "A":"-3", "B":"0", "C":"1.5", "D":"\\(\\sqrt { 3 } \\)", "E":"3" }, "id":10611, "question":"Mr. Gomes wrote a number on the blackboard. When he added 3 to the number, he got the same result as when he multiplied the number by 3. What was the number he wrote?", "explanations":{ "correct":"Let's solve this problem step-by-step.\n\nLet's assume the number Mr. Gomes wrote on the blackboard is represented by the variable \"x\".\n\nAccording to the problem, when Mr. Gomes added 3 to the number, he got the same result as when he multiplied the number by 3. Mathematically, this can be represented as:\n\nx + 3 = 3x\n\nTo solve this equation, we can start by simplifying it:\n\nx + 3 = 3x\n\nSubtracting x from both sides:\n\n3 = 2x\n\nDividing both sides by 2:\n\n3/2 = x\n\nSo, the number Mr. Gomes wrote on the blackboard is 3/2, which is equivalent to 1.5.\n\nTherefore, the answer is C) 1.5." } }, { "answer":"E", "choices":{ "A":"22 minutes and 28 minutes", "B":"28 minutes and 48 minutes", "C":"28 minutes and 58 minutes", "D":"42 minutes and 48 minutes", "E":"42 minutes and 58 minutes" }, "id":10614, "question":"Depending on the cycle, washing a load of clothes takes from 22 to 28 minutes. Drying takes an additional 20 to 30 minutes. What are the minimum and maximum total times to complete a load of laundry?", "explanations":{ "correct":"To find the minimum and maximum total times to complete a load of laundry, we need to consider the minimum and maximum times for both washing and drying.\n\nThe minimum time for washing is 22 minutes, and the minimum time for drying is 20 minutes. Adding these two minimum times together, we get 22 + 20 = 42 minutes.\n\nThe maximum time for washing is 28 minutes, and the maximum time for drying is 30 minutes. Adding these two maximum times together, we get 28 + 30 = 58 minutes.\n\nTherefore, the minimum and maximum total times to complete a load of laundry are 42 minutes and 58 minutes, respectively.\n\nThe answer is E) 42 minutes and 58 minutes." } }, { "answer":"E", "choices":{ "A":"mode < mean < median", "B":"median < mode < mean", "C":"median < mean < mode", "D":"mean < median < mean", "E":"mode < median < mean" }, "id":10615, "question":"\\begin { gather* } 1, 2, 2, 2, 3, 3, 4, 5, 8, 10 \\end { gather* } For the list of 10 numbers above, which of the following must be true?", "explanations":{ "correct":"To determine which statement must be true for the given list of numbers, we need to understand the definitions of mode, mean, and median.\n\nThe mode is the value that appears most frequently in a set of numbers. In this case, the mode is 2 because it appears three times, which is more than any other number in the list.\n\nThe mean is the average of a set of numbers. To find the mean, we add up all the numbers in the list and divide the sum by the total count of numbers. In this case, the sum of the numbers is 1 + 2 + 2 + 2 + 3 + 3 + 4 + 5 + 8 + 10 = 40. Dividing 40 by 10 (the total count of numbers) gives us a mean of 4.\n\nThe median is the middle value in a set of numbers when they are arranged in ascending or descending order. In this case, the numbers are already in ascending order, so the median is the middle number, which is 3.\n\nNow, let's evaluate each statement:\n\nA) mode < mean < median\nThe mode is 2, the mean is 4, and the median is 3. Since 2 < 4 < 3 is not true, this statement is false.\n\nB) median < mode < mean\nThe median is 3, the mode is 2, and the mean is 4. Since 3 < 2 < 4 is not true, this statement is false.\n\nC) median < mean < mode\nThe median is 3, the mean is 4, and the mode is 2. Since 3 < 4 < 2 is not true, this statement is false.\n\nD) mean < median < mean\nThe mean is 4, the median is 3, and the mode is 2. Since 4 < 3 < 4 is not true, this statement is false.\n\nE) mode < median < mean\nThe mode is 2, the median is 3, and the mean is 4. Since 2 < 3 < 4 is true, this statement must be true.\n\nTherefore, the answer is E) mode < median < mean." } }, { "answer":"E", "choices":{ "A":"\\(2 \\frac { 1 } { 3 } \\)", "B":"7", "C":"\\(9 \\frac { 2 } { 3 } \\)", "D":"14", "E":"21" }, "id":10619, "question":"What is nine times y if 4 more than three times y is equal to 11 ?", "explanations":{ "correct":"To solve this problem, we need to set up an equation based on the given information and then solve for the value of y. \n\nThe problem states that \"4 more than three times y is equal to 11.\" We can represent this as an equation: \n\n3y + 4 = 11\n\nTo solve for y, we need to isolate the variable. We can do this by subtracting 4 from both sides of the equation: \n\n3y = 11 - 4\n3y = 7\n\nNext, we divide both sides of the equation by 3 to solve for y: \n\ny = 7/3\n\nNow that we have the value of y, we can find nine times y by multiplying y by 9: \n\n9 * (7/3) = 63/3 = 21\n\nTherefore, the answer is E) 21." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I, II, and III" }, "id":10623, "question":"If k is an integer and \\(k = \\frac { m } { 3 } \\) which of the following must be true? \\(\\newline\\)I. m is an even number. \\(\\newline\\)II. m is a multiple of 3. \\(\\newline\\)III. m is an odd number.", "explanations":{ "correct":"To determine which statements must be true, let's analyze the given equation: \\(k = \\frac{m}{3}\\).\n\nSince \\(k\\) is an integer, \\(m\\) must be divisible by 3. This means that statement II, \"m is a multiple of 3,\" must be true.\n\nNow, let's consider statement I, \"m is an even number.\" To determine if this statement must be true, we need to examine the relationship between \\(m\\) and \\(k\\). Since \\(k\\) is equal to \\(m\\) divided by 3, we can rewrite the equation as \\(m = 3k\\).\n\\(\\newline\\)If \\(m\\) is divisible by 3, then \\(3k\\) is divisible by 3 as well. This means that \\(k\\) must be an integer. However, it does not necessarily mean that \\(m\\) must be an even number. For example, if \\(k\\) is 1, then \\(m\\) would be 3, which is an odd number. Therefore, statement I is not necessarily true.\n\nLastly, let's consider statement III, \"m is an odd number.\" Similar to our analysis for statement I, we can see that \\(m\\) does not have to be an odd number. If \\(k\\) is 1, then \\(m\\) would be 3, which is an odd number. However, if \\(k\\) is 2, then \\(m\\) would be 6, which is an even number. Therefore, statement III is not necessarily true.\n\\(\\newline\\)In conclusion, the only statement that must be true is statement II, \"m is a multiple of 3.\" Therefore, the answer is B) II only." } }, { "answer":"A", "choices":{ "A":"\\(10\\sqrt { 10 } \\)", "B":"\\(15\\sqrt { 15 } \\)", "C":"\\(60\\sqrt { 60 } \\)", "D":"1000", "E":"3375" }, "id":10624, "question":"What is the volume, in cubic inches, of a cube whose surface area is 60 square inches?", "explanations":{ "correct":"To find the volume of a cube, we need to know the length of one side of the cube. However, in this question, we are only given the surface area of the cube, which is 60 square inches. \n\nThe surface area of a cube is given by the formula \\(6s^2\\), where \\(s\\) is the length of one side of the cube. \n\nSo, we can set up the equation \\(6s^2 = 60\\) to find the length of one side of the cube. \n\nDividing both sides of the equation by 6, we get \\(s^2 = 10\\). \n\nTaking the square root of both sides, we find \\(s = \\sqrt{10}\\). \n\nNow that we know the length of one side of the cube, we can find the volume by cubing the length of one side. \n\nSo, the volume of the cube is \\((\\sqrt{10})^3 = 10\\sqrt{10}\\). \n\nTherefore, the answer is A) \\(10\\sqrt{10}\\)." } }, { "answer":"E", "choices":{ "A":"3", "B":"6", "C":"7", "D":"8", "E":"12" }, "id":10627, "question":"Faisel watches television for a total of three hours. During the first hour, he has his choice of three shows. For each of the second and third hours, he can choose from two shows. How many different schedules of shows can Faisel watch?", "explanations":{ "correct":"To find the number of different schedules of shows Faisel can watch, we need to consider the choices he has for each hour.\n\nDuring the first hour, Faisel has 3 shows to choose from.\n\nFor the second hour, he has 2 shows to choose from.\n\nFor the third hour, he also has 2 shows to choose from.\n\nTo find the total number of different schedules, we multiply the number of choices for each hour together.\n\n3 choices for the first hour * 2 choices for the second hour * 2 choices for the third hour = 12 different schedules.\n\nTherefore, the answer is E) 12." } }, { "answer":"C", "choices":{ "A":"4", "B":"5", "C":"6", "D":"7", "E":"8" }, "id":10631, "question":"Theresa is comparing the length of four magazine articles. The first article begins on page 1 and ends on page 7. The second article begins on page 17 and ends on page 24. The third article begins on page 28 and ends on page 31, and the fourth article begins on page 52 and ends on page 56. What is the median number of pages in these articles?", "explanations":{ "correct":"To find the median number of pages in these articles, we need to arrange the number of pages in ascending order. \n\nThe first article has 7 - 1 + 1 = 7 pages.\nThe second article has 24 - 17 + 1 = 8 pages.\nThe third article has 31 - 28 + 1 = 4 pages.\nThe fourth article has 56 - 52 + 1 = 5 pages.\n\nArranging these numbers in ascending order, we have: 4, 5, 7, 8.\n\nSince we have an even number of values, the median is the average of the two middle values. In this case, the two middle values are 5 and 7. Adding them together and dividing by 2, we get (5 + 7) / 2 = 12 / 2 = 6.\n\nTherefore, the median number of pages in these articles is 6.\n\nThe answer is C) 6." } }, { "answer":"A", "choices":{ "A":"\\(3.5d\\)", "B":"\\(365d\\)", "C":"\\(d + 3.5\\)", "D":"\\(\\frac { d } { 3.5 } \\)", "E":"\\(\\frac { 3.5 } { d } \\)" }, "id":10632, "question":"Mr. Barua teaches for 3.5 hours on each day that he is scheduled to teach. If he teaches \\(d\\) days per year, then which of the following is an expression for the total number of hours he teaches per year?", "explanations":{ "correct":"To find the total number of hours Mr. Barua teaches per year, we need to multiply the number of hours he teaches per day by the number of days he teaches in a year.\n\nGiven that Mr. Barua teaches for 3.5 hours on each day, and he teaches \\(d\\) days per year, the expression for the total number of hours he teaches per year is \\(3.5d\\).\n\nTherefore, the answer is A) \\(3.5d\\)." } }, { "answer":"D", "choices":{ "A":"3", "B":"4", "C":"6", "D":"8", "E":"14" }, "id":10634, "question":"A jar contained red and green marbles in a ratio of 3 to 4. After 6 red marbles are added to the jar, the ratio becomes 3 to 2. How many green marbles does the jar now contain?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\n1. Let's assume the initial number of red marbles in the jar is 3x and the initial number of green marbles is 4x, where x is a positive integer.\n\n2. The ratio of red to green marbles is given as 3 to 4, so we have the equation: (3x)/(4x).\n\n3. After adding 6 red marbles, the new number of red marbles becomes 3x + 6.\n\n4. The new ratio of red to green marbles is given as 3 to 2, so we have the equation: (3x + 6)/(4x).\n\n5. We can set up an equation using the given information: (3x + 6)/(4x) = 3/2.\n\n6. Cross-multiplying, we get: 2(3x + 6) = 3(4x).\n\n7. Expanding, we get: 6x + 12 = 12x.\n\n8. Subtracting 6x from both sides, we get: 12 = 6x.\n\n9. Dividing both sides by 6, we get: x = 2.\n\n10. Now, we can find the number of green marbles in the jar by substituting x = 2 into the initial ratio: 4x = 4(2) = 8.\n\nTherefore, the jar now contains 8 green marbles.\n\nThe answer is D) 8." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{50}{5h}\\)", "B":"\\(50 - 5h\\)", "C":"250h", "D":"\\(50 - \\frac{5}{h}\\)", "E":"\\(5h - 50\\)" }, "id":10641, "question":"While biking on a 50-mile path, Jerry averages 5 miles per hour for the first h hours. In terms of h, where \\(h < 10\\), how many miles remain to be traveled?", "explanations":{ "correct":"To find out how many miles remain to be traveled, we need to subtract the distance Jerry has already traveled from the total distance of 50 miles.\n\nJerry's average speed is 5 miles per hour for the first h hours. So, the distance he has traveled in h hours is 5h miles.\n\nTo find the remaining distance, we subtract the distance traveled from the total distance:\nRemaining distance = Total distance - Distance traveled\nRemaining distance = 50 miles - 5h miles\n\nSimplifying the expression, we get:\nRemaining distance = 50 - 5h\n\nTherefore, the answer is B) \\(50 - 5h\\)." } }, { "answer":"A", "choices":{ "A":"\\(2\\sqrt { \\frac { A } { \\pi } } \\)", "B":"\\(\\sqrt { \\frac { A } { \\pi } } \\)", "C":"\\(\\frac { A } { 2\\pi } \\)", "D":"\\(\\frac { A } { \\pi } \\)", "E":"\\(\\frac { 2\\sqrt { A } } { \\pi } \\)" }, "id":10646, "question":"What is the diameter of a circle whose area is A?", "explanations":{ "correct":"To find the diameter of a circle whose area is A, we need to use the formula for the area of a circle, which is given by \\(A = \\pi r^2\\), where A is the area and r is the radius of the circle.\n\nTo find the diameter, we need to find the radius first. We can rearrange the formula for the area to solve for r:\n\n\\(A = \\pi r^2\\)\n\nDividing both sides of the equation by \\(\\pi\\), we get:\n\n\\(\\frac {A}{\\pi} = r^2\\)\n\nTaking the square root of both sides, we get:\n\n\\(\\sqrt{\\frac {A}{\\pi}} = r\\)\n\nNow that we have the radius, we can find the diameter by multiplying the radius by 2:\n\n\\(d = 2r = 2\\sqrt{\\frac {A}{\\pi}}\\)\n\nTherefore, the correct answer is A) \\(2\\sqrt{\\frac {A}{\\pi}}\\).\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"0%", "B":"1%", "C":"2%", "D":"20%", "E":"No change" }, "id":10647, "question":"If the price of a jacket was increased by 10% last week and then decreased 10% this week, what is the overall percent change from the original price?", "explanations":{ "correct":"To find the overall percent change from the original price, we need to calculate the net effect of the two changes: the 10% increase and the 10% decrease.\n\nLet's assume the original price of the jacket is \\$ 100.\n\nStep 1: Increase by 10%\nThe price of the jacket increased by 10%, which means it increased by 10% of \\$ 100, or \\$ 10. The new price after the increase is \\$ 100 + \\$ 10 = \\$ 110.\n\nStep 2: Decrease by 10%\nNow, we need to calculate the decrease of 10% from the increased price of \\$ 110. The decrease is 10% of \\$ 110, which is \\$ 11. The new price after the decrease is \\$ 110 - \\$ 11 = \\$ 99.\n\nStep 3: Calculate the overall percent change\nTo find the overall percent change, we need to compare the new price of \\$ 99 to the original price of \\$ 100. The difference is \\$ 100 - \\$ 99 = \\$ 1.\n\nTo express this difference as a percentage of the original price, we divide the difference by the original price and multiply by 100: ($ 1 / \\$ 100) * 100 = 1%.\n\nTherefore, the overall percent change from the original price is 1%.\n\nThe answer is B) 1%." } }, { "answer":"D", "choices":{ "A":"\\(\\frac{2}{5}\\)", "B":"\\(\\frac{5}{12}\\)", "C":"\\(\\frac{5}{6}\\)", "D":"\\(\\frac{6}{5}\\)", "E":"\\(\\frac{12}{5}\\)" }, "id":10649, "question":"If \\(\\frac{2x}{3} = \\frac{4}{5}\\), then \\(x =\\)", "explanations":{ "correct":"To solve the equation \\(\\frac{2x}{3} = \\frac{4}{5}\\) for \\(x\\), we can cross-multiply. \n\nFirst, we multiply both sides of the equation by 3 to eliminate the fraction in the left-hand side:\n\n\\(\\frac{2x}{3} \\cdot 3 = \\frac{4}{5} \\cdot 3\\)\n\nThis simplifies to:\n\n\\(2x = \\frac{12}{5}\\)\n\nNext, we isolate \\(x\\) by dividing both sides of the equation by 2:\n\n\\(\\frac{2x}{2} = \\frac{12}{5} \\div 2\\)\n\nThis simplifies to:\n\n\\(x = \\frac{6}{5}\\)\n\nTherefore, the answer is D) \\(\\frac{6}{5}\\)." } }, { "answer":"D", "choices":{ "A":"0.01", "B":"0.1", "C":"1", "D":"10", "E":"100" }, "id":10650, "question":"If \\((0.0013)x = 0.013\\), then \\(x =\\)", "explanations":{ "correct":"To solve the equation \\\\((0.0013)x = 0.013\\\\) for \\\\(x\\\\), we need to isolate \\\\(x\\\\) on one side of the equation. \n\nFirst, we can divide both sides of the equation by \\\\(0.0013\\\\) to eliminate the coefficient of \\\\(x\\\\). This gives us:\n\n\\\\(x = \\\\frac{0.013}{0.0013}\\\\)\n\nTo simplify the right side of the equation, we divide \\\\(0.013\\\\) by \\\\(0.0013\\\\):\n\n\\\\(x = 10\\\\)\n\nTherefore, the value of \\\\(x\\\\) is 10.\n\nThe answer is D) 10." } }, { "answer":"C", "choices":{ "A":"-12", "B":"0", "C":"12", "D":"24", "E":"36" }, "id":10655, "question":"If \\((w + 12) - 12 = 12\\), \\(w\\) =", "explanations":{ "correct":"To solve the equation \\((w + 12) - 12 = 12\\), we need to simplify the expression on the left side step-by-step.\n\nFirst, we can simplify the expression inside the parentheses by combining like terms. The \\(+12\\) and \\(-12\\) cancel each other out, leaving us with \\(w\\) on the left side of the equation.\n\nSo, the equation becomes \\(w = 12\\).\n\nTherefore, the value of \\(w\\) that satisfies the equation is 12.\n\nThe answer is C) 12." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { 1 } { 2 } \\)", "B":"\\(\\frac { 1 } { B } \\)", "C":"\\(\\frac { A } { B } \\)", "D":"\\(\\frac { A - B } { B } \\)", "E":"\\(\\frac { B - A } { B } \\)" }, "id":10656, "question":"The volume of pitcher I is A ounces, and the volume of pitcher II is B ounces, with B > A. If pitcher II is full of water and pitcher I is empty, and if just enough water is poured from pitcher II to fill pitcher I, what fraction of pitcher II is now full?", "explanations":{ "correct":"To find the fraction of pitcher II that is now full, we need to determine the amount of water poured from pitcher II to fill pitcher \\(\\newline\\)I.\n\nSince pitcher II is full of water and pitcher I is empty, the initial volume of water in pitcher II is B ounces. After pouring enough water to fill pitcher I, the volume of water remaining in pitcher II is B - A ounces.\n\nTo find the fraction of pitcher II that is now full, we need to compare the remaining volume of water in pitcher II (B - A ounces) to the initial volume of pitcher II (B ounces).\n\nThe fraction can be calculated as (B - A) / B.\n\nTherefore, the answer is E) \\(\\frac { B - A } { B }\\)." } }, { "answer":"B", "choices":{ "A":"-3", "B":"0", "C":"3", "D":"6", "E":"9" }, "id":10664, "question":"If \\(p + q - 2 = 5 \\), what is the value of \\(p + q - 7\\)?", "explanations":{ "correct":"To find the value of \\(p + q - 7\\), we need to substitute the given equation \\(p + q - 2 = 5\\) into the expression.\n\nStep 1: Start with the given equation \\(p + q - 2 = 5\\).\n\nStep 2: Add 7 to both sides of the equation to isolate \\(p + q\\):\n\\(p + q - 2 + 7 = 5 + 7\\).\nThis simplifies to \\(p + q + 5 = 12\\).\n\nStep 3: Subtract 5 from both sides of the equation to solve for \\(p + q\\):\n\\(p + q + 5 - 5 = 12 - 5\\).\nThis simplifies to \\(p + q = 7\\).\n\nTherefore, the value of \\(p + q - 7\\) is equal to 7 minus 7, which is 0.\n\nThe answer is B) 0." } }, { "answer":"A", "choices":{ "A":"200", "B":"250", "C":"300", "D":"350", "E":"400" }, "id":10671, "question":"According to the function \\(p(n) = \\frac { 2 } { 3 } n + 100\\), if \\(n\\) increases by 300, by how much does \\(p\\) increase?", "explanations":{ "correct":"To find out how much the function \\(p\\) increases when \\(n\\) increases by 300, we need to evaluate the function for \\(n+300\\) and subtract the original value of \\(p(n)\\).\n\nGiven the function \\(p(n) = \\frac{2}{3}n + 100\\), we can substitute \\(n+300\\) into the function to find \\(p(n+300)\\):\n\n\\(p(n+300) = \\frac{2}{3}(n+300) + 100\\)\n\nExpanding and simplifying:\n\n\\(p(n+300) = \\frac{2}{3}n + 200 + 100\\)\n\n\\(p(n+300) = \\frac{2}{3}n + 300\\)\n\nNow, we subtract the original value of \\(p(n)\\) from \\(p(n+300)\\) to find the increase:\n\n\\(p(n+300) - p(n) = \\left(\\frac{2}{3}n + 300\\right) - \\left(\\frac{2}{3}n + 100\\right)\\)\n\nSimplifying further:\n\n\\(p(n+300) - p(n) = \\frac{2}{3}n + 300 - \\frac{2}{3}n - 100\\)\n\nThe \\(n\\) terms cancel out, leaving us with:\n\n\\(p(n+300) - p(n) = 300 - 100\\)\n\n\\(p(n+300) - p(n) = 200\\)\n\nTherefore, when \\(n\\) increases by 300, \\(p\\) increases by 200.\n\nThe answer is A) 200." } }, { "answer":"E", "choices":{ "A":"\\((0, 3)\\)", "B":"\\((5, 5)\\)", "C":"\\((0, 0)\\)", "D":"\\((3, -2)\\)", "E":"\\((-2, 5)\\)" }, "id":10673, "question":"Line l has an undefined slope and contains the point \\((-2,3)\\). Which of the following points is also on line l?", "explanations":{ "correct":"To determine which of the given points is also on line l, we need to consider the characteristics of a line with an undefined slope. \n\nWhen a line has an undefined slope, it means that the line is vertical. A vertical line passes through all points with the same x-coordinate. In this case, since the line passes through the point (-2,3), any point with x-coordinate -2 will also be on line l.\n\nLet's check each of the given points:\n\nA) (0, 3): The x-coordinate is 0, not -2. Therefore, this point is not on line l.\n\nB) (5, 5): The x-coordinate is 5, not -2. Therefore, this point is not on line l.\n\nC) (0, 0): The x-coordinate is 0, not -2. Therefore, this point is not on line l.\n\nD) (3, -2): The x-coordinate is 3, not -2. Therefore, this point is not on line l.\n\nE) (-2, 5): The x-coordinate is -2, which matches the x-coordinate of the given point (-2,3). Therefore, this point is on line l.\n\nBased on the reasoning above, the answer is E." } }, { "answer":"B", "choices":{ "A":"0", "B":"1", "C":"2", "D":"3", "E":"more than 3" }, "id":10675, "question":"For how many positive numbers \\(a\\) is it true that \\(a \\times a \\times a = a + a + a\\) ?", "explanations":{ "correct":"To solve this problem, we need to find the values of \\\\(a\\\\) that satisfy the equation \\\\(a \\\\times a \\\\times a = a + a + a\\\\).\n\nFirst, we simplify the equation by combining like terms on the right side:\n\\\\(a \\\\times a \\\\times a = 3a\\\\).\n\nNext, we can divide both sides of the equation by \\\\(a\\\\) to isolate the variable:\n\\\\(a \\\\times a = 3\\\\).\n\nTaking the square root of both sides, we get:\n\\\\(a = \\sqrt{3}\\\\).\n\nHowever, we need to find positive values of \\\\(a\\\\). The square root of 3 is approximately 1.732, which is positive. Therefore, there is only one positive value of \\\\(a\\\\) that satisfies the equation.\n\nThe answer is B) 1." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 1 } { x } \\)", "B":"\\(\\frac { x + 1 } { x - 1 } \\)", "C":"\\(\\frac { x + 1 } { x } \\)", "D":"\\(\\frac { x - 1 } { x + 1 } \\)", "E":"\\(\\frac { x - 1 } { x } \\)" }, "id":10676, "question":"If \\(x \\neq 0\\), 1, which of the following is equivalent to \\(\\frac { x^2 - 1 } { x^2 - x } \\)?", "explanations":{ "correct":"To simplify the expression \\(\\frac { x^2 - 1 } { x^2 - x }\\), we can factor the numerator and denominator.\n\nThe numerator \\(x^2 - 1\\) can be factored as \\((x + 1)(x - 1)\\).\n\nThe denominator \\(x^2 - x\\) can be factored as \\(x(x - 1)\\).\n\nNow, we can cancel out the common factor of \\(x - 1\\) in the numerator and denominator.\n\n\\(\\frac { x^2 - 1 } { x^2 - x } = \\frac { (x + 1)(x - 1) } { x(x - 1) }\\)\n\nAfter canceling out the common factor of \\(x - 1\\), we are left with:\n\n\\(\\frac { x + 1 } { x }\\)\n\nTherefore, the answer is C) \\(\\frac { x + 1 } { x }\\)." } }, { "answer":"C", "choices":{ "A":"6", "B":"12", "C":"34", "D":"38", "E":"40" }, "id":10678, "question":"If \\(3x + 1 = 19\\), then \\(6x - 2 \\)=", "explanations":{ "correct":"To find the value of \\(6x - 2\\), we first need to solve the equation \\(3x + 1 = 19\\) for \\(x\\).\n\nStep 1: Subtract 1 from both sides of the equation:\n\\(3x + 1 - 1 = 19 - 1\\)\n\\(3x = 18\\)\n\nStep 2: Divide both sides of the equation by 3 to isolate \\(x\\):\n\\(\\frac{3x}{3} = \\frac{18}{3}\\)\n\\(x = 6\\)\n\nNow that we have found the value of \\(x\\), we can substitute it into the expression \\(6x - 2\\) to find the final answer.\n\n\\(6x - 2 = 6(6) - 2\\)\n\\(6x - 2 = 36 - 2\\)\n\\(6x - 2 = 34\\)\n\nTherefore, the answer is C) 34." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 4\\sqrt { 3 } } { 9 } \\)", "B":"\\(\\frac { 4 } { 9 } \\)", "C":"\\(\\frac { 3 } { 4 } \\)", "D":"\\(\\frac { 1 } { 1 } \\)", "E":"\\(\\frac { 4 } { 3 } \\)" }, "id":10680, "question":"If a square and an equilateral triangle have equal perimeters, what is the ratio of the area of the triangle to the area of the square?", "explanations":{ "correct":"Let's assume that the side length of the square is \\(s\\) and the side length of the equilateral triangle is \\(t\\).\n\nThe perimeter of the square is equal to 4 times the side length, so it is \\(4s\\).\nThe perimeter of the equilateral triangle is equal to 3 times the side length, so it is \\(3t\\).\n\nSince the perimeters are equal, we can set up the equation \\(4s = 3t\\) and solve for \\(t\\):\n\\[t = \\frac{4s}{3}\\]\n\nThe area of a square is given by \\(A_{\\text{square}} = s^2\\).\nThe area of an equilateral triangle is given by \\(A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4}t^2\\).\n\nSubstituting the value of \\(t\\) from the perimeter equation into the area equation for the triangle, we get:\n\\[A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4}\\left(\\frac{4s}{3}\\right)^2 = \\frac{\\sqrt{3}}{4}\\left(\\frac{16s^2}{9}\\right) = \\frac{4\\sqrt{3}}{9}s^2\\]\n\nNow, we can find the ratio of the area of the triangle to the area of the square:\n\\[\\frac{A_{\\text{triangle}}}{A_{\\text{square}}} = \\frac{\\frac{4\\sqrt{3}}{9}s^2}{s^2} = \\frac{4\\sqrt{3}}{9}\\]\n\nTherefore, the ratio of the area of the triangle to the area of the square is \\(\\frac{4\\sqrt{3}}{9}\\).\n\nThe answer is A) \\(\\frac{4\\sqrt{3}}{9}\\)." } }, { "answer":"E", "choices":{ "A":"4", "B":"6", "C":"8", "D":"10", "E":"12" }, "id":10683, "question":"Two farmers, working together, can plow a field in 3 days. One farmer working alone can plow the field in 4 days. How many days will it take the second farmer, working alone, to plow the entire field?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume that the field is 1 unit in size.\n\\(\\newline\\)If two farmers can plow the field in 3 days, it means that their combined plowing rate is 1/3 of the field per day. \n\\(\\newline\\)If one farmer can plow the field in 4 days, it means that his plowing rate is 1/4 of the field per day.\n\nLet's denote the plowing rate of the second farmer as x (in units per day).\n\nSince the two farmers are working together, their combined plowing rate is the sum of their individual rates. So, we have the equation:\n\n1/3 + 1/x = 1/4\n\nTo solve this equation, we need to find a common denominator. The common denominator is 12x. Multiplying both sides of the equation by 12x, we get:\n\n4x + 12 = 3x\n\nSubtracting 3x from both sides, we get:\n\nx = 12\n\nTherefore, the second farmer's plowing rate is 1/12 of the field per day.\n\nTo find out how many days it will take the second farmer to plow the entire field, we can use the formula:\n\nTime = Work / Rate\n\nSince the field is 1 unit in size, the work is 1 unit. Plugging in the second farmer's plowing rate of 1/12, we get:\n\nTime = 1 / (1/12) = 12\n\nSo, it will take the second farmer 12 days to plow the entire field.\n\nThe answer is E) 12." } }, { "answer":"B", "choices":{ "A":"\\(m - \\sqrt{n} = (m + 2)^2\\)", "B":"\\(m - n^2 = \\sqrt{m + n}\\)", "C":"\\(m - n^2 = \\sqrt{m} + \\sqrt{n}\\)", "D":"\\(m - \\sqrt{n} = m^2 + n ^2\\)", "E":"\\(m^2 - n^2 = \\sqrt{m + n}\\)" }, "id":10685, "question":"The difference between m and the square of n is equal to the square root of the sum of m and n. Which of the following expressions is the same as the statement above?", "explanations":{ "correct":"To solve this problem, we need to carefully analyze the given statement and compare it with each of the answer choices.\n\nThe given statement is: \"The difference between m and the square of n is equal to the square root of the sum of m and n.\"\n\nLet's break down the given statement into an equation:\n\n\\(m - n^2 = \\sqrt{m + n}\\)\n\nNow, let's compare this equation with each of the answer choices:\n\nA) \\(m - \\sqrt{n} = (m + 2)^2\\)\nThis answer choice does not match the given equation. It has a different expression on the right side of the equation.\n\nB) \\(m - n^2 = \\sqrt{m + n}\\)\nThis answer choice matches the given equation exactly. The left side of the equation is \\(m - n^2\\), which is the difference between m and the square of n. The right side of the equation is \\(\\sqrt{m + n}\\), which is the square root of the sum of m and n. Therefore, this answer choice is correct.\n\nC) \\(m - n^2 = \\sqrt{m} + \\sqrt{n}\\)\nThis answer choice does not match the given equation. It has the sum of square roots on the right side of the equation, instead of just a single square root.\n\nD) \\(m - \\sqrt{n} = m^2 + n ^2\\)\nThis answer choice does not match the given equation. It has a different expression on the right side of the equation.\n\nE) \\(m^2 - n^2 = \\sqrt{m + n}\\)\nThis answer choice does not match the given equation. It has the difference of squares on the left side of the equation, instead of just a single variable.\n\nAfter carefully analyzing each answer choice, we can conclude that the correct answer is B) \\(m - n^2 = \\sqrt{m + n}\\). \n\nThe answer is B." } }, { "answer":"E", "choices":{ "A":"\\(ab\\)", "B":"\\(\\frac { a + b } { 2 } \\)", "C":"\\(\\frac { a - b } { 2 } \\)", "D":"\\(\\frac { a^2 - b^2 } { 4 } \\)", "E":"\\(\\frac { a^2 - b^2 } { 8 } \\)" }, "id":10686, "question":"If \\(x + 2y = a\\) and \\(x - 2y = b\\), which of the following is an expression for \\(xy\\)?", "explanations":{ "correct":"To find an expression for \\(xy\\), we need to eliminate one of the variables, either \\(x\\) or \\(y\\), from the given equations.\n\nLet's add the two equations together:\n\\((x + 2y) + (x - 2y) = a + b\\)\n\\(2x = a + b\\)\n\\(x = \\frac{a + b}{2}\\)\n\nNow, substitute this value of \\(x\\) into one of the original equations. Let's use the first equation:\n\\(\\frac{a + b}{2} + 2y = a\\)\n\\(2y = a - \\frac{a + b}{2}\\)\n\\(2y = \\frac{2a - a - b}{2}\\)\n\\(2y = \\frac{a - b}{2}\\)\n\\(y = \\frac{a - b}{4}\\)\n\nFinally, we can find the expression for \\(xy\\) by multiplying \\(x\\) and \\(y\\):\n\\(xy = \\frac{a + b}{2} \\cdot \\frac{a - b}{4}\\)\n\\(xy = \\frac{(a + b)(a - b)}{8}\\)\n\\(xy = \\frac{a^2 - b^2}{8}\\)\n\nTherefore, the answer is E) \\(\\frac{a^2 - b^2}{8}\\)." } }, { "answer":"E", "choices":{ "A":"202 degrees", "B":"250 degrees", "C":"350 degrees", "D":"363 degrees", "E":"405 degrees" }, "id":10689, "question":"The earth makes one complete rotation about its axis every 24 hours. Assuming it rotates at a constant rate, through how many degrees would Goannaville, Australia rotate from 1:00 PM on January 2 to 4:00 PM on January 3?", "explanations":{ "correct":"To find the number of degrees that Goannaville, Australia rotates from 1:00 PM on January 2 to 4:00 PM on January 3, we need to calculate the number of hours that have passed and then convert it to degrees.\n\nFrom 1:00 PM on January 2 to 1:00 PM on January 3, 24 hours have passed. From 1:00 PM to 4:00 PM on January 3, an additional 3 hours have passed. Therefore, a total of 27 hours have passed.\n\nSince the earth makes one complete rotation every 24 hours, we can calculate the number of degrees it rotates per hour. \n\nTo find this, we divide the total number of degrees in a full rotation (360 degrees) by the number of hours in a full rotation (24 hours):\n\n360 degrees / 24 hours = 15 degrees per hour\n\nNow, we can calculate the number of degrees that Goannaville, Australia rotates by multiplying the number of hours (27) by the number of degrees per hour (15):\n\n27 hours * 15 degrees per hour = 405 degrees\n\nTherefore, the answer is E) 405 degrees." } }, { "answer":"C", "choices":{ "A":"4", "B":"12", "C":"20", "D":"64", "E":"68" }, "id":10690, "question":"If \\(\\frac { 5^a } { 5^4 } = 5^16\\)", "explanations":{ "correct":"To solve this problem, we can use the property of exponents that states: \\(\\frac{a^m}{a^n} = a^{m-n}\\).\n\nGiven that \\(\\frac{5^a}{5^4} = 5^{16}\\), we can rewrite the left side of the equation using the exponent property:\n\n\\(5^a \\cdot 5^{-4} = 5^{16}\\).\n\nNow, we can simplify the equation by combining the exponents on the left side:\n\n\\(5^{a-4} = 5^{16}\\).\n\nSince the bases are the same, the exponents must be equal:\n\n\\(a-4 = 16\\).\n\nTo solve for \\(a\\), we can add 4 to both sides of the equation:\n\n\\(a = 16 + 4 = 20\\).\n\nTherefore, the answer is C) 20." } }, { "answer":"D", "choices":{ "A":"\\(x < 5\\)", "B":"\\(x > 5\\)", "C":"\\(-5 < x < 5\\)", "D":"\\(x \\geq 5\\) or \\(x \\leq -5\\)", "E":"\\(-5 \\leq x \\leq 5\\)" }, "id":10694, "question":"Let the function \\(f\\) be defined as \\(f(x) = \\sqrt { (x - 5)(x + 5) } \\). Which of the following could represent the domain of the function \\(f\\)?", "explanations":{ "correct":"To determine the domain of the function \\(f(x) = \\sqrt{(x - 5)(x + 5)}\\), we need to consider the values of \\(x\\) that make the expression inside the square root non-negative.\n\nThe expression inside the square root, \\((x - 5)(x + 5)\\), represents the product of two factors. For the product to be non-negative, either both factors must be non-negative or both factors must be non-positive.\n\nLet's consider each option:\n\nA) \\(x < 5\\): If \\(x\\) is less than 5, then both \\((x - 5)\\) and \\((x + 5)\\) are negative. The product of two negative numbers is positive, so this option does not satisfy the condition.\n\nB) \\(x > 5\\): If \\(x\\) is greater than 5, then both \\((x - 5)\\) and \\((x + 5)\\) are positive. The product of two positive numbers is positive, so this option satisfies the condition.\n\nC) \\(-5 < x < 5\\): If \\(x\\) is between -5 and 5, then \\((x - 5)\\) is negative and \\((x + 5)\\) is positive. The product of a negative number and a positive number is negative, so this option does not satisfy the condition.\n\nD) \\(x \\geq 5\\) or \\(x \\leq -5\\): If \\(x\\) is greater than or equal to 5 or less than or equal to -5, then either \\((x - 5)\\) and \\((x + 5)\\) are both non-negative or both non-positive. The product of two non-negative numbers is non-negative, and the product of two non-positive numbers is non-negative, so this option satisfies the condition.\n\nE) \\(-5 \\leq x \\leq 5\\): If \\(x\\) is between -5 and 5 (inclusive), then \\((x - 5)\\) is negative and \\((x + 5)\\) is positive. The product of a negative number and a positive number is negative, so this option does not satisfy the condition.\n\nBased on the reasoning above, the correct answer is D) \\(x \\geq 5\\) or \\(x \\leq -5\\)." } }, { "answer":"B", "choices":{ "A":"200", "B":"250", "C":"400", "D":"500", "E":"600" }, "id":10695, "question":"Steve took a bike trip in which he covered half the total distance on Monday. After going 100 kilometers on Tuesday, he determined that he still had 10% of the trip to complete. What was the total length, in kilometers, of the trip?", "explanations":{ "correct":"Let's solve this problem step-by-step.\n\nLet's assume the total length of the trip is \"x\" kilometers.\n\nAccording to the problem, Steve covered half the total distance on Monday. So, he traveled x/2 kilometers on Monday.\n\nOn Tuesday, he traveled an additional 100 kilometers. Therefore, the total distance covered by Steve after Tuesday is x/2 + 100 kilometers.\n\nThe problem states that after Tuesday, Steve determined that he still had 10% of the trip to complete. This means that the distance he covered after Tuesday is 90% of the total trip distance.\n\nWe can set up the following equation to represent this information:\n\nx/2 + 100 = 0.9x\n\nTo solve for x, we can multiply both sides of the equation by 2 to eliminate the fraction:\n\n2(x/2 + 100) = 2(0.9x)\nx + 200 = 1.8x\n\nNext, we can subtract x from both sides of the equation:\n\n200 = 1.8x - x\n200 = 0.8x\n\nTo isolate x, we can divide both sides of the equation by 0.8:\n\n200/0.8 = x\n250 = x\n\nTherefore, the total length of the trip is 250 kilometers.\n\nThe answer is B) 250." } }, { "answer":"D", "choices":{ "A":"4", "B":"4y", "C":"\\(4y^2\\)", "D":"\\(y^4\\)", "E":"\\(y^5\\)" }, "id":10697, "question":"If \\(y = 2^{\\frac{3}{4}}\\), which of the following expressions is equal to \\(2^3\\)?", "explanations":{ "correct":"To find the value of \\(2^3\\), we need to simplify the expression \\(2^3\\).\n\nWe know that \\(2^3\\) means multiplying 2 by itself 3 times. So, \\(2^3 = 2 \\times 2 \\times 2\\), which equals 8.\n\nNow, let's evaluate the given options:\n\nA) 4: This is not equal to 8, so it is not the correct answer.\n\nB) 4y: We are given that \\(y = 2^{\\frac{3}{4}}\\), so substituting this value, we have \\(4y = 4 \\times 2^{\\frac{3}{4}}\\). This is not equal to 8, so it is not the correct answer.\n\nC) \\(4y^2\\): Substituting the value of \\(y = 2^{\\frac{3}{4}}\\), we have \\(4y^2 = 4 \\times (2^{\\frac{3}{4}})^2\\). Simplifying this expression, we get \\(4 \\times 2^{\\frac{6}{4}} = 4 \\times 2^{\\frac{3}{2}}\\). This is not equal to 8, so it is not the correct answer.\n\nD) \\(y^4\\): Substituting the value of \\(y = 2^{\\frac{3}{4}}\\), we have \\(y^4 = (2^{\\frac{3}{4}})^4\\). Simplifying this expression, we get \\(2^3\\), which we know is equal to 8. Therefore, this is the correct answer.\n\nE) \\(y^5\\): Substituting the value of \\(y = 2^{\\frac{3}{4}}\\), we have \\(y^5 = (2^{\\frac{3}{4}})^5\\). Simplifying this expression, we get \\(2^{\\frac{15}{4}}\\), which is not equal to 8. So, it is not the correct answer.\n\nAfter evaluating all the options, we can conclude that the correct answer is D) \\(y^4\\).\n\nThe answer is D)." } }, { "answer":"A", "choices":{ "A":"3", "B":"4", "C":"5", "D":"6", "E":"8" }, "id":10699, "question":"If \\(m^a \\times m^b = m^5\\) and \\((m^2a)^b = m^12\\), which of the following could be a value of \\(a\\) ?", "explanations":{ "correct":"To solve this problem, we will use the properties of exponents. \n\nFirst, let's simplify the equation \\(m^a \\times m^b = m^5\\). According to the product of powers property, when multiplying two powers with the same base, we add their exponents. Therefore, we can rewrite the equation as \\(m^{a+b} = m^5\\).\n\nNext, let's simplify the equation \\((m^{2a})^b = m^{12}\\). According to the power of a power property, when raising a power to another power, we multiply the exponents. Therefore, we can rewrite the equation as \\(m^{2ab} = m^{12}\\).\n\nNow, we have two equations:\n1) \\(m^{a+b} = m^5\\)\n2) \\(m^{2ab} = m^{12}\\)\n\nSince the bases are the same in both equations, we can equate the exponents:\n1) \\(a+b = 5\\)\n2) \\(2ab = 12\\)\n\nTo find the possible value of \\(a\\), we need to solve these equations simultaneously.\n\nFrom equation 1), we can isolate \\(b\\) by subtracting \\(a\\) from both sides: \n\\(b = 5 - a\\)\n\nSubstituting this value of \\(b\\) into equation 2), we get:\n\\(2a(5-a) = 12\\)\n\nExpanding and rearranging the equation, we have:\n\\(10a - 2a^2 = 12\\)\n\nSimplifying further, we get:\n\\(2a^2 - 10a + 12 = 0\\)\n\nFactoring the quadratic equation, we have:\n\\((2a - 4)(a - 3) = 0\\)\n\nSetting each factor equal to zero, we find two possible values for \\(a\\):\n\\(2a - 4 = 0\\) or \\(a - 3 = 0\\)\n\nSolving these equations, we get:\n\\(2a = 4\\) or \\(a = 3\\)\n\nTherefore, the possible value of \\(a\\) is 3. \n\nThe answer is A) 3." } }, { "answer":"B", "choices":{ "A":"0", "B":"2", "C":"3", "D":"5", "E":"6" }, "id":10702, "question":"For a linear function \\(f\\), \\(f(1) = 8\\) and \\(f(7) = -10\\). If \\(f(k) = 5\\), what is the value of \\(k\\)?", "explanations":{ "correct":"To find the value of \\(k\\) when \\(f(k) = 5\\), we need to determine the equation of the linear function \\(f\\).\n\nWe are given that \\(f(1) = 8\\) and \\(f(7) = -10\\). \n\nUsing the two points \\((1, 8)\\) and \\((7, -10)\\), we can find the slope of the line:\n\n\\[\n\\text{{Slope}} = \\frac{{\\text{{change in }} y}}{{\\text{{change in }} x}} = \\frac{{-10 - 8}}{{7 - 1}} = \\frac{{-18}}{{6}} = -3\n\\]\n\nNow that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line:\n\n\\[\ny - y_1 = m(x - x_1)\n\\]\n\nwhere \\((x_1, y_1)\\) is any point on the line and \\(m\\) is the slope.\n\nUsing the point \\((1, 8)\\), we have:\n\n\\[\ny - 8 = -3(x - 1)\n\\]\n\nSimplifying the equation:\n\n\\[\ny - 8 = -3x + 3\n\\]\n\n\\[\ny = -3x + 11\n\\]\n\nNow, we can substitute \\(f(k) = 5\\) into the equation to find the value of \\(k\\):\n\n\\[\n5 = -3k + 11\n\\]\n\nSolving for \\(k\\):\n\n\\[\n-3k = 5 - 11\n\\]\n\n\\[\n-3k = -6\n\\]\n\n\\[\nk = \\frac{{-6}}{{-3}} = 2\n\\]\n\nTherefore, the value of \\(k\\) is 2.\n\nThe answer is B." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 1 } { 2 } \\)", "B":"4", "C":"8", "D":"12", "E":"16" }, "id":10704, "question":"The average (arithmetic mean) of the numbers in set A is 8. If a second set of numbers, B, is created by dividing each of the numbers in set A by two, then what is the average of the numbers in set B?", "explanations":{ "correct":"To find the average of the numbers in set B, we need to divide each number in set A by 2 and then calculate the average.\n\nGiven that the average of the numbers in set A is 8, we know that the sum of all the numbers in set A is \\(8 \\times \\text{{number of elements in set A}}\\).\n\nLet's assume that set A has n elements. Therefore, the sum of the numbers in set A is \\(8n\\).\n\nNow, when we divide each number in set A by 2 to create set B, the sum of the numbers in set B will be \\(\\frac{{8n}}{2} = 4n\\).\n\nSince set B has the same number of elements as set A, the average of the numbers in set B is \\(\\frac{{4n}}{{\\text{{number of elements in set B}}}} = \\frac{{4n}}{{n}} = 4\\).\n\nTherefore, the answer is B) 4." } }, { "answer":"D", "choices":{ "A":"16", "B":"22", "C":"23", "D":"32", "E":"36" }, "id":10710, "question":"The percent increase from 6 to 16 is equal to the percent increase from 12 to what number?", "explanations":{ "correct":"To find the percent increase from 6 to 16, we can use the formula:\n\nPercent Increase = (New Value - Original Value) / Original Value * 100\n\nLet's calculate the percent increase from 6 to 16:\n\nPercent Increase = (16 - 6) / 6 * 100\nPercent Increase = 10 / 6 * 100\nPercent Increase = 1.6667 * 100\nPercent Increase = 166.67%\n\nNow, we need to find the number that has the same percent increase when going from 12 to that number.\n\nLet's set up the equation:\n\n166.67% = (New Value - 12) / 12 * 100\n\nTo solve for the new value, we can cross multiply:\n\n166.67 * 12 = New Value - 12\n\n2000.04 = New Value - 12\n\nNew Value = 2000.04 + 12\nNew Value = 2012.04\n\nTherefore, the number that has the same percent increase from 12 to that number as the percent increase from 6 to 16 is 2012.04.\n\nThe answer is D) 32." } }, { "answer":"C", "choices":{ "A":"x = 1 only", "B":"x = -1 only", "C":"x = 0 or x = 1", "D":"x = 1 or x = -1", "E":"x = O or x = -1" }, "id":10715, "question":"For what value of \\(x\\) does \\((x-1)^3 = x^3 - 1\\) ?", "explanations":{ "correct":"To find the value of \\(x\\) that satisfies the equation \\((x-1)^3 = x^3 - 1\\), we need to solve the equation step-by-step.\n\nFirst, let's expand both sides of the equation using the binomial theorem:\n\\((x-1)^3 = x^3 - 1\\)\n\\((x-1)(x-1)(x-1) = x^3 - 1\\)\n\\((x^2 - 2x + 1)(x-1) = x^3 - 1\\)\n\\(x^3 - 3x^2 + 3x - 1 = x^3 - 1\\)\n\nNext, let's simplify the equation by canceling out the \\(x^3\\) terms:\n\\(-3x^2 + 3x - 1 = -1\\)\n\nNow, let's simplify further by adding 1 to both sides of the equation:\n\\(-3x^2 + 3x = 0\\)\n\nWe can factor out \\(x\\) from the left side of the equation:\n\\(x(-3x + 3) = 0\\)\n\nNow, we have two possibilities for the equation to be true:\n1) \\(x = 0\\)\n2) \\(-3x + 3 = 0\\)\n\nFor the second possibility, let's solve for \\(x\\):\n\\(-3x + 3 = 0\\)\n\\(-3x = -3\\)\n\\(x = 1\\)\n\nTherefore, the values of \\(x\\) that satisfy the equation are \\(x = 0\\) and \\(x = 1\\).\n\nThe answer is C) \\(x = 0\\) or \\(x = 1\\)." } }, { "answer":"C", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I and III only" }, "id":10717, "question":"If \\(\\frac { m + m } { m \\times m } = \\frac { 1 } { 5 } \\), which of the following must be true? \\(\\newline\\)I. m = 0 \\(\\newline\\)II. m = 5 \\(\\newline\\)III. m = 10", "explanations":{ "correct":"To solve this problem, we need to simplify the given equation and determine the possible values of \\(m\\) that satisfy the equation.\n\nWe are given the equation \\(\\frac{m + m}{m \\times m} = \\frac{1}{5}\\).\n\nFirst, we simplify the left side of the equation by combining like terms in the numerator: \\(\\frac{2m}{m \\times m} = \\frac{1}{5}\\).\n\nNext, we simplify further by canceling out one \\(m\\) in the numerator and denominator: \\(\\frac{2}{m} = \\frac{1}{5}\\).\n\nTo solve for \\(m\\), we can cross-multiply: \\(2 \\times 5 = m \\times 1\\).\n\nThis simplifies to \\(10 = m\\).\n\nTherefore, the only value of \\(m\\) that satisfies the equation is \\(m = 10\\).\n\nSince only statement III, \"m = 10,\" is true, the answer is C) III only." } }, { "answer":"B", "choices":{ "A":"8", "B":"9", "C":"10", "D":"11", "E":"12" }, "id":10719, "question":"A right circular cylinder vase with an inner base radius of 6 inches is filled with water to a height of 20 inches. After a smaller solid right circular cylinder with base radius of 4 inches is then completely submerged in the water, the height of water is 24 inches. What is the height, in inches, of smaller cylinder?", "explanations":{ "correct":"To solve this problem, we can use the concept of volume. \n\nThe volume of the water in the vase before the smaller cylinder is submerged is equal to the volume of the vase itself. The volume of a right circular cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.\n\nThe volume of the water in the vase before the smaller cylinder is submerged is π(6^2)(20) = 720π cubic inches.\n\nAfter the smaller cylinder is submerged, the volume of the water in the vase increases to π(6^2)(24) = 864π cubic inches.\n\nThe volume of the smaller cylinder is given by π(4^2)(h), where h is the height of the smaller cylinder.\n\nTo find the height of the smaller cylinder, we can set up the equation:\n\n720π + π(4^2)(h) = 864π\n\nSimplifying the equation, we have:\n\n720π + 16πh = 864π\n\nDividing both sides of the equation by π, we get:\n\n720 + 16h = 864\n\nSubtracting 720 from both sides of the equation, we have:\n\n16h = 144\n\nDividing both sides of the equation by 16, we get:\n\nh = 9\n\nTherefore, the height of the smaller cylinder is 9 inches.\n\nThe answer is B." } }, { "answer":"B", "choices":{ "A":"4.4", "B":"11", "C":"22", "D":"33", "E":"44" }, "id":10721, "question":"It takes 5 complete turns of the crank to raise a fishing rod hook 2 feet. At this rate, how many turns will it take to raise the hook 4.4 feet?", "explanations":{ "correct":"To solve this problem, we need to find the number of turns it will take to raise the hook 4.4 feet, given that it takes 5 complete turns to raise the hook 2 feet.\n\nFirst, we can find the number of turns it takes to raise the hook 1 foot. Since it takes 5 turns to raise the hook 2 feet, it will take half as many turns to raise the hook 1 foot. Therefore, it takes 2.5 turns to raise the hook 1 foot.\n\nNext, we can find the number of turns it takes to raise the hook 4.4 feet. To do this, we can multiply the number of turns it takes to raise the hook 1 foot by 4.4. \n\n2.5 turns/foot * 4.4 feet = 11 turns\n\nTherefore, it will take 11 turns to raise the hook 4.4 feet.\n\nThe answer is B) 11." } }, { "answer":"C", "choices":{ "A":"3", "B":"4", "C":"5", "D":"6", "E":"7" }, "id":10722, "question":"A sequence of numbers begins 1, 1, 1, 2, 2, 3 and then repeats this pattern of six numbers forever. What is the sum of the 135th, 136th, and 137th numbers in the sequence?", "explanations":{ "correct":"To find the sum of the 135th, 136th, and 137th numbers in the sequence, we need to determine the pattern of the sequence and then calculate the values for those positions.\n\nThe given sequence begins with three 1's, followed by two 2's, and then a 3. This pattern repeats every six numbers. \n\nTo find the position of the numbers we are interested in, we can divide 135, 136, and 137 by 6 and observe the remainders.\n\n135 divided by 6 gives a quotient of 22 and a remainder of 3. This means that the 135th number is the same as the 3rd number in the pattern, which is 1.\n\n136 divided by 6 gives a quotient of 22 and a remainder of 4. This means that the 136th number is the same as the 4th number in the pattern, which is 2.\n\n137 divided by 6 gives a quotient of 22 and a remainder of 5. This means that the 137th number is the same as the 5th number in the pattern, which is 2.\n\nNow, we can calculate the sum of these three numbers: 1 + 2 + 2 = 5.\n\nTherefore, the sum of the 135th, 136th, and 137th numbers in the sequence is 5.\n\nThe answer is C) 5." } }, { "answer":"B", "choices":{ "A":"0", "B":"9", "C":"12", "D":"16", "E":"18" }, "id":10732, "question":"If the sum of three numbers is 27, what is the average (arithmetic mean) of the three numbers?", "explanations":{ "correct":"To find the average (arithmetic mean) of three numbers, we need to divide the sum of the three numbers by 3.\n\nGiven that the sum of the three numbers is 27, we can divide 27 by 3 to find the average.\n\n27 ÷ 3 = 9\n\nTherefore, the average of the three numbers is 9.\n\nThe answer is B) 9." } }, { "answer":"C", "choices":{ "A":"8.5", "B":"5", "C":"4.5", "D":"4", "E":"3" }, "id":10735, "question":"\\(h(x) = \\frac { 3 } { 2 } x + 1\\) \\(f(x) = \\frac { 4 } { 3 } x\\) The functions h and f are defined above. What is the value of \\(h(5) - f(3)\\)?", "explanations":{ "correct":"To find the value of \\(h(5) - f(3)\\), we need to substitute the values of \\(x\\) into the given functions and then subtract the results.\n\nFirst, let's find \\(h(5)\\). We substitute \\(x = 5\\) into the function \\(h(x) = \\frac{3}{2}x + 1\\):\n\n\\[h(5) = \\frac{3}{2}(5) + 1 = \\frac{15}{2} + 1 = \\frac{15}{2} + \\frac{2}{2} = \\frac{17}{2}\\]\n\nNext, let's find \\(f(3)\\). We substitute \\(x = 3\\) into the function \\(f(x) = \\frac{4}{3}x\\):\n\n\\[f(3) = \\frac{4}{3}(3) = \\frac{12}{3} = 4\\]\n\nNow, we can subtract \\(f(3)\\) from \\(h(5)\\):\n\n\\[h(5) - f(3) = \\frac{17}{2} - 4 = \\frac{17}{2} - \\frac{8}{2} = \\frac{9}{2} = 4.5\\]\n\nTherefore, the value of \\(h(5) - f(3)\\) is 4.5.\n\nThe answer is C) 4.5." } }, { "answer":"E", "choices":{ "A":"5", "B":"10", "C":"15", "D":"20", "E":"25" }, "id":10747, "question":"If you add some number j to 50, and then divide this sum by j, the result is 3. What is the value of j?", "explanations":{ "correct":"Let's solve the problem step-by-step:\n\n1. We are given that if we add some number j to 50 and then divide this sum by j, the result is 3. So, we can write this as an equation: (50 + j) / j = 3.\n\n2. To solve this equation, we can start by multiplying both sides by j to eliminate the denominator: 50 + j = 3j.\n\n3. Next, we can simplify the equation by subtracting j from both sides: 50 = 2j.\n\n4. To isolate j, we can divide both sides by 2: j = 50 / 2.\n\n5. Simplifying the right side of the equation, we find that j = 25.\n\nTherefore, the value of j is 25.\n\nThe answer is E." } }, { "answer":"A", "choices":{ "A":"-2", "B":"0", "C":"2", "D":"\\(\\frac { a + b } { a - b } \\)", "E":"\\(\\frac { a - b } { b - a } \\)" }, "id":10749, "question":"If \\(a^2 \\neq b^2\\), then \\(\\frac { a^2 - b^2 } { b^2 - a^2 } + \\frac { a - b } { b - a } \\) =", "explanations":{ "correct":"To simplify the expression \\(\\frac { a^2 - b^2 } { b^2 - a^2 } + \\frac { a - b } { b - a }\\), we can start by factoring the numerator and denominator of the first fraction.\n\nThe numerator \\(a^2 - b^2\\) can be factored as \\((a - b)(a + b)\\), and the denominator \\(b^2 - a^2\\) can be factored as \\((b - a)(b + a)\\).\n\nNow, let's substitute these factorizations back into the expression:\n\n\\(\\frac { (a - b)(a + b) } { (b - a)(b + a) } + \\frac { a - b } { b - a }\\)\n\nNext, we can simplify the expression by canceling out common factors. Notice that \\((a - b)\\) appears in both terms, so we can factor it out:\n\n\\(\\frac { (a - b)(a + b) + (a - b) } { (b - a)(b + a) }\\)\n\nNow, we can combine the two terms in the numerator:\n\n\\(\\frac { (a - b)(a + b + 1) } { (b - a)(b + a) }\\)\n\nSince \\((b - a) = -(a - b)\\), we can simplify further:\n\n\\(\\frac { (a - b)(a + b + 1) } { -(a - b)(b + a) }\\)\n\nNow, we can cancel out the common factor \\((a - b)\\):\n\n\\(\\frac { a + b + 1 } { -(b + a) }\\)\n\nFinally, we can simplify the expression by rearranging the terms:\n\n\\(\\frac { a + b + 1 } { -2(a + b) }\\)\n\nTherefore, the answer is A) -2." } }, { "answer":"E", "choices":{ "A":"4", "B":"8", "C":"15", "D":"20", "E":"25" }, "id":10750, "question":"In a certain neighborhood, the number of cats is inversely proportional to the number of dogs. At one point, there were 10 cats and 50 dogs in the neighborhood. If today there are 20 dogs in the neighborhood, how many cats are in the neighborhood?", "explanations":{ "correct":"To solve this problem, we can use the concept of inverse proportionality. Inverse proportionality means that as one quantity increases, the other quantity decreases, and vice versa, while their product remains constant.\n\nLet's denote the number of cats as C and the number of dogs as D. We are given that the number of cats is inversely proportional to the number of dogs, so we can write the equation:\n\nC * D = k\n\nwhere k is a constant.\n\nWe are also given that at one point, there were 10 cats and 50 dogs in the neighborhood. We can use this information to find the value of k:\n\n10 * 50 = k\nk = 500\n\nNow, we are asked to find the number of cats in the neighborhood when there are 20 dogs. We can use the equation C * D = k and substitute D = 20:\n\nC * 20 = 500\nC = 500 / 20\nC = 25\n\nTherefore, the answer is E) 25." } }, { "answer":"C", "choices":{ "A":"0", "B":"\\(2\\sqrt{2}\\)", "C":"4", "D":"\\(4\\sqrt{2}\\)", "E":"\\(\\sqrt{34}\\)" }, "id":10752, "question":"If for \\(x \\geq 1\\), \\(\\star x \\star\\) is defined as \\(\\star x \\star = \\sqrt{(x - 1)}\\), which of the following is equal to \\(\\star 37 \\star - \\star 5 \\star\\)?", "explanations":{ "correct":"To find the value of \\(\\star 37 \\star - \\star 5 \\star\\), we need to substitute the given definition of \\(\\star x \\star\\) into the expression.\n\nFirst, let's find the value of \\(\\star 37 \\star\\):\n\\(\\star 37 \\star = \\sqrt{(37 - 1)} = \\sqrt{36} = 6\\)\n\nNext, let's find the value of \\(\\star 5 \\star\\):\n\\(\\star 5 \\star = \\sqrt{(5 - 1)} = \\sqrt{4} = 2\\)\n\nNow, we can substitute these values back into the expression:\n\\(\\star 37 \\star - \\star 5 \\star = 6 - 2 = 4\\)\n\nTherefore, the answer is C) 4." } }, { "answer":"B", "choices":{ "A":"93", "B":"130", "C":"145", "D":"153", "E":"160" }, "id":10755, "question":"The length of a playground is 3 feet more than its width. If the length of the playground is 13 feet, what is the area of the playground in square feet?", "explanations":{ "correct":"To find the area of the playground, we need to know the width of the playground. We are given that the length of the playground is 13 feet, and it is 3 feet more than its width. \n\nLet's assume the width of the playground is \"w\" feet. Since the length is 3 feet more than the width, the length can be represented as \"w + 3\" feet. \n\nGiven that the length is 13 feet, we can set up the equation: \nw + 3 = 13\n\nSubtracting 3 from both sides of the equation, we get: \nw = 10\n\nSo, the width of the playground is 10 feet. \n\nTo find the area of the playground, we multiply the length by the width: \nArea = length × width\nArea = 13 × 10\nArea = 130 square feet\n\nTherefore, the area of the playground is 130 square feet.\n\nThe answer is B) 130." } }, { "answer":"C", "choices":{ "A":"\\$ 7.00", "B":"\\$ 7.20", "C":"\\$ 7.40", "D":"\\$ 8.00", "E":"\\$ 8.20" }, "id":10759, "question":"A telephone call cost 2 dollars for the first three minutes and 20 cents each additional minute. If a call started at 10:20 a.m. and ended at 10:50 a.m., how much did the call cost?", "explanations":{ "correct":"To find the cost of the call, we need to determine the total number of minutes the call lasted. \n\nThe call started at 10:20 a.m. and ended at 10:50 a.m., which means it lasted for 30 minutes. \n\nThe first three minutes of the call cost \\$ 2.00, and each additional minute costs \\$ 0.20. \n\nTo find the cost of the additional minutes, we subtract the first three minutes from the total duration of the call: 30 minutes - 3 minutes = 27 minutes. \n\nThe cost of the additional minutes is then calculated by multiplying the number of additional minutes (27) by the cost per minute ($ 0.20): 27 minutes * \\$ 0.20 = \\$ 5.40. \n\nAdding the cost of the first three minutes ($ 2.00) to the cost of the additional minutes ($ 5.40), we get the total cost of the call: \\$ 2.00 + \\$ 5.40 = \\$ 7.40. \n\nTherefore, the call cost \\$ 7.40. \n\nThe answer is C) \\$ 7.40." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { w - g } { w } \\)", "B":"\\(\\frac { w - g } { g } \\)", "C":"\\(\\frac { g - w } { g } \\)", "D":"\\(\\frac { g } { g - w } \\)", "E":"\\(\\frac { g - w } { w } \\)" }, "id":10760, "question":"If a team played g games and won w of them, what fraction of the games played did the team lose?", "explanations":{ "correct":"To find the fraction of games the team lost, we need to determine the number of games lost and divide it by the total number of games played.\n\nThe number of games lost can be found by subtracting the number of games won (w) from the total number of games played (g). Therefore, the number of games lost is g - w.\n\nTo find the fraction, we divide the number of games lost by the total number of games played. So the fraction of games lost is \\(\\frac { g - w } { g }\\).\n\nTherefore, the answer is C) \\(\\frac { g - w } { g }\\)." } }, { "answer":"C", "choices":{ "A":"\\((r + s)^2 - q^2 = (t + r)^2\\)", "B":"\\((r + s)^2 + q^2 = (t - r)^2\\)", "C":"\\(\\sqrt{r + s} - q^2 = (t + r)^2\\)", "D":"\\(q^2 - \\sqrt{r + s} = (t - r)^2\\)", "E":"\\(q^2 - \\sqrt{(r + s)^2} = (t + r)^2\\)" }, "id":10765, "question":"Which of the following equations represents the statement ``When the square of q is subtracted from the square root of the sum of r and s, the result is equal to the square of the sum of t and r''?", "explanations":{ "correct":"To solve this problem, let's break down the given statement step-by-step:\n\n1. \"When the square of q is subtracted from the square root of the sum of r and s\" can be represented as \\(\\sqrt{r + s} - q^2\\).\n\n2. \"The result is equal to the square of the sum of t and r\" can be represented as \\((t + r)^2\\).\n\nNow, let's compare the given options with the equation we derived:\n\nA) \\((r + s)^2 - q^2 = (t + r)^2\\)\nB) \\((r + s)^2 + q^2 = (t - r)^2\\)\nC) \\(\\sqrt{r + s} - q^2 = (t + r)^2\\) (Matches our equation)\nD) \\(q^2 - \\sqrt{r + s} = (t - r)^2\\)\nE) \\(q^2 - \\sqrt{(r + s)^2} = (t + r)^2\\)\n\nComparing the equation we derived with the given options, we can see that option C) \\(\\sqrt{r + s} - q^2 = (t + r)^2\\) matches our equation. \n\nTherefore, the answer is C)." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { a } { 3 } \\)", "B":"\\(\\frac { 2a } { 3 } \\)", "C":"\\(3a\\)", "D":"\\(6a\\)", "E":"\\(9a\\)" }, "id":10767, "question":"For the function \\(f\\), \\(f(x)\\) is inversely proportional to \\(x\\). If \\(f(k) = a\\), what is the value of \\(f(\\frac { k } { 3 } )\\) ?", "explanations":{ "correct":"To solve this problem, we need to understand the concept of inverse proportionality. Inverse proportionality means that as one variable increases, the other variable decreases, and vice versa, while their product remains constant.\n\nGiven that \\(f(x)\\) is inversely proportional to \\(x\\), we can write the equation as \\(f(x) = \\frac{k}{x}\\), where \\(k\\) is the constant of proportionality.\n\nWe are given that \\(f(k) = a\\), so we can substitute \\(k\\) into the equation to get \\(f(k) = \\frac{k}{k} = 1\\). Therefore, \\(a = 1\\).\n\nNow, we need to find the value of \\(f\\left(\\frac{k}{3}\\right)\\). Substituting \\(\\frac{k}{3}\\) into the equation, we get \\(f\\left(\\frac{k}{3}\\right) = \\frac{k}{\\frac{k}{3}} = 3\\).\n\nSince \\(a = 1\\), we have \\(f\\left(\\frac{k}{3}\\right) = 3a = 3\\cdot1 = 3\\).\n\nTherefore, the answer is C) \\(3a\\)." } }, { "answer":"D", "choices":{ "A":"30", "B":"40", "C":"60", "D":"90", "E":"120" }, "id":10769, "question":"Stephan takes 240 minutes to draw 20 pictures. Pavel draws three times as fast as Stephan. How many pictures can Pavel draw in 6 hours?", "explanations":{ "correct":"To find out how many pictures Pavel can draw in 6 hours, we first need to determine how many minutes are in 6 hours. Since there are 60 minutes in an hour, we can calculate that there are 6 * 60 = 360 minutes in 6 hours.\n\nNext, we need to determine how many pictures Stephan can draw in 360 minutes. We know that Stephan takes 240 minutes to draw 20 pictures. To find out how many pictures he can draw in 360 minutes, we can set up a proportion:\n\n240 minutes / 20 pictures = 360 minutes / x pictures\n\nCross-multiplying, we get:\n\n240x = 20 * 360\n\nSimplifying, we have:\n\n240x = 7200\n\nDividing both sides by 240, we find:\n\nx = 7200 / 240\n\nx = 30\n\nTherefore, Stephan can draw 30 pictures in 360 minutes.\n\nSince Pavel draws three times as fast as Stephan, we can multiply the number of pictures Stephan can draw by 3 to find out how many pictures Pavel can draw in the same amount of time:\n\n30 pictures * 3 = 90 pictures\n\nTherefore, Pavel can draw 90 pictures in 360 minutes, which is equivalent to 6 hours.\n\nThe answer is D) 90." } }, { "answer":"D", "choices":{ "A":"22", "B":"24", "C":"28", "D":"30", "E":"33" }, "id":10770, "question":"If \\(m\\) and \\(n\\) are integers and \\((x + m)(x + n) = x^2 + 11x + c\\), what is the greatest possible value of \\(c\\) ?", "explanations":{ "correct":"To find the greatest possible value of \\(c\\), we need to consider the expansion of \\((x + m)(x + n)\\). \n\nUsing the distributive property, we can expand the expression as follows:\n\n\\((x + m)(x + n) = x^2 + nx + mx + mn\\)\n\nSimplifying further, we have:\n\n\\(x^2 + (n + m)x + mn\\)\n\nComparing this to the given expression \\(x^2 + 11x + c\\), we can see that the coefficient of \\(x\\) in both expressions must be the same. Therefore, we have:\n\n\\(n + m = 11\\)\n\nTo maximize the value of \\(c\\), we need to find the maximum value of \\(mn\\). Since \\(m\\) and \\(n\\) are integers, the maximum value of their product occurs when they are consecutive integers. \n\nLet's consider two cases:\n\nCase 1: \\(m = 5\\) and \\(n = 6\\)\\(\\newline\\)In this case, \\(mn = 5 \\times 6 = 30\\)\n\nCase 2: \\(m = 6\\) and \\(n = 5\\)\\(\\newline\\)In this case, \\(mn = 6 \\times 5 = 30\\)\n\nBoth cases yield the same maximum value of \\(mn\\), which is 30.\n\nTherefore, the greatest possible value of \\(c\\) is 30.\n\nThe answer is D." } }, { "answer":"D", "choices":{ "A":"30", "B":"45", "C":"60", "D":"120", "E":"135" }, "id":10771, "question":"By noon, \\(\\frac{1}{4}\\) of Company X's employees had arrived to work. An hour later, 30 more employees had arrived, raising total attendance to \\(\\frac{1}{2}\\) of the total staff. How many employees work for Company X?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nLet's assume the total number of employees at Company X is \"x\".\n\nAccording to the problem, by noon, 1/4 of the employees had arrived. So, the number of employees who arrived by noon is (1/4)x.\n\nAn hour later, 30 more employees arrived, raising the total attendance to 1/2 of the total staff. So, the number of employees who arrived in that hour is 30.\n\nThe total number of employees who arrived by that time is (1/4)x + 30.\n\nAccording to the problem, this total attendance is 1/2 of the total staff, so we can set up the equation:\n\n(1/4)x + 30 = (1/2)x\n\nTo solve this equation, we can multiply both sides by 4 to eliminate the fraction:\n\n4 * [(1/4)x + 30] = 4 * (1/2)x\n\nx + 120 = 2x\n\nNow, we can subtract x from both sides:\n\nx + 120 - x = 2x - x\n\n120 = x\n\nSo, the total number of employees at Company X is 120.\n\nThe answer is D) 120." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { 2 } { 3 } y\\)", "B":"\\(y\\)", "C":"\\(\\frac { 3 } { 2 } y\\)", "D":"\\(2y\\)", "E":"\\(3y\\)" }, "id":10774, "question":"If \\(x = \\frac { 2 } { 3 } (x + y)\\), which of the following is an expression for \\(x\\) in terms of \\(y\\) ?", "explanations":{ "correct":"To find an expression for \\(x\\) in terms of \\(y\\), we need to isolate \\(x\\) on one side of the equation.\n\nGiven the equation \\(x = \\frac{2}{3}(x + y)\\), we can start by distributing \\(\\frac{2}{3}\\) to \\(x\\) and \\(y\\):\n\n\\(x = \\frac{2}{3}x + \\frac{2}{3}y\\)\n\nNext, we can subtract \\(\\frac{2}{3}x\\) from both sides of the equation to move all terms with \\(x\\) to one side:\n\n\\(x - \\frac{2}{3}x = \\frac{2}{3}y\\)\n\nSimplifying the left side of the equation, we have:\n\n\\(\\frac{1}{3}x = \\frac{2}{3}y\\)\n\nTo isolate \\(x\\), we can multiply both sides of the equation by \\(\\frac{3}{1}\\) (or simply multiply by 3):\n\n\\(3 \\cdot \\frac{1}{3}x = 3 \\cdot \\frac{2}{3}y\\)\n\nThis simplifies to:\n\n\\(x = 2y\\)\n\nTherefore, the expression for \\(x\\) in terms of \\(y\\) is \\(2y\\).\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"13", "B":"46", "C":"77", "D":"103", "E":"113" }, "id":10777, "question":"If \\(f(c) = \\frac { 9 } { 5 } c + 32\\), then what is the value of \\(f(25)\\)?", "explanations":{ "correct":"To find the value of \\(f(25)\\), we need to substitute \\(c = 25\\) into the given function \\(f(c) = \\frac{9}{5}c + 32\\).\n\nSubstituting \\(c = 25\\) into the function, we have:\n\\(f(25) = \\frac{9}{5}(25) + 32\\)\n\nSimplifying the expression inside the parentheses:\n\\(f(25) = \\frac{9}{5} \\cdot 25 + 32\\)\n\nMultiplying \\(\\frac{9}{5}\\) by 25:\n\\(f(25) = \\frac{225}{5} + 32\\)\n\nSimplifying \\(\\frac{225}{5}\\):\n\\(f(25) = 45 + 32\\)\n\nAdding 45 and 32:\n\\(f(25) = 77\\)\n\nTherefore, the value of \\(f(25)\\) is 77.\n\nThe answer is C." } }, { "answer":"A", "choices":{ "A":"16", "B":"13", "C":"\\(6 + 2\\sqrt{7}\\)", "D":"11", "E":"\\(6 + 2\\sqrt{5}\\)" }, "id":10778, "question":"What is the perimeter of a triangle with vertices \\((-1, 1)\\), \\((5, 1)\\), and \\((2, 5)\\)?", "explanations":{ "correct":"To find the perimeter of a triangle, we need to calculate the sum of the lengths of all three sides.\n\nFirst, let's find the length of the side connecting the points \\((-1, 1)\\) and \\((5, 1)\\). We can use the distance formula, which states that the distance between two points \\((x_1, y_1)\\) and \\((x_2, y_2)\\) is given by:\n\n\\(\\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\)\n\nUsing this formula, we have:\n\n\\(\\sqrt{(5 - (-1))^2 + (1 - 1)^2} = \\sqrt{6^2 + 0^2} = \\sqrt{36} = 6\\)\n\nNext, let's find the length of the side connecting the points \\((-1, 1)\\) and \\((2, 5)\\):\n\n\\(\\sqrt{(2 - (-1))^2 + (5 - 1)^2} = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5\\)\n\nFinally, let's find the length of the side connecting the points \\((5, 1)\\) and \\((2, 5)\\):\n\n\\(\\sqrt{(2 - 5)^2 + (5 - 1)^2} = \\sqrt{(-3)^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5\\)\n\nNow, we can calculate the perimeter by adding up the lengths of all three sides:\n\n\\(6 + 5 + 5 = 16\\)\n\nTherefore, the perimeter of the triangle is 16.\n\nThe answer is A) 16." } }, { "answer":"E", "choices":{ "A":"\\(7b + a\\)", "B":"\\(21b + a\\)", "C":"\\(7(b + a)\\)", "D":"\\(7(b + 3a)\\)", "E":"\\(7(3b + a)\\)" }, "id":10780, "question":"Marvin eats 3 bananas and 1 apple every day. If the cost of each banana is b dollars and the cost of each apple is a dollars, which of the following represents the total cost, in dollars, of the bananas and apples Marvin eats in 1 week? (1 week = 7 days).", "explanations":{ "correct":"To find the total cost of the bananas and apples Marvin eats in 1 week, we need to calculate the cost per day and then multiply it by the number of days in a week.\n\nMarvin eats 3 bananas and 1 apple every day. The cost of each banana is b dollars, so the cost of 3 bananas per day is 3b dollars. The cost of each apple is a dollars, so the cost of 1 apple per day is a dollars.\n\nTo find the total cost per day, we add the cost of the bananas and the cost of the apple: 3b + a.\n\nSince there are 7 days in a week, we multiply the total cost per day by 7 to find the total cost in 1 week: 7(3b + a).\n\nTherefore, the answer is E) \\(7(3b + a)\\)." } }, { "answer":"A", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":10782, "question":"The function \\(h\\) is defined by \\(h(x) = 3x - 2\\) and the function \\(g\\) is defined by \\(g(x) = 3h(x )- 1\\). If \\(g(k) = 11\\), what is the value of \\(k\\)?", "explanations":{ "correct":"To find the value of \\(k\\), we need to substitute \\(g(k)\\) into the equation for \\(g(x)\\) and solve for \\(k\\).\n\nGiven that \\(g(x) = 3h(x) - 1\\) and \\(g(k) = 11\\), we can substitute \\(g(k)\\) into the equation:\n\n\\(11 = 3h(k) - 1\\)\n\nNow, we need to substitute \\(h(x)\\) into the equation for \\(h(k)\\):\n\n\\(11 = 3(3k - 2) - 1\\)\n\nSimplifying the equation:\n\n\\(11 = 9k - 6 - 1\\)\n\n\\(11 = 9k - 7\\)\n\nAdding 7 to both sides of the equation:\n\n\\(18 = 9k\\)\n\nDividing both sides of the equation by 9:\n\n\\(2 = k\\)\n\nTherefore, the value of \\(k\\) is 2.\n\nThe answer is A) 2." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 1 } { 25 } \\)", "B":"\\(\\frac { 1 } { 12 } \\)", "C":"\\(\\frac { 1 } { 8 } \\)", "D":"\\(\\frac { 1 } { 6 } \\)", "E":"\\(\\frac { 1 } { 3 } \\)" }, "id":10783, "question":"A certain fraction is equivalent to n%, where n is an integer. Which of the following could be the value of that fraction?", "explanations":{ "correct":"To determine which of the given fractions could be equivalent to n%, we need to convert n% to a fraction and compare it to the given options.\n\nStep 1: Convert n% to a fraction\nTo convert a percentage to a fraction, we divide the percentage by 100. So, n% can be written as n/100.\n\nStep 2: Compare the converted fraction to the given options\nLet's compare the converted fraction n/100 to each of the given options:\n\nA) \\(\\frac { 1 } { 25 } \\)\nTo check if n/100 is equal to 1/25, we need to find an integer n such that n/100 = 1/25. Cross-multiplying, we get n = 100/25 = 4. So, n/100 is not equal to 1/25.\n\nB) \\(\\frac { 1 } { 12 } \\)\nTo check if n/100 is equal to 1/12, we need to find an integer n such that n/100 = 1/12. Cross-multiplying, we get n = 100/12 = 8.33. Since n must be an integer, n/100 is not equal to 1/12.\n\nC) \\(\\frac { 1 } { 8 } \\)\nTo check if n/100 is equal to 1/8, we need to find an integer n such that n/100 = 1/8. Cross-multiplying, we get n = 100/8 = 12.5. Since n must be an integer, n/100 is not equal to 1/8.\n\nD) \\(\\frac { 1 } { 6 } \\)\nTo check if n/100 is equal to 1/6, we need to find an integer n such that n/100 = 1/6. Cross-multiplying, we get n = 100/6 = 16.67. Since n must be an integer, n/100 is not equal to 1/6.\n\nE) \\(\\frac { 1 } { 3 } \\)\nTo check if n/100 is equal to 1/3, we need to find an integer n such that n/100 = 1/3. Cross-multiplying, we get n = 100/3 = 33.33. Since n must be an integer, n/100 is not equal to 1/3.\n\nBased on the above comparisons, the only fraction that could be equivalent to n% is \\(\\frac { 1 } { 25 } \\).\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"78", "B":"81", "C":"82", "D":"83", "E":"84" }, "id":10784, "question":"Maja has the following scores on 7 quizzes in French class: 81,76,80,84,78,91,84. What was the median score of her French quizzes?", "explanations":{ "correct":"To find the median score of Maja's French quizzes, we need to arrange the scores in ascending order: 76, 78, 80, 81, 84, 84, 91.\n\nSince there are 7 scores, the median will be the middle value. In this case, the middle value is the 4th score, which is 81.\n\nTherefore, the median score of Maja's French quizzes is 81.\n\nThe answer is B) 81." } }, { "answer":"A", "choices":{ "A":"\\(-10q\\)", "B":"\\(10q\\)", "C":"\\(14q - 12\\)", "D":"\\(28 - 6q\\)", "E":"\\(36q\\)" }, "id":10785, "question":"If \\(p = 2\\), what is \\(2q(7 - 6p)\\) in terms of q?", "explanations":{ "correct":"To find the value of \\(2q(7 - 6p)\\) in terms of \\(q\\), we need to substitute the given value of \\(p\\) into the expression and simplify.\n\nGiven that \\(p = 2\\), we substitute \\(2\\) for \\(p\\) in the expression:\n\n\\(2q(7 - 6p) = 2q(7 - 6(2))\\)\n\nSimplifying the expression inside the parentheses:\n\n\\(2q(7 - 6(2)) = 2q(7 - 12)\\)\n\nFurther simplifying:\n\n\\(2q(7 - 12) = 2q(-5)\\)\n\nFinally, multiplying \\(2q\\) by \\(-5\\):\n\n\\(2q(-5) = -10q\\)\n\nTherefore, the answer is A) \\(-10q\\)." } }, { "answer":"D", "choices":{ "A":"-36", "B":"-6", "C":"6", "D":"36", "E":"There is no value of x that satisfies the equation" }, "id":10790, "question":"If \\(25 - 3\\sqrt { x } = 7\\), what is the value of \\(x\\) ?", "explanations":{ "correct":"To find the value of \\(x\\), we need to solve the equation \\(25 - 3\\sqrt{x} = 7\\).\n\nFirst, let's isolate the term with the square root by subtracting 25 from both sides of the equation:\n\n\\(25 - 3\\sqrt{x} - 25 = 7 - 25\\)\n\nThis simplifies to:\n\n\\(-3\\sqrt{x} = -18\\)\n\nNext, let's get rid of the coefficient in front of the square root by dividing both sides of the equation by -3:\n\n\\(\\frac{-3\\sqrt{x}}{-3} = \\frac{-18}{-3}\\)\n\nThis simplifies to:\n\n\\(\\sqrt{x} = 6\\)\n\nTo solve for \\(x\\), we need to square both sides of the equation:\n\n\\((\\sqrt{x})^2 = 6^2\\)\n\nThis simplifies to:\n\n\\(x = 36\\)\n\nTherefore, the value of \\(x\\) that satisfies the equation is 36.\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"0", "B":"5", "C":"10", "D":"20", "E":"30" }, "id":10794, "question":"What is the greatest value of \\(x\\) that is a solution of the following equation? \\(|x - 5| + 10 = 15\\)", "explanations":{ "correct":"To find the greatest value of \\(x\\) that is a solution of the equation \\(|x - 5| + 10 = 15\\), we need to isolate the absolute value expression and solve for \\(x\\).\n\nStep 1: Subtract 10 from both sides of the equation:\n\\[|x - 5| = 15 - 10\\]\n\\[|x - 5| = 5\\]\n\nStep 2: Since the absolute value of a number is always non-negative, we can rewrite the equation as two separate equations, one with the positive value and one with the negative value:\n\\[x - 5 = 5 \\quad \\text{and} \\quad -(x - 5) = 5\\]\n\nStep 3: Solve the first equation:\n\\[x - 5 = 5\\]\nAdding 5 to both sides:\n\\[x = 10\\]\n\nStep 4: Solve the second equation:\n\\[-(x - 5) = 5\\]\nDistribute the negative sign:\n\\[-x + 5 = 5\\]\nSubtract 5 from both sides:\n\\[-x = 0\\]\nMultiply both sides by -1 to isolate \\(x\\):\n\\[x = 0\\]\n\nStep 5: Compare the solutions \\(x = 10\\) and \\(x = 0\\) to find the greatest value. Since 10 is greater than 0, the greatest value of \\(x\\) that is a solution of the equation is 10.\n\nTherefore, the answer is C) 10." } }, { "answer":"B", "choices":{ "A":"1", "B":"2", "C":"4", "D":"6", "E":"8" }, "id":10796, "question":"After 4 new people entered the elevator and 2 people left the elevator, there were 2 times as many people as before. How many people were in the elevator before the changes?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume that the number of people in the elevator before the changes is \"x\".\n\nAfter 4 new people entered the elevator, the number of people in the elevator becomes \"x + 4\".\n\nAfter 2 people left the elevator, the number of people in the elevator becomes \"x + 4 - 2\", which simplifies to \"x + 2\".\n\nAccording to the problem, the number of people in the elevator after the changes is 2 times the number of people before the changes. So we can write the equation:\n\nx + 2 = 2x\n\nTo solve this equation, we can subtract \"x\" from both sides:\n\n2 = x\n\nTherefore, the number of people in the elevator before the changes is 2.\n\nThe answer is B) 2." } }, { "answer":"B", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":10799, "question":"A bicyclist riding at 12 miles an hour for 2 hours travels twice the distance a hiker travels walking at 4 miles an hour for how many hours?", "explanations":{ "correct":"To find the answer, we need to compare the distances traveled by the bicyclist and the hiker. \n\nThe distance traveled by the bicyclist can be calculated by multiplying the speed (12 miles per hour) by the time (2 hours):\nDistance = Speed × Time = 12 miles/hour × 2 hours = 24 miles.\n\nThe distance traveled by the hiker can be calculated in the same way:\nDistance = Speed × Time = 4 miles/hour × Time.\n\nSince the bicyclist travels twice the distance of the hiker, we can set up the equation:\n24 miles = 2 × (4 miles/hour × Time).\n\nSimplifying the equation, we have:\n24 miles = 8 miles/hour × Time.\n\nTo solve for Time, we divide both sides of the equation by 8 miles/hour:\nTime = 24 miles / 8 miles/hour = 3 hours.\n\nTherefore, the hiker travels for 3 hours.\n\nThe answer is B) 3." } }, { "answer":"A", "choices":{ "A":"-2", "B":"-1", "C":"0", "D":"2", "E":"3" }, "id":10802, "question":"\\begin { gather* } 5, 5k, 5k^2 , 5k^3 , ...\\end { gather* } The first term in the sequence above is 5, and each term after the first is \\(k\\) times the preceding term. If the sum of the first three terms is 15, which of the following could be the value of \\(k\\) ?", "explanations":{ "correct":"To find the sum of the first three terms, we can use the formula for the sum of a geometric series:\n\n\\[S = \\frac{a(1 - r^n)}{1 - r}\\]\n\nwhere:\n- \\(S\\) is the sum of the terms,\n- \\(a\\) is the first term,\n- \\(r\\) is the common ratio, and\n- \\(n\\) is the number of terms.\n\\(\\newline\\)In this case, we are given that the first term is 5, and each term after the first is \\(k\\) times the preceding term. So, the common ratio is \\(k\\).\n\nWe are also given that the sum of the first three terms is 15. Plugging in the values into the formula, we have:\n\n\\[15 = \\frac{5(1 - k^3)}{1 - k}\\]\n\nTo solve for \\(k\\), we can multiply both sides of the equation by \\(1 - k\\) to eliminate the denominator:\n\n\\[15(1 - k) = 5(1 - k^3)\\]\n\nExpanding both sides of the equation:\n\n\\[15 - 15k = 5 - 5k^3\\]\n\nRearranging the equation:\n\n\\[5k^3 - 15k + 10 = 0\\]\n\nDividing both sides of the equation by 5:\n\n\\[k^3 - 3k + 2 = 0\\]\n\nNow, we can try each of the answer choices to see if they satisfy this equation.\n\nA) For \\(k = -2\\):\n\\((-2)^3 - 3(-2) + 2 = -8 + 6 + 2 = 0\\)\nSo, \\(k = -2\\) is a valid solution.\n\nB) For \\(k = -1\\):\n\\((-1)^3 - 3(-1) + 2 = -1 + 3 + 2 = 4\\)\nSo, \\(k = -1\\) is not a valid solution.\n\nC) For \\(k = 0\\):\n\\(0^3 - 3(0) + 2 = 2\\)\nSo, \\(k = 0\\) is not a valid solution.\n\nD) For \\(k = 2\\):\n\\(2^3 - 3(2) + 2 = 8 - 6 + 2 = 4\\)\nSo, \\(k = 2\\) is not a valid solution.\n\nE) For \\(k = 3\\):\n\\(3^3 - 3(3) + 2 = 27 - 9 + 2 = 20\\)\nSo, \\(k = 3\\) is not a valid solution.\n\nTherefore, the only value of \\(k\\) that could be the solution is -2.\n\nThe answer is A." } }, { "answer":"A", "choices":{ "A":"\\(-\\frac { 14 } { 3 } \\)", "B":"\\(-\\frac { 2 } { 3 } \\)", "C":"\\(\\frac { 4 } { 7 } \\)", "D":"\\(\\frac { 2 } { 3 } \\)", "E":"\\(\\frac { 14 } { 3 } \\)" }, "id":10806, "question":"When a certain number is increased by 4, the result is equal to that number divided by 7. What is the number?", "explanations":{ "correct":"Let's solve the problem step-by-step:\n\nLet's assume the certain number is represented by the variable \\(x\\).\n\nAccording to the problem, when this number is increased by 4, the result is equal to that number divided by 7. Mathematically, we can represent this as:\n\n\\(x + 4 = \\frac{x}{7}\\)\n\nTo solve this equation, we can start by multiplying both sides of the equation by 7 to eliminate the fraction:\n\n\\(7(x + 4) = x\\)\n\nExpanding the left side of the equation:\n\n\\(7x + 28 = x\\)\n\nNext, we can simplify the equation by subtracting \\(x\\) from both sides:\n\n\\(7x - x + 28 = 0\\)\n\nSimplifying further:\n\n\\(6x + 28 = 0\\)\n\nNow, we can isolate \\(x\\) by subtracting 28 from both sides:\n\n\\(6x = -28\\)\n\nFinally, we can solve for \\(x\\) by dividing both sides by 6:\n\n\\(x = -\\frac{28}{6} = -\\frac{14}{3}\\)\n\nTherefore, the answer is A) \\(-\\frac{14}{3}\\)." } }, { "answer":"D", "choices":{ "A":"1", "B":"2", "C":"3", "D":"4", "E":"5" }, "id":10812, "question":"\\begin { gather* } a^2 - b^2 = 12 \\\\ a + b > 4 \\end { gather* } If \\(a\\) and \\(b\\) are positive integers in the inequalities above and \\(a > b\\), what is the value of \\(a\\)?", "explanations":{ "correct":"To find the value of \\(a\\), we need to solve the given system of equations:\n\n\\[\n\\begin{gather*}\na^2 - b^2 = 12 \\\\\na + b > 4\n\\end{gather*}\n\\]\n\nSince \\(a > b\\), we can rewrite the first equation as:\n\n\\[\n(a + b)(a - b) = 12\n\\]\n\nNow, let's consider the possible values of \\(a + b\\) and \\(a - b\\) that satisfy this equation:\n\n\\[\n\\begin{align*}\na + b &= 12 \\quad \\text{(1)} \\\\\na - b &= 1 \\quad \\text{(2)} \\\\\n\\end{align*}\n\\]\n\nSolving equations (1) and (2) simultaneously, we can add them to eliminate \\(b\\):\n\n\\[\n\\begin{align*}\n(a + b) + (a - b) &= 12 + 1 \\\\\n2a &= 13 \\\\\na &= \\frac{13}{2}\n\\end{align*}\n\\]\n\nSince \\(a\\) and \\(b\\) are positive integers, \\(\\frac{13}{2}\\) is not a valid solution. Therefore, we need to consider other possible values of \\(a + b\\) and \\(a - b\\):\n\n\\[\n\\begin{align*}\na + b &= 6 \\quad \\text{(3)} \\\\\na - b &= 2 \\quad \\text{(4)} \\\\\n\\end{align*}\n\\]\n\nAdding equations (3) and (4), we get:\n\n\\[\n\\begin{align*}\n(a + b) + (a - b) &= 6 + 2 \\\\\n2a &= 8 \\\\\na &= 4\n\\end{align*}\n\\]\n\nSince \\(a = 4\\) is a positive integer and satisfies the given conditions, the value of \\(a\\) is 4.\n\nTherefore, the answer is D) 4." } }, { "answer":"A", "choices":{ "A":"\\(606 \\times 10^3\\)", "B":"\\(606 \\times 10^2\\)", "C":"\\(120 \\times 10^6\\)", "D":"\\(66 \\times 10^4\\)", "E":"\\(60 \\times 10^6\\)" }, "id":10824, "question":"\\((60 \\times 10^4) + (60 \\times 10^2)\\) =", "explanations":{ "correct":"To solve the given expression \\((60 \\times 10^4) + (60 \\times 10^2)\\), we need to simplify each term separately and then add them together.\n\nFirst, let's simplify the first term: \\(60 \\times 10^4\\). To do this, we multiply 60 by 10 raised to the power of 4. When we multiply a number by 10 raised to a power, we move the decimal point to the right by that many places. In this case, since the power is 4, we move the decimal point four places to the right. So, \\(60 \\times 10^4\\) is equal to 600,000.\n\nNext, let's simplify the second term: \\(60 \\times 10^2\\). Again, we multiply 60 by 10 raised to the power of 2. This time, since the power is 2, we move the decimal point two places to the right. So, \\(60 \\times 10^2\\) is equal to 6,000.\n\nNow, we can add the two simplified terms together: 600,000 + 6,000 = 606,000.\n\nTherefore, the answer is A) \\(606 \\times 10^3\\)." } }, { "answer":"D", "choices":{ "A":"\\(-22\\)", "B":"\\(-18\\)", "C":"\\(-17\\)", "D":"18", "E":"22" }, "id":10825, "question":"If \\((2x^2 + 5x - 3)(4x - 1) = ax^3 + bx^2 + cx + d\\) for all values of x, what is the value of b?", "explanations":{ "correct":"To find the value of b, we need to expand the given expression \\((2x^2 + 5x - 3)(4x - 1)\\) and compare it with the general form \\(ax^3 + bx^2 + cx + d\\).\n\nFirst, let's use the distributive property to expand the expression:\n\\((2x^2 + 5x - 3)(4x - 1) = 2x^2 \\cdot 4x + 2x^2 \\cdot (-1) + 5x \\cdot 4x + 5x \\cdot (-1) - 3 \\cdot 4x - 3 \\cdot (-1)\\)\nSimplifying this, we get:\n\\(8x^3 - 2x^2 + 20x^2 - 5x - 12x + 3\\)\nCombining like terms, we have:\n\\(8x^3 + 18x^2 + 3x + 3\\)\n\nNow, we can compare this expression with the general form \\(ax^3 + bx^2 + cx + d\\).\nFrom the comparison, we can see that the coefficient of \\(x^2\\) in the expanded expression is 18. Therefore, the value of b is 18.\n\nThe answer is D) 18." } }, { "answer":"B", "choices":{ "A":"\\$ 5,000", "B":"\\$ 10,000", "C":"\\$ 64,000", "D":"\\$ 160,000", "E":"\\(\\$ 320,000 \\times 2^5\\)" }, "id":10827, "question":"Thirty years ago, Mr. and Mrs. Lopez purchased a house. On average, the value of the house has doubled every 6 years. If the house is worth \\$ 320,000 today, what did they pay for it 30 years ago?", "explanations":{ "correct":"To find out what Mr. and Mrs. Lopez paid for the house 30 years ago, we need to determine the initial value of the house and then calculate the growth over the 30-year period.\n\nGiven that the value of the house has doubled every 6 years, we can use the formula for exponential growth: \n\nFinal Value = Initial Value × (1 + Growth Rate)^Number of Periods\n\nLet's denote the initial value of the house as \"x\". After 30 years, the value of the house is \\$ 320,000. So we have:\n\n$ 320,000 = x × (1 + 100%)^(30/6)\n\nSimplifying the equation, we get:\n\n$ 320,000 = x × (2)^5\n\nNow, we can solve for x by dividing both sides of the equation by (2)^5:\n\n$ 320,000 / (2)^5 = x\n\n$ 320,000 / 32 = x\n\n$ 10,000 = x\n\nTherefore, Mr. and Mrs. Lopez paid \\$ 10,000 for the house 30 years ago.\n\nThe answer is B) \\$ 10,000." } }, { "answer":"A", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":10829, "question":"The function \\(h\\) is defined by \\(h(x) = 3x - 2\\) and the function \\(g\\) is defined by \\(g(x) = 3h(x )- 1\\). If \\(g(k) = 11\\), what is the value of \\(k\\)?", "explanations":{ "correct":"To find the value of \\(k\\), we need to substitute \\(g(k)\\) into the equation for \\(g(x)\\) and solve for \\(k\\).\n\nGiven that \\(g(x) = 3h(x) - 1\\) and \\(g(k) = 11\\), we can substitute \\(g(k)\\) into the equation:\n\n\\(11 = 3h(k) - 1\\)\n\nNow, we need to substitute \\(h(x)\\) into the equation for \\(h(k)\\):\n\n\\(11 = 3(3k - 2) - 1\\)\n\nSimplifying the equation:\n\n\\(11 = 9k - 6 - 1\\)\n\n\\(11 = 9k - 7\\)\n\nAdding 7 to both sides of the equation:\n\n\\(18 = 9k\\)\n\nDividing both sides of the equation by 9:\n\n\\(2 = k\\)\n\nTherefore, the value of \\(k\\) is 2.\n\nThe answer is A) 2." } }, { "answer":"D", "choices":{ "A":"\\(-4\\)", "B":"\\(-\\sqrt[6]{4}\\)", "C":"\\(\\sqrt[6]{4}\\)", "D":"4", "E":"\\(4^6\\)" }, "id":10831, "question":"If \\(x^6 + 4 = x^6 + w\\), then \\(w =\\)", "explanations":{ "correct":"To find the value of \\(w\\), we need to solve the equation \\(x^6 + 4 = x^6 + w\\).\n\nStep 1: Subtract \\(x^6\\) from both sides of the equation to isolate \\(w\\):\n\\(x^6 + 4 - x^6 = x^6 + w - x^6\\)\nSimplifying, we get:\n\\(4 = w\\)\n\nStep 2: We can see that \\(w\\) is equal to 4.\n\nTherefore, the answer is D) 4." } }, { "answer":"B", "choices":{ "A":"0", "B":"10", "C":"11", "D":"19", "E":"30" }, "id":10840, "question":"If set A is \\{3,5,7,11,19\\} and set B consists of all the even numbers between 1 and 11, how many elements are in the union of the two sets?", "explanations":{ "correct":"To find the union of two sets, we need to combine all the elements from both sets without any repetition. \n\nSet A consists of the elements {3, 5, 7, 11, 19}. \nSet B consists of all the even numbers between 1 and 11, which are {2, 4, 6, 8, 10}. \n\nTo find the union, we need to combine all the elements from both sets. \n\nCombining the elements from both sets, we have {2, 3, 4, 5, 6, 7, 8, 10, 11, 19}. \n\nThere are 10 elements in the union of the two sets. \n\nTherefore, the answer is B) 10." } }, { "answer":"C", "choices":{ "A":"9", "B":"18", "C":"27", "D":"36", "E":"48" }, "id":10842, "question":"If \\(m\\) and \\(n\\) are constants, what is the value of \\(n\\) if the equation \\((x + 9)(x + m) = x^2 + 4mx + n\\) is true for all values of \\(x\\)?", "explanations":{ "correct":"To find the value of \\(n\\), we need to expand the left side of the equation and compare it to the right side.\n\nExpanding \\((x + 9)(x + m)\\) using the distributive property, we get:\n\\(x^2 + mx + 9x + 9m\\)\n\nComparing this to the right side of the equation \\(x^2 + 4mx + n\\), we can see that the coefficients of the \\(x^2\\) terms, \\(x\\) terms, and constant terms must be equal.\n\nComparing the coefficients of the \\(x^2\\) terms:\nThe coefficient of \\(x^2\\) on the left side is 1.\nThe coefficient of \\(x^2\\) on the right side is also 1.\n\nComparing the coefficients of the \\(x\\) terms:\nThe coefficient of \\(x\\) on the left side is \\(m + 9\\).\nThe coefficient of \\(x\\) on the right side is \\(4m\\).\n\nSince these coefficients must be equal, we have the equation:\n\\(m + 9 = 4m\\)\n\nSolving for \\(m\\):\n\\(9 = 3m\\)\n\\(m = 3\\)\n\nNow, substituting \\(m = 3\\) back into the expanded equation, we have:\n\\(x^2 + 3x + 9x + 27 = x^2 + 12x + n\\)\n\nComparing the constant terms:\nThe constant term on the left side is \\(27\\).\nThe constant term on the right side is \\(n\\).\n\nSince these constant terms must be equal, we have the equation:\n\\(27 = n\\)\n\nTherefore, the value of \\(n\\) is 27.\n\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"5", "B":"5.5", "C":"6.5", "D":"23.5", "E":"1408" }, "id":10843, "question":"If an alarm beeps at a constant rate of 16 beeps per minute, how many minutes will it take to beep 88 times?", "explanations":{ "correct":"To find the number of minutes it will take for the alarm to beep 88 times, we can set up a proportion. \n\nWe know that the alarm beeps at a constant rate of 16 beeps per minute. So, we can set up the proportion:\n\n16 beeps / 1 minute = 88 beeps / x minutes\n\nTo solve for x, we can cross-multiply:\n\n16 * x = 88 * 1\n\n16x = 88\n\nNow, we can solve for x by dividing both sides of the equation by 16:\n\nx = 88 / 16\n\nx = 5.5\n\nTherefore, it will take 5.5 minutes for the alarm to beep 88 times.\n\nThe answer is B) 5.5." } }, { "answer":"A", "choices":{ "A":"45", "B":"50", "C":"53", "D":"55", "E":"60" }, "id":10850, "question":"A car going 40 miles per hour set out on an 80-mile trip at 9:00 A.M. Exactly 10 minutes later, a second car left from the same place and followed the same route. How fast, in miles per hour, was the second car going if it caught up with the first car at 10:30 A.M.?", "explanations":{ "correct":"To solve this problem, we need to find the speed of the second car. \n\nLet's start by finding the time it took for the first car to travel 80 miles. The first car traveled for 1 hour and 30 minutes (from 9:00 A.M. to 10:30 A.M.). \n\nNow, let's find the time it took for the second car to catch up with the first car. Since the second car left 10 minutes after the first car, it traveled for 1 hour and 20 minutes (from 9:10 A.M. to 10:30 A.M.). \n\nSince both cars traveled the same distance (80 miles), we can set up the following equation to find the speed of the second car:\n\nSpeed of first car * Time of first car = Speed of second car * Time of second car\n\n40 mph * 1.5 hours = Speed of second car * 1.33 hours\n\nSimplifying the equation, we have:\n\n60 = Speed of second car * 1.33\n\nDividing both sides of the equation by 1.33, we find:\n\nSpeed of second car = 60 / 1.33\n\nCalculating this, we get:\n\nSpeed of second car ≈ 45.11 mph\n\nRounding to the nearest whole number, the speed of the second car is approximately 45 mph.\n\nTherefore, the answer is A) 45." } }, { "answer":"A", "choices":{ "A":"10", "B":"12", "C":"14", "D":"16", "E":"18" }, "id":10852, "question":"If a = 5b + 3, how much greater is a when b = 9 than when b = 7?", "explanations":{ "correct":"To find out how much greater a is when b = 9 compared to when b = 7, we need to substitute the values of b into the equation a = 5b + 3.\n\nWhen b = 7:\na = 5(7) + 3\na = 35 + 3\na = 38\n\nWhen b = 9:\na = 5(9) + 3\na = 45 + 3\na = 48\n\nTo find the difference, we subtract the value of a when b = 7 from the value of a when b = 9:\n48 - 38 = 10\n\nTherefore, the answer is A) 10." } }, { "answer":"E", "choices":{ "A":"0", "B":"\\(-q\\)", "C":"1", "D":"s", "E":"q" }, "id":10854, "question":"If\\( q = \\frac { 1 } { s } \\) and \\(qs \\neq 0\\), what is the result of \\(\\frac { 1 + q } { 1 + s } \\) ?", "explanations":{ "correct":"To find the result of \\(\\frac{{1+q}}{{1+s}}\\), we need to substitute the given value of \\(q\\) into the expression.\n\nGiven: \\(q = \\frac{1}{s}\\)\n\nSubstituting this value into the expression, we get:\n\n\\(\\frac{{1+\\frac{1}{s}}}{{1+s}}\\)\n\nTo simplify this expression, we need to find a common denominator for the numerator. The common denominator is \\(s\\), so we multiply the numerator and denominator by \\(s\\):\n\n\\(\\frac{{s(1+\\frac{1}{s})}}{{s(1+s)}}\\)\n\nSimplifying the numerator, we get:\n\n\\(\\frac{{s+\\frac{s}{s}}}{{s(1+s)}}\\)\n\nThe term \\(\\frac{s}{s}\\) simplifies to 1, so the numerator becomes:\n\n\\(\\frac{{s+1}}{{s(1+s)}}\\)\n\nNow, we can cancel out the common factor of \\(s\\) in the numerator and denominator:\n\n\\(\\frac{{s+1}}{{s \\cdot 1 + s \\cdot s}}\\)\n\nSimplifying further, we get:\n\n\\(\\frac{{s+1}}{{s+s^2}}\\)\n\nTherefore, the result of \\(\\frac{{1+q}}{{1+s}}\\) is \\(\\frac{{s+1}}{{s+s^2}}\\).\n\nDouble-checking our reasoning, we can see that we followed the correct steps of substitution, simplification, and cancellation. We have also accounted for the condition \\(qs \\neq 0\\) in the given question.\n\nThe answer is E) \\(q\\)." } }, { "answer":"B", "choices":{ "A":"4", "B":"6", "C":"7", "D":"8", "E":"10" }, "id":10857, "question":"Set A contains the letters A, E, I, O, and U. Set B contains the letters A, B, C, D, and E. What is the difference between the number of elements in the union of the two sets and the number of elements in their intersection?", "explanations":{ "correct":"To find the difference between the number of elements in the union of two sets and the number of elements in their intersection, we need to first determine the number of elements in each set.\n\nSet A contains the letters A, E, I, O, and U, which means it has 5 elements.\n\nSet B contains the letters A, B, C, D, and E, which means it also has 5 elements.\n\nTo find the union of the two sets, we need to combine all the unique elements from both sets. In this case, the union of Set A and Set B would be A, E, I, O, U, B, C, and D. This gives us a total of 8 elements.\n\nTo find the intersection of the two sets, we need to identify the elements that are common to both sets. In this case, the intersection of Set A and Set B would be A and E. This gives us a total of 2 elements.\n\nNow, to find the difference between the number of elements in the union and the number of elements in the intersection, we subtract the number of elements in the intersection from the number of elements in the union: 8 - 2 = 6.\n\nTherefore, the answer is B) 6." } }, { "answer":"D", "choices":{ "A":"-1", "B":"1", "C":"10", "D":"100", "E":"1,000" }, "id":10860, "question":"If \\(\\sqrt { x } = \\frac { x } { 10 } \\) and \\(x > 0\\), what is the value of \\(x\\) ?", "explanations":{ "correct":"To find the value of \\(x\\), we can start by squaring both sides of the equation \\(\\sqrt{x} = \\frac{x}{10}\\). \n\n\\((\\sqrt{x})^2 = \\left(\\frac{x}{10}\\right)^2\\)\n\n\\(x = \\frac{x^2}{100}\\)\n\nNext, we can multiply both sides of the equation by 100 to eliminate the fraction:\n\n\\(100x = x^2\\)\n\nRearranging the equation, we have:\n\n\\(x^2 - 100x = 0\\)\n\nFactoring out \\(x\\), we get:\n\n\\(x(x - 100) = 0\\)\n\nFrom this equation, we can see that either \\(x = 0\\) or \\(x - 100 = 0\\). Since the question states that \\(x > 0\\), we can conclude that \\(x\\) must be equal to 100.\n\nTherefore, the answer is D) 100." } }, { "answer":"C", "choices":{ "A":"4", "B":"10", "C":"18", "D":"24", "E":"60" }, "id":10861, "question":"Each of the following is a factor of 120 EXCEPT", "explanations":{ "correct":"To determine which of the given numbers is not a factor of 120, we need to find the factors of 120 and then eliminate the options that are factors.\n\nThe factors of 120 are the numbers that divide evenly into 120 without leaving a remainder. We can find the factors by listing all the numbers that divide 120:\n\n1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120\n\nNow, let's go through each option and check if it is a factor of 120:\n\nA) 4: 120 ÷ 4 = 30, so 4 is a factor of 120.\nB) 10: 120 ÷ 10 = 12, so 10 is a factor of 120.\nC) 18: 120 ÷ 18 = 6.6667, so 18 is not a factor of 120.\nD) 24: 120 ÷ 24 = 5, so 24 is a factor of 120.\nE) 60: 120 ÷ 60 = 2, so 60 is a factor of 120.\n\nFrom the above analysis, we can see that the only option that is not a factor of 120 is C) 18.\n\nTherefore, the answer is C." } }, { "answer":"D", "choices":{ "A":"9", "B":"12", "C":"15", "D":"18", "E":"21" }, "id":10862, "question":"At a certain restaurant, there are 25 tables and each table has either 2 or 4 chairs. If there is a total of 86 chairs accompanying the 25 tables, how many tables have exactly 4 chairs?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume the number of tables with 2 chairs is x, and the number of tables with 4 chairs is y.\n\nAccording to the problem, there are 25 tables in total. So we can write the equation:\n\nx + y = 25 ---(1)\n\nWe are also given that the total number of chairs is 86. Since each table with 2 chairs contributes 2 chairs and each table with 4 chairs contributes 4 chairs, we can write another equation:\n\n2x + 4y = 86 ---(2)\n\nNow we have a system of two equations with two variables. We can solve this system to find the values of x and y.\n\nMultiplying equation (1) by 2, we get:\n\n2x + 2y = 50 ---(3)\n\nSubtracting equation (3) from equation (2), we eliminate x:\n\n2x + 4y - (2x + 2y) = 86 - 50\n2y = 36\ny = 18\n\nNow we can substitute the value of y into equation (1) to find x:\n\nx + 18 = 25\nx = 7\n\nTherefore, the number of tables with exactly 4 chairs is 18.\n\nThe answer is D) 18." } }, { "answer":"E", "choices":{ "A":"864", "B":"576", "C":"336", "D":"144", "E":"48" }, "id":10866, "question":"If a rectangular rug measures 72 inches by 96 inches, what is its area in square feet? (1 foot = 12 inches)", "explanations":{ "correct":"To find the area of the rectangular rug in square feet, we need to convert the measurements from inches to feet. \n\nGiven that 1 foot is equal to 12 inches, we can divide the length and width of the rug by 12 to convert them to feet.\n\nLength in feet = 72 inches / 12 = 6 feet\nWidth in feet = 96 inches / 12 = 8 feet\n\nNow, we can find the area of the rug by multiplying the length and width in feet.\n\nArea = Length x Width = 6 feet x 8 feet = 48 square feet\n\nTherefore, the area of the rectangular rug is 48 square feet.\n\nThe answer is E) 48." } }, { "answer":"A", "choices":{ "A":"-4", "B":"-2", "C":"-\\(\\frac { 3 } { 8 } \\)", "D":"0", "E":"\\(\\frac { 3 } { 8 } \\)" }, "id":10868, "question":"If \\(f(x) = 3x + 8\\), for what value of a is \\(f(a) = a\\)?", "explanations":{ "correct":"To find the value of \\(a\\) for which \\(f(a) = a\\), we need to substitute \\(a\\) into the function \\(f(x) = 3x + 8\\) and solve for \\(a\\).\n\nSubstituting \\(a\\) into the function, we have:\n\\(f(a) = 3a + 8\\)\n\nNow, we set this expression equal to \\(a\\) and solve for \\(a\\):\n\\(3a + 8 = a\\)\n\nTo isolate \\(a\\), we subtract \\(a\\) from both sides:\n\\(2a + 8 = 0\\)\n\nNext, we subtract 8 from both sides:\n\\(2a = -8\\)\n\nFinally, we divide both sides by 2 to solve for \\(a\\):\n\\(a = -4\\)\n\nTherefore, the value of \\(a\\) for which \\(f(a) = a\\) is -4.\n\nThe answer is A) -4." } }, { "answer":"B", "choices":{ "A":"3", "B":"7", "C":"11", "D":"14", "E":"17" }, "id":10869, "question":"After 21 students were added to a class, there were four times as many students as before. How many students were in the class before the addition?", "explanations":{ "correct":"Let's solve this problem step-by-step:\n\nLet's assume the number of students in the class before the addition is \"x\".\n\nAfter 21 students were added, the total number of students in the class became \"x + 21\".\n\nAccording to the problem, after the addition, there were four times as many students as before. So, we can set up the equation:\n\nx + 21 = 4x\n\nNow, let's solve for x:\n\nSubtract x from both sides of the equation:\n\n21 = 3x\n\nDivide both sides of the equation by 3:\n\n7 = x\n\nTherefore, the number of students in the class before the addition was 7.\n\nThe answer is B) 7." } }, { "answer":"A", "choices":{ "A":"ac", "B":"ab", "C":"bc", "D":"\\(\\frac{a}{c}\\)", "E":"\\(\\frac{c}{a}\\)" }, "id":10873, "question":"If a, b, and c are positive numbers and \\(abc = b^2\\), which of the following must equal b?", "explanations":{ "correct":"To find the value that must equal b, we can start by rearranging the given equation \\(abc = b^2\\) to isolate b. \n\nDividing both sides of the equation by b, we get \\(ac = b\\). \n\nTherefore, the value that must equal b is ac. \n\nThe answer is A) ac." } }, { "answer":"D", "choices":{ "A":"\\(-4\\)", "B":"10", "C":"18", "D":"26", "E":"36" }, "id":10875, "question":"If \\(a - 2b = 10\\), \\(b - 3c = 5\\), and \\(c = 1\\), what is the value of a?", "explanations":{ "correct":"To find the value of \\(a\\), we need to substitute the given values of \\(b\\) and \\(c\\) into the equation \\(a - 2b = 10\\).\n\nGiven:\n\\(a - 2b = 10\\) ...(1)\n\\(b - 3c = 5\\) ...(2)\n\\(c = 1\\) ...(3)\n\nSubstituting the value of \\(c\\) from equation (3) into equation (2), we get:\n\\(b - 3(1) = 5\\)\n\\(b - 3 = 5\\)\n\\(b = 5 + 3\\)\n\\(b = 8\\)\n\nNow, substituting the value of \\(b\\) and \\(c\\) into equation (1), we get:\n\\(a - 2(8) = 10\\)\n\\(a - 16 = 10\\)\n\\(a = 10 + 16\\)\n\\(a = 26\\)\n\nTherefore, the value of \\(a\\) is 26.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"\\(x^2\\)", "B":"\\(xy\\)", "C":"\\(xy^2\\)", "D":"\\((x + y)^2\\)", "E":"\\((x - y)^2\\)" }, "id":10877, "question":"If \\(x\\) and \\(y\\) are distinct prime numbers, which of the following has exactly four distinct factors?", "explanations":{ "correct":"To determine which of the given options has exactly four distinct factors, we need to consider the prime factorization of each option.\n\nA) \\(x^2\\) has the prime factorization \\(x \\cdot x\\). Since \\(x\\) is a distinct prime number, the only factors of \\(x^2\\) are 1, \\(x\\), \\(x^2\\), resulting in three distinct factors. Therefore, option A does not have exactly four distinct factors.\n\nB) \\(xy\\) has the prime factorization \\(x \\cdot y\\). Since \\(x\\) and \\(y\\) are distinct prime numbers, the factors of \\(xy\\) are 1, \\(x\\), \\(y\\), and \\(xy\\). This gives us four distinct factors. Therefore, option B has exactly four distinct factors.\n\nC) \\(xy^2\\) has the prime factorization \\(x \\cdot y \\cdot y\\). The factors of \\(xy^2\\) are 1, \\(x\\), \\(y\\), \\(y^2\\), \\(xy\\), and \\(xy^2\\). This gives us six distinct factors. Therefore, option C does not have exactly four distinct factors.\n\nD) \\((x + y)^2\\) can be expanded as \\(x^2 + 2xy + y^2\\). Since \\(x\\) and \\(y\\) are distinct prime numbers, the factors of \\((x + y)^2\\) are 1, \\(x\\), \\(y\\), \\(x^2\\), \\(xy\\), \\(y^2\\), \\(2xy\\), and \\((x + y)^2\\). This gives us eight distinct factors. Therefore, option D does not have exactly four distinct factors.\n\nE) \\((x - y)^2\\) can be expanded as \\(x^2 - 2xy + y^2\\). Since \\(x\\) and \\(y\\) are distinct prime numbers, the factors of \\((x - y)^2\\) are 1, \\(x\\), \\(y\\), \\(x^2\\), \\(xy\\), \\(y^2\\), \\(-2xy\\), and \\((x - y)^2\\). This gives us eight distinct factors. Therefore, option E does not have exactly four distinct factors.\n\nBased on the analysis above, the answer is B." } }, { "answer":"C", "choices":{ "A":"\\$ 3.00", "B":"\\$ 4.50", "C":"\\$ 6.00", "D":"\\$ 7.50", "E":"\\$ 30.00" }, "id":10880, "question":"The only way to purchase Brand X muffins is to buy one or more boxes that each contain 6 muffins. Each box costs \\$ 1.50. If Alejandro needs at least 20 muffins, what is the least amount of money he could spend?", "explanations":{ "correct":"To find the least amount of money Alejandro could spend, we need to determine the minimum number of boxes of muffins he needs to purchase.\n\nSince each box contains 6 muffins, we can divide the total number of muffins Alejandro needs (20) by the number of muffins in each box (6) to find the minimum number of boxes he needs to buy.\n\n20 ÷ 6 = 3 remainder 2\n\nThis means that Alejandro needs to buy at least 3 boxes of muffins to get 18 muffins. However, he still needs 2 more muffins to reach a total of 20.\n\nSince each box costs \\$ 1.50, Alejandro needs to buy an additional box to get the remaining 2 muffins. Therefore, he needs to spend an additional \\$ 1.50.\n\nThe total amount of money Alejandro needs to spend is \\$ 1.50 (for the additional box) + \\$ 4.50 (for the 3 boxes) = \\$ 6.00.\n\nTherefore, the least amount of money Alejandro could spend is \\$ 6.00.\n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"6", "B":"8", "C":"10", "D":"12", "E":"14" }, "id":10882, "question":"A bowl contains punch made from pineapple juice, orange juice, and seltzer, in ratio of 2 : 3 : 1, respectively. If the bowl contains 2 liters of seltzer, how many liters of punch does the bowl contain?", "explanations":{ "correct":"To find the total liters of punch in the bowl, we need to determine the total ratio of the pineapple juice, orange juice, and seltzer.\n\nThe given ratio is 2 : 3 : 1 for pineapple juice, orange juice, and seltzer, respectively. Since the ratio is in terms of liters, we can assume that the total ratio represents the total liters of punch in the bowl.\n\nLet's calculate the total ratio:\n2 + 3 + 1 = 6\n\nThe total ratio is 6, which means the bowl contains 6 liters of punch.\n\nSince the ratio of seltzer is 1, and we know that there are 2 liters of seltzer in the bowl, we can calculate the amount of punch by multiplying the total ratio by the amount of seltzer:\n6 * 2 = 12\n\nTherefore, the bowl contains 12 liters of punch.\n\nThe answer is D) 12." } }, { "answer":"C", "choices":{ "A":"\\(8x\\)", "B":"\\(4x\\)", "C":"\\(2x\\)", "D":"\\(x\\)", "E":"\\(\\frac { x } { 2 } \\)" }, "id":10883, "question":"If \\(10a + 8b = 4x\\), then, in terms of \\(x\\), \\(5a + 4b\\) =", "explanations":{ "correct":"To find the value of \\(5a + 4b\\) in terms of \\(x\\), we can start by dividing the given equation \\(10a + 8b = 4x\\) by 2. This gives us:\n\n\\(\\frac{{10a + 8b}}{2} = \\frac{{4x}}{2}\\)\n\nSimplifying both sides of the equation, we have:\n\n\\(5a + 4b = 2x\\)\n\nTherefore, the value of \\(5a + 4b\\) in terms of \\(x\\) is \\(2x\\).\n\nThe answer is C) \\(2x\\)." } }, { "answer":"C", "choices":{ "A":"10", "B":"15", "C":"20", "D":"25", "E":"30" }, "id":10885, "question":"C, D, and E are points on a line, in that order. If the length of segment CE is 30, and the length of segment CD is twice the length of DE, what is the length of segment CD?", "explanations":{ "correct":"Let's denote the length of segment DE as x. According to the given information, the length of segment CD is twice the length of DE, so it is 2x.\n\nWe also know that the length of segment CE is 30. Since segment CE is the sum of segment CD and segment DE, we can write the equation:\n\nCE = CD + DE\n\nSubstituting the values we have:\n\n30 = 2x + x\n\nCombining like terms:\n\n30 = 3x\n\nDividing both sides by 3:\n\n10 = x\n\nSo, the length of segment DE is 10. \n\nSince the length of segment CD is twice the length of DE, we can calculate it:\n\nCD = 2 * DE\nCD = 2 * 10\nCD = 20\n\nTherefore, the length of segment CD is 20.\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"24", "B":"30", "C":"36", "D":"40", "E":"48" }, "id":10888, "question":"A local store donated some markers to Mrs. Kettz's fourth-grade class. If each student takes 2 markers, there will be 16 markers left. If 4 students do not take any markers and the rest of the students take 6 markers, there will be no markers left. How many markers were donated to the class?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nLet's assume the total number of students in Mrs. Kettz's fourth-grade class is \"x\".\n\nAccording to the problem, if each student takes 2 markers, there will be 16 markers left. So, the number of markers left can be represented as:\n\n16 = Total number of markers - (Number of students * Number of markers each student takes)\n16 = Total number of markers - (x * 2)\n\nSimilarly, if 4 students do not take any markers and the rest of the students take 6 markers, there will be no markers left. So, the number of markers left can be represented as:\n\n0 = Total number of markers - (Number of students who take 6 markers * Number of markers each student takes) - (Number of students who do not take any markers * Number of markers each student takes)\n0 = Total number of markers - ((x - 4) * 6) - (4 * 0)\n\nNow, we have two equations:\n\n16 = Total number of markers - (x * 2)\n0 = Total number of markers - ((x - 4) * 6) - (4 * 0)\n\nLet's solve these equations simultaneously to find the value of \"x\" (total number of students) and \"Total number of markers\".\n\nFrom the first equation, we can rewrite it as:\n\nTotal number of markers = 16 + (x * 2)\n\nSubstituting this value in the second equation, we get:\n\n0 = (16 + (x * 2)) - ((x - 4) * 6) - (4 * 0)\n0 = 16 + 2x - 6x + 24\n0 = 40 - 4x\n\nRearranging the equation, we get:\n\n4x = 40\nx = 40/4\nx = 10\n\nNow, substituting the value of \"x\" in the first equation to find the total number of markers:\n\nTotal number of markers = 16 + (10 * 2)\nTotal number of markers = 16 + 20\nTotal number of markers = 36\n\nTherefore, the answer is C) 36." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 1 } { 12 } \\)", "B":"\\(\\frac { 1 } { 6 } \\)", "C":"\\(\\frac { 1 } { 3 } \\)", "D":"\\(\\frac { 1 } { 2 } \\)", "E":"\\(\\frac { 2 } { 3 } \\)" }, "id":10889, "question":"If \\(x\\) is \\(\\frac { 2 } { 5 } \\)of \\(y\\) and \\(y\\) is \\(\\frac { 5 } { 12 } \\)of \\(z\\), what is the value of \\(\\frac { x } { z } \\)?", "explanations":{ "correct":"To find the value of \\(\\frac{x}{z}\\), we need to express \\(x\\) and \\(y\\) in terms of \\(z\\).\n\nGiven that \\(x\\) is \\(\\frac{2}{5}\\) of \\(y\\), we can write this as:\n\n\\(x = \\frac{2}{5}y\\)\n\nSimilarly, given that \\(y\\) is \\(\\frac{5}{12}\\) of \\(z\\), we can write this as:\n\n\\(y = \\frac{5}{12}z\\)\n\nNow, substitute the value of \\(y\\) in terms of \\(z\\) into the equation for \\(x\\):\n\n\\(x = \\frac{2}{5}\\left(\\frac{5}{12}z\\)\\)\n\nSimplifying this expression, we get:\n\n\\(x = \\frac{1}{6}z\\)\n\nTherefore, the value of \\(\\frac{x}{z}\\) is \\(\\frac{1}{6}\\).\n\nThe answer is B) \\(\\frac{1}{6}\\)." } }, { "answer":"E", "choices":{ "A":"If an integer is positive, it is in set T.", "B":"If an integer is negative, it is in set T.", "C":"All integers in set T are positive.", "D":"All integers in set T are negative.", "E":"Not all integers in set T are positive." }, "id":10891, "question":"Some integers in set T are negative. If the statement above is true, which of the following must also be true?", "explanations":{ "correct":"To determine which statement must also be true if some integers in set T are negative, we need to carefully analyze the given information.\n\nThe statement tells us that \"some integers in set T are negative.\" This means that there is at least one negative integer in set T, but it does not imply that all integers in set T are negative.\n\nLet's evaluate each option:\n\nA) If an integer is positive, it is in set T.\nThis statement cannot be concluded from the given information. We only know that some integers in set T are negative, but there is no information about positive integers in set T. Therefore, option A cannot be determined as true.\n\nB) If an integer is negative, it is in set T.\nThis statement aligns with the given information. The statement explicitly states that some integers in set T are negative. Therefore, if an integer is negative, it must be in set T. This statement is true based on the given information.\n\nC) All integers in set T are positive.\nThis statement contradicts the given information. The statement explicitly states that some integers in set T are negative, which means that not all integers in set T can be positive. Therefore, option C is not true.\n\nD) All integers in set T are negative.\nThis statement cannot be concluded from the given information. The statement only states that some integers in set T are negative, but it does not imply that all integers in set T are negative. Therefore, option D cannot be determined as true.\n\nE) Not all integers in set T are positive.\nThis statement aligns with the given information. The statement explicitly states that some integers in set T are negative, which means that not all integers in set T can be positive. Therefore, option E is true based on the given information.\n\\(\\newline\\)In conclusion, the statement that must also be true if some integers in set T are negative is: \"Not all integers in set T are positive.\"\n\nThe answer is E." } }, { "answer":"C", "choices":{ "A":"2", "B":"8", "C":"18", "D":"36", "E":"72" }, "id":10892, "question":"If \\(\\frac{a}{b} = 4\\), \\(a = 8c\\), and \\(c = 9\\), what is the value of b?", "explanations":{ "correct":"To find the value of b, we can use the given information and solve for b step-by-step.\n\nGiven:\n\\(\\frac{a}{b} = 4\\)\n\\(a = 8c\\)\n\\(c = 9\\)\n\nSubstitute the value of c into the equation \\(a = 8c\\):\n\\(a = 8 \\cdot 9\\)\n\\(a = 72\\)\n\nNow, substitute the value of a into the equation \\(\\frac{a}{b} = 4\\):\n\\(\\frac{72}{b} = 4\\)\n\nTo solve for b, we can cross-multiply:\n\\(72 = 4b\\)\n\nDivide both sides of the equation by 4 to isolate b:\n\\(\\frac{72}{4} = b\\)\n\\(18 = b\\)\n\nTherefore, the value of b is 18.\n\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"4", "B":"5", "C":"6", "D":"7", "E":"8" }, "id":10898, "question":"Admission to the local movie theater for a groups of 12 people is \\$ 3 for each child and \\$ 7 for each adult. If the group pays \\$ 64, which of the following could be the number of children in the group?", "explanations":{ "correct":"Let's solve this problem step-by-step.\n\nLet's assume the number of children in the group is \"x\". Since there are 12 people in total, the number of adults in the group would be 12 - x.\n\nThe cost for each child is \\$ 3, so the total cost for the children would be 3x.\nThe cost for each adult is \\$ 7, so the total cost for the adults would be 7(12 - x).\n\nAccording to the problem, the group pays \\$ 64 in total. Therefore, we can set up the following equation:\n\n3x + 7(12 - x) = 64\n\nNow, let's solve for x:\n\n3x + 84 - 7x = 64\n-4x + 84 = 64\n-4x = -20\nx = 5\n\nSo, the number of children in the group could be 5.\n\nThe answer is B) 5." } }, { "answer":"A", "choices":{ "A":"10", "B":"16", "C":"24", "D":"36", "E":"72" }, "id":10900, "question":"If two coins are selected from a collection consisting of 2 pennies, 2 nickels, 2 dimes, and 2 quarters, how many different sums are possible?", "explanations":{ "correct":"To find the number of different sums possible when two coins are selected from the given collection, we need to consider all possible combinations of coins.\n\nFirst, let's list all the possible combinations of coins:\n1. Penny + Penny\n2. Penny + Nickel\n3. Penny + Dime\n4. Penny + Quarter\n5. Nickel + Nickel\n6. Nickel + Dime\n7. Nickel + Quarter\n8. Dime + Dime\n9. Dime + Quarter\n10. Quarter + Quarter\n\nNow, let's calculate the sums for each combination:\n1. 1 cent + 1 cent = 2 cents\n2. 1 cent + 5 cents = 6 cents\n3. 1 cent + 10 cents = 11 cents\n4. 1 cent + 25 cents = 26 cents\n5. 5 cents + 5 cents = 10 cents\n6. 5 cents + 10 cents = 15 cents\n7. 5 cents + 25 cents = 30 cents\n8. 10 cents + 10 cents = 20 cents\n9. 10 cents + 25 cents = 35 cents\n10. 25 cents + 25 cents = 50 cents\n\nFrom the above calculations, we can see that there are 10 different sums possible.\n\nTherefore, the answer is A) 10." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { 1000 } { r } + k\\)", "B":"\\(\\frac { 1000 } { r } + tk\\)", "C":"\\(\\frac { 1000 - k - t } { r } \\) ", "D":"\\(\\frac { 1000 - k } { r } + t\\) ", "E":"\\(\\frac { 1000 + k } { r } + kt\\)" }, "id":10901, "question":"The Maxim Telephone Company charges \\(k\\) cents for the first \\(t\\) minutes of a call and charges for any additional time at the rate of \\(r\\) cents per minute. If a certain customer pays \\$ 10, which of the following could be the length of that phone call in minutes?", "explanations":{ "correct":"To determine the length of the phone call in minutes, we need to consider the charges for the first \\(t\\) minutes and any additional time beyond that.\n\nLet's break down the information given:\n- The Maxim Telephone Company charges \\(k\\) cents for the first \\(t\\) minutes of a call.\n- Any additional time beyond \\(t\\) minutes is charged at the rate of \\(r\\) cents per minute.\n- The customer pays \\$ 10 for the phone call.\n\nTo find the length of the phone call, we can set up an equation based on the given information.\n\nThe cost of the first \\(t\\) minutes is \\(k\\) cents per minute, so the total cost for the first \\(t\\) minutes is \\(kt\\) cents.\n\nAny additional time beyond \\(t\\) minutes is charged at the rate of \\(r\\) cents per minute. Let's assume the length of the additional time is \\(x\\) minutes. The cost for the additional time would be \\(rx\\) cents.\n\nThe total cost of the phone call is \\$ 10, which is equal to 1000 cents. Therefore, we can set up the equation:\n\n\\(kt + rx = 1000\\)\n\nNow, let's rearrange the equation to solve for \\(x\\), the length of the additional time:\n\n\\(rx = 1000 - kt\\)\n\nDividing both sides of the equation by \\(r\\), we get:\n\n\\(x = \\frac{{1000 - kt}}{r}\\)\n\nSo, the length of the phone call in minutes is \\(\\frac{{1000 - kt}}{r}\\).\n\nTherefore, the answer is D) \\(\\frac{{1000 - k}}{r} + t\\)." } }, { "answer":"E", "choices":{ "A":"\\(x < 0\\)", "B":"\\(x > 0\\)", "C":"\\(-3 < x < 0\\)", "D":"\\(-2 < x < 1\\)", "E":"\\(-1 < x < 2\\)" }, "id":10904, "question":"If \\(|x - \\frac { 1 } { 2 } | < \\frac { 3 } { 2 } \\), which of the following must be true?", "explanations":{ "correct":"To solve the inequality \\(|x - \\frac{1}{2}| < \\frac{3}{2}\\), we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.\n\nCase 1: \\(x - \\frac{1}{2} > 0\\)\\(\\newline\\)In this case, the absolute value expression becomes \\(x - \\frac{1}{2}\\). So we have \\(x - \\frac{1}{2} < \\frac{3}{2}\\). Adding \\(\\frac{1}{2}\\) to both sides gives us \\(x < 2\\).\n\nCase 2: \\(x - \\frac{1}{2} < 0\\)\\(\\newline\\)In this case, the absolute value expression becomes \\(-x + \\frac{1}{2}\\). So we have \\(-x + \\frac{1}{2} < \\frac{3}{2}\\). Subtracting \\(\\frac{1}{2}\\) from both sides gives us \\(-x < 1\\). Multiplying both sides by -1 (and flipping the inequality sign) gives us \\(x > -1\\).\n\nCombining the results from both cases, we have \\(-1 < x < 2\\). Therefore, the correct answer is E) \\(-1 < x < 2\\)." } }, { "answer":"E", "choices":{ "A":"-2", "B":"-1", "C":"0", "D":"1", "E":"2" }, "id":10912, "question":"If \\(f(x) = |x| + 2\\), which of the following is the LEAST possible value of \\(f(x)\\) ?", "explanations":{ "correct":"To find the least possible value of \\(f(x)\\), we need to determine the smallest possible value for \\(|x|\\) and then add 2 to it.\n\nSince \\(|x|\\) represents the absolute value of \\(x\\), it will always be non-negative. This means that the smallest possible value for \\(|x|\\) is 0, which occurs when \\(x = 0\\).\n\nTherefore, the least possible value of \\(f(x)\\) is \\(0 + 2 = 2\\).\n\nThe answer is E) 2." } }, { "answer":"E", "choices":{ "A":"\\(-6\\)", "B":"\\(-3\\)", "C":"0", "D":"2", "E":"3" }, "id":10916, "question":"If \\(g(t) = 2t - 6\\), then at what value of t does the graph of g(t) cross the x-axis?", "explanations":{ "correct":"To find the value of \\(t\\) at which the graph of \\(g(t)\\) crosses the x-axis, we need to find the value of \\(t\\) that makes \\(g(t) = 0\\). \n\nGiven that \\(g(t) = 2t - 6\\), we can set \\(g(t)\\) equal to zero and solve for \\(t\\):\n\n\\(2t - 6 = 0\\)\n\nAdding 6 to both sides:\n\n\\(2t = 6\\)\n\nDividing both sides by 2:\n\n\\(t = 3\\)\n\nTherefore, the graph of \\(g(t)\\) crosses the x-axis at \\(t = 3\\).\n\nThe answer is E) 3." } }, { "answer":"B", "choices":{ "A":"\\(a^2 + b^2\\)", "B":"\\((a + b)^2\\)", "C":"\\(a^2 + b\\)", "D":"\\(2a + 2b\\)", "E":"\\(a + b^2\\)" }, "id":10917, "question":"Which of the following represents \"the square of the sum of a and b\"?", "explanations":{ "correct":"To find the expression that represents \"the square of the sum of a and b,\" we need to break down the given phrase and simplify it step-by-step.\n\n\"The square of the sum of a and b\" means we need to square the sum of a and b. \n\nThe sum of a and b is represented by \\(a + b\\). \n\nTo square \\(a + b\\), we multiply it by itself: \\((a + b) \\times (a + b)\\).\n\nExpanding this expression using the distributive property, we get: \\(a \\times a + a \\times b + b \\times a + b \\times b\\).\n\nSimplifying further, we have: \\(a^2 + ab + ab + b^2\\).\n\nCombining like terms, we get: \\(a^2 + 2ab + b^2\\).\n\nTherefore, the expression that represents \"the square of the sum of a and b\" is \\((a + b)^2\\).\n\nThe answer is B." } }, { "answer":"C", "choices":{ "A":"-120", "B":"-15", "C":"0", "D":"15", "E":"120" }, "id":10918, "question":"What is the absolute value of the product of all the integers from -6 to 3, inclusive?", "explanations":{ "correct":"To find the absolute value of the product of all the integers from -6 to 3, inclusive, we need to multiply all the integers together and then take the absolute value of the result.\n\nFirst, let's list out all the integers from -6 to 3: -6, -5, -4, -3, -2, -1, 0, 1, 2, 3.\n\nNext, let's multiply all these integers together: (-6) * (-5) * (-4) * (-3) * (-2) * (-1) * 0 * 1 * 2 * 3.\n\nWhen we multiply a negative number with a positive number, the result is always negative. Since there are six negative numbers in the list, the product will be negative.\n\nAlso, any number multiplied by 0 is always 0. Therefore, the entire product will be 0.\n\nFinally, we need to take the absolute value of 0, which is 0.\n\nTherefore, the absolute value of the product of all the integers from -6 to 3, inclusive, is 0.\n\nThe answer is C) 0." } }, { "answer":"C", "choices":{ "A":"\\$ 1200", "B":"\\$ 1500", "C":"\\$ 1600", "D":"\\$ 2400", "E":"\\$ 2500" }, "id":10921, "question":"Last year Jose sold a painting for \\$ 2000. If he made a 25% profit on the sale, how much had he paid for the painting?", "explanations":{ "correct":"To find out how much Jose had paid for the painting, we need to determine the original cost of the painting before the 25% profit was added.\n\nLet's assume the original cost of the painting is x dollars.\n\nSince Jose made a 25% profit on the sale, the selling price of the painting is 100% + 25% = 125% of the original cost.\n\nWe can set up the equation:\n125% of x = \\$ 2000\n\nTo solve for x, we need to convert 125% to decimal form by dividing it by 100:\n1.25x = \\$ 2000\n\nNow, we can solve for x by dividing both sides of the equation by 1.25:\nx = \\$ 2000 / 1.25\nx = \\$ 1600\n\nTherefore, Jose had paid \\$ 1600 for the painting.\n\nThe answer is C) \\$ 1600." } }, { "answer":"B", "choices":{ "A":"2", "B":"3", "C":"5", "D":"6", "E":"12" }, "id":10927, "question":"If \\(\\frac{q}{r} = 4\\), \\(\\frac{q}{s} = 6\\), and \\(s = 2\\), what is the value of r?", "explanations":{ "correct":"To find the value of \\(r\\), we can use the given information and solve for \\(r\\) step-by-step.\n\nGiven:\n\\(\\frac{q}{r} = 4\\)\n\\(\\frac{q}{s} = 6\\)\n\\(s = 2\\)\n\nFirst, let's substitute the value of \\(s\\) into the second equation:\n\\(\\frac{q}{2} = 6\\)\n\nTo solve for \\(q\\), we can multiply both sides of the equation by 2:\n\\(q = 6 \\times 2\\)\n\\(q = 12\\)\n\nNow that we know the value of \\(q\\), we can substitute it into the first equation:\n\\(\\frac{12}{r} = 4\\)\n\nTo solve for \\(r\\), we can multiply both sides of the equation by \\(r\\):\n\\(12 = 4r\\)\n\nNext, divide both sides of the equation by 4:\n\\(\\frac{12}{4} = \\frac{4r}{4}\\)\n\\(3 = r\\)\n\nTherefore, the value of \\(r\\) is 3.\n\nThe answer is B." } }, { "answer":"A", "choices":{ "A":"9:8", "B":"5:7", "C":"4:5", "D":"3:4", "E":"2:3" }, "id":10932, "question":"The spare change on a dresser is composed of pennies, nickels, and dimes. If the ratio of pennies to nickels is 2:3 and the ratio of pennies to dimes is 3:4, what is the ratio of nickels to dimes?", "explanations":{ "correct":"To find the ratio of nickels to dimes, we need to compare the ratios of pennies to nickels and pennies to dimes. \n\nGiven that the ratio of pennies to nickels is 2:3, we can assume that for every 2 pennies, there are 3 nickels. \n\nSimilarly, the ratio of pennies to dimes is 3:4, meaning for every 3 pennies, there are 4 dimes. \n\nTo compare the ratio of nickels to dimes, we need to find a common factor between the two ratios. In this case, the common factor is the number of pennies. \n\nSince the ratio of pennies to nickels is 2:3, and the ratio of pennies to dimes is 3:4, we can multiply the ratio of pennies to nickels by 3 and the ratio of pennies to dimes by 2 to make the number of pennies equal in both ratios. \n\nMultiplying the ratio of pennies to nickels by 3, we get 6:9. \nMultiplying the ratio of pennies to dimes by 2, we get 6:8. \n\nNow, we can compare the ratio of nickels to dimes. \nThe ratio of nickels to dimes is 9:8. \n\nTherefore, the answer is A) 9:8." } }, { "answer":"D", "choices":{ "A":"6", "B":"5", "C":"4", "D":"3", "E":"2" }, "id":10935, "question":"If the perimeter of a regular polygon is 21, which of the following could be the length of one side of the polygon?", "explanations":{ "correct":"To find the length of one side of a regular polygon, we need to divide the perimeter by the number of sides. In this case, the perimeter is given as 21.\n\nLet's consider each option and see if it could be the length of one side of the polygon:\n\nA) If one side of the polygon is 6, then the number of sides would be 21/6 = 3.5, which is not a whole number. Since a regular polygon must have a whole number of sides, option A is not valid.\n\nB) If one side of the polygon is 5, then the number of sides would be 21/5 = 4.2, which is not a whole number. Therefore, option B is not valid.\n\nC) If one side of the polygon is 4, then the number of sides would be 21/4 = 5.25, which is not a whole number. Hence, option C is not valid.\n\nD) If one side of the polygon is 3, then the number of sides would be 21/3 = 7, which is a whole number. Therefore, option D is a valid possibility.\n\nE) If one side of the polygon is 2, then the number of sides would be 21/2 = 10.5, which is not a whole number. Thus, option E is not valid.\n\nBased on the reasoning above, the only valid option is D. Therefore, the answer is D." } }, { "answer":"E", "choices":{ "A":"I only", "B":"III only", "C":"I and II only", "D":"II and III only", "E":"I, II and III" }, "id":10938, "question":"Points A, B and C are three vertices of a triangle. The location of point X is such that AX = BX. Which of the following could be true? \\(\\newline\\)I. X is on \\(\\overline { AB } \\) \\(\\newline\\)II. X is inside \\(\\triangle { ABC } \\) \\(\\newline\\)III. X is outside \\(\\triangle { ABC } \\)", "explanations":{ "correct":"To determine the possible locations of point X, we need to consider the given information that AX = BX. \n\\(\\newline\\)I. X is on \\(\\overline{AB}\\):\\(\\newline\\)If AX = BX, it means that point X is equidistant from points A and B. This implies that point X lies on the perpendicular bisector of segment AB. Therefore, option I is possible.\n\\(\\newline\\)II. X is inside \\(\\triangle{ABC}\\):\\(\\newline\\)If point X is inside \\(\\triangle{ABC}\\), it must lie within the region enclosed by the three sides of the triangle. Since AX = BX, point X must lie on the perpendicular bisector of segment AB. This perpendicular bisector will intersect the interior of \\(\\triangle{ABC}\\) at a point. Therefore, option II is possible.\n\\(\\newline\\)III. X is outside \\(\\triangle{ABC}\\):\\(\\newline\\)If point X is outside \\(\\triangle{ABC}\\), it must lie outside the region enclosed by the three sides of the triangle. Since AX = BX, point X must lie on the perpendicular bisector of segment AB. This perpendicular bisector will extend beyond the triangle and intersect the exterior of \\(\\triangle{ABC}\\) at a point. Therefore, option III is possible.\n\nBased on the above reasoning, all three options I, II, and III are possible. Therefore, the answer is E) I, II, and \\(\\newline\\)III." } }, { "answer":"C", "choices":{ "A":"0", "B":"\\(\\frac { 1 } { 2 } \\)", "C":"1", "D":"b", "E":"\\(\\frac { 1 } { b } \\)" }, "id":10940, "question":"If \\(ab = k\\), \\(b = ka\\),and \\(k \\neq 0\\), which of the following could be equal to \\(a\\)?", "explanations":{ "correct":"To find the value of \\(a\\), we can substitute the given equations into each other and solve for \\(a\\).\n\nGiven: \\(ab = k\\) and \\(b = ka\\)\n\nSubstituting the value of \\(b\\) from the second equation into the first equation, we get:\n\n\\(a(ka) = k\\)\n\nSimplifying, we have:\n\n\\(a^2k = k\\)\n\nDividing both sides of the equation by \\(k\\), we get:\n\n\\(a^2 = 1\\)\n\nTaking the square root of both sides, we have:\n\n\\(a = \\pm 1\\)\n\nTherefore, the possible values of \\(a\\) are 1 and -1.\n\nLooking at the answer choices:\n\nA) 0: This is not a possible value for \\(a\\) since we found that \\(a\\) can be 1 or -1, not 0.\n\nB) \\(\\frac { 1 } { 2 } \\): This is not a possible value for \\(a\\) since we found that \\(a\\) can be 1 or -1, not \\(\\frac { 1 } { 2 } \\).\n\nC) 1: This is a possible value for \\(a\\) since we found that \\(a\\) can be 1.\n\nD) b: This is not a possible value for \\(a\\) since \\(a\\) and \\(b\\) are not necessarily equal.\n\nE) \\(\\frac { 1 } { b } \\): This is not a possible value for \\(a\\) since we found that \\(a\\) can be 1 or -1, not \\(\\frac { 1 } { b } \\).\n\nTherefore, the answer is C) 1." } }, { "answer":"E", "choices":{ "A":"5", "B":"6", "C":"7", "D":"7.5", "E":"8" }, "id":10941, "question":"If the length of one side of a triangle is 6 and the length of another side is 2, which of the following CANNOT be the length of the third side of the triangle?", "explanations":{ "correct":"To determine which length cannot be the length of the third side of the triangle, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.\n\nLet's consider the given lengths:\n- One side has a length of 6.\n- Another side has a length of 2.\n\nTo find the range of possible lengths for the third side, we can use the triangle inequality theorem:\n\n6 + 2 > third side length\n8 > third side length\n\nSo, the third side must be less than 8 units long.\n\nNow, let's evaluate each answer choice:\n\nA) 5: 5 is less than 8, so it is a possible length for the third side.\nB) 6: 6 is equal to one of the given side lengths, so it is a possible length for the third side.\nC) 7: 7 is less than 8, so it is a possible length for the third side.\nD) 7.5: 7.5 is less than 8, so it is a possible length for the third side.\nE) 8: 8 is equal to the sum of the two given side lengths, which violates the triangle inequality theorem. Therefore, 8 cannot be the length of the third side.\n\nBased on the reasoning above, the answer is E." } }, { "answer":"A", "choices":{ "A":"\\(r < s\\)", "B":"\\(s < r\\)", "C":"\\(r > 1\\)", "D":"\\(s > 1\\)", "E":"r and s can be any numbers as long as \\(r \\neq s\\)" }, "id":10951, "question":"If r and s are positive numbers satisfying the inequality \\(\\frac { r } { s } < \\frac { r + 1 } { s + 1 } \\), which of the following must be true?", "explanations":{ "correct":"To determine which of the given options must be true, let's start by simplifying the inequality:\n\n\\(\\frac{r}{s} < \\frac{r+1}{s+1}\\)\n\nTo eliminate the fractions, we can cross-multiply:\n\n\\(r(s+1) < s(r+1)\\)\n\nExpanding both sides of the inequality:\n\n\\(rs + r < rs + s\\)\n\nThe \\(rs\\) terms cancel out:\n\n\\(r < s\\)\n\nTherefore, the inequality simplifies to \\(r < s\\). This means that option A) \\(r < s\\) must be true.\n\nThe answer is A)." } }, { "answer":"C", "choices":{ "A":"25", "B":"34", "C":"57", "D":"87", "E":"It is not possible for Adam to do this." }, "id":10960, "question":"So far this year, Adam has played 30 games of chess and has won only 6 of them. What is the minimum number of additional games he must play, given that he is sure to lose at least one-third of them, so that for the year he will have won more games than he lost?", "explanations":{ "correct":"To find the minimum number of additional games Adam must play in order to have won more games than he lost for the year, we need to determine the number of games he can afford to lose.\n\nAdam has played 30 games of chess so far and has won 6 of them. This means he has lost 30 - 6 = 24 games.\n\nWe are told that Adam is sure to lose at least one-third of the additional games he plays. Let's assume he plays x additional games. Since he is sure to lose at least one-third of them, he will lose at least (1/3)x games.\n\nTo ensure that Adam has won more games than he lost, we need to set up an inequality. The number of games he has won should be greater than the number of games he has lost.\n\n6 + (1/3)x > 24 + (1/3)x\n\nSimplifying the inequality, we get:\n\n(1/3)x > 18\n\nMultiplying both sides of the inequality by 3, we get:\n\nx > 54\n\nThis means Adam must play more than 54 additional games to have won more games than he lost for the year.\n\nLooking at the answer choices, the minimum number of additional games Adam must play is 57.\n\nTherefore, the answer is C) 57." } }, { "answer":"B", "choices":{ "A":"\\(2c\\)", "B":"\\(\\frac { 2c } { 3 } \\)", "C":"\\(\\frac { a + b } { c } \\)", "D":"\\(\\frac { 3c } { a + b } \\)", "E":"\\(c - (a + b)\\)" }, "id":10964, "question":"If the sum of \\(a\\) and \\(b\\) is \\(c\\), what is the average (arithmetic mean) of \\(a\\), \\(b\\), and \\(c\\)?", "explanations":{ "correct":"To find the average (arithmetic mean) of \\(a\\), \\(b\\), and \\(c\\), we need to add up the three numbers and then divide the sum by 3.\n\nThe sum of \\(a\\) and \\(b\\) is given as \\(c\\). So, we have \\(a + b = c\\).\n\nTo find the average, we add up \\(a\\), \\(b\\), and \\(c\\):\n\\(a + b + c = c + c = 2c\\).\n\nNow, we divide the sum by 3 to find the average:\n\\(\\frac{2c}{3}\\).\n\nTherefore, the answer is B) \\(\\frac{2c}{3}\\)." } }, { "answer":"C", "choices":{ "A":"\\(b + 3\\)", "B":"\\(b + 12\\)", "C":"3b", "D":"\\(3b + 3\\)", "E":"\\(3b + 12\\)" }, "id":10965, "question":"If \\(2x + 4 = b\\), then \\(6x + 12 =\\)", "explanations":{ "correct":"To find the value of \\(6x + 12\\) when \\(2x + 4 = b\\), we need to substitute the value of \\(b\\) into the expression.\n\nGiven that \\(2x + 4 = b\\), we can solve for \\(x\\) by subtracting 4 from both sides of the equation:\n\\[2x + 4 - 4 = b - 4\\]\n\\[2x = b - 4\\]\nNext, we divide both sides of the equation by 2 to isolate \\(x\\):\n\\[\\frac{2x}{2} = \\frac{b - 4}{2}\\]\n\\[x = \\frac{b}{2} - 2\\]\n\nNow, we substitute this value of \\(x\\) into the expression \\(6x + 12\\):\n\\[6\\left(\\frac{b}{2} - 2\\right) + 12\\]\nUsing the distributive property, we can simplify this expression:\n\\[3b - 12 + 12\\]\nThe -12 and +12 cancel each other out, leaving us with:\n\\[3b\\]\n\nTherefore, the answer is C) \\(3b\\)." } }, { "answer":"B", "choices":{ "A":"10", "B":"12", "C":"33", "D":"248", "E":"468.75" }, "id":10977, "question":"The force needed to stretch a spring varies directly with the distance the spring is stretched from its equilibrium position. If 50 pounds of force stretch a spring 8 inches from equilibrium, how much, in inches, will the spring be stretched by a force of 75 pounds?", "explanations":{ "correct":"To solve this problem, we can set up a proportion to find the unknown distance the spring will be stretched by a force of 75 pounds.\n\nWe know that the force needed to stretch the spring varies directly with the distance the spring is stretched. This means that the force and distance have a constant ratio.\n\nLet's set up the proportion using the given information:\n50 pounds / 8 inches = 75 pounds / x inches\n\nTo solve for x, we can cross-multiply:\n50x = 8 * 75\n\nSimplifying the right side:\n50x = 600\n\nDividing both sides by 50:\nx = 600 / 50\nx = 12\n\nTherefore, the spring will be stretched by 12 inches when a force of 75 pounds is applied.\n\nThe answer is B) 12." } }, { "answer":"A", "choices":{ "A":"\\(n = 2d - 10\\)", "B":"\\(n = 2(d - 10)\\)", "C":"\\(n = 10 - 2d\\)", "D":"\\(n = 2(10 - d)\\)", "E":"\\(n = 10 - (d +2)\\)" }, "id":10978, "question":"Nora has 10 fewer than twice the number of CDs that Deborah has. If n represents the number of Nora's CDs, and d represents the number of Deborah's CDs, which of the following is a correct equation relating n and d ?", "explanations":{ "correct":"To solve this problem, we need to carefully analyze the given information and translate it into an equation.\n\nThe problem states that Nora has 10 fewer than twice the number of CDs that Deborah has. Let's break this down step-by-step:\n\n1. Let's start by representing the number of Nora's CDs as 'n' and the number of Deborah's CDs as 'd'.\n\n2. The problem states that Nora has 10 fewer than twice the number of CDs that Deborah has. This can be translated into the equation: n = 2d - 10.\n\nNow, let's compare this equation to the answer choices:\n\nA) \\(n = 2d - 10\\) - This matches the equation we derived from the problem statement.\n\nB) \\(n = 2(d - 10)\\) - This equation represents Nora having twice the number of CDs that Deborah has, but it does not account for Nora having 10 fewer CDs.\n\nC) \\(n = 10 - 2d\\) - This equation represents Nora having 10 CDs and Deborah having twice as many CDs, but it does not account for Nora having 10 fewer CDs.\n\nD) \\(n = 2(10 - d)\\) - This equation represents Nora having twice the number of CDs that Deborah has, but it does not account for Nora having 10 fewer CDs.\n\nE) \\(n = 10 - (d +2)\\) - This equation represents Nora having 10 CDs and Deborah having 2 fewer CDs, but it does not account for Nora having twice the number of CDs that Deborah has.\n\nFrom our analysis, we can conclude that the correct equation relating n and d is A) \\(n = 2d - 10\\).\n\nThe answer is A." } }, { "answer":"D", "choices":{ "A":"r > 0", "B":"s < 0", "C":"t < 0", "D":"su > 0", "E":"st > 0" }, "id":10982, "question":"If \\(r^2s^3t^4u > 0\\), then which of the following must be true?", "explanations":{ "correct":"To determine which statement must be true when \\(r^2s^3t^4u > 0\\), we need to analyze the given inequality step-by-step.\n\nFirst, we know that \\(r^2\\) is always positive since any number squared is positive. Therefore, the sign of \\(r\\) does not affect the inequality. So, statement A) r > 0 is not necessarily true.\n\nNext, let's consider \\(s^3\\). Since \\(s\\) is raised to an odd power, the sign of \\(s^3\\) will be the same as the sign of \\(s\\). Therefore, if \\(s\\) is positive, \\(s^3\\) will also be positive, and if \\(s\\) is negative, \\(s^3\\) will be negative. So, statement B) s < 0 is not necessarily true.\n\nMoving on to \\(t^4\\), we know that \\(t\\) raised to an even power will always be positive, regardless of the sign of \\(t\\). Therefore, the sign of \\(t\\) does not affect the inequality. So, statement C) t < 0 is not necessarily true.\n\nNow, let's consider \\(u\\). Since we have \\(r^2s^3t^4u > 0\\), we know that the product of all the terms on the left side is positive. In order for a product to be positive, either all the terms must be positive or an even number of terms must be negative. Since we have already determined that \\(r^2\\), \\(s^3\\), and \\(t^4\\) are positive, \\(u\\) must also be positive. Therefore, statement D) su > 0 is true.\n\nLastly, let's consider \\(st\\). Since both \\(s\\) and \\(t\\) are variables, we cannot determine their signs individually. However, we know that \\(s^3\\) and \\(t^4\\) are positive, so the product \\(st\\) must also be positive. Therefore, statement E) st > 0 is true.\n\\(\\newline\\)In conclusion, the statements that must be true are D) su > 0 and E) st > 0.\n\nThe answer is D." } }, { "answer":"A", "choices":{ "A":"21", "B":"28", "C":"49", "D":"63", "E":"81" }, "id":10983, "question":"If \\(3(a + 2b) = 7\\), what is the value of \\(9a + 18b\\)?", "explanations":{ "correct":"To find the value of \\(9a + 18b\\), we need to first solve the given equation \\(3(a + 2b) = 7\\) for either \\(a\\) or \\(b\\). Let's solve for \\(a\\):\n\nStart by distributing the 3 to both terms inside the parentheses:\n\\(3a + 6b = 7\\)\n\nNext, isolate \\(a\\) by subtracting \\(6b\\) from both sides of the equation:\n\\(3a = 7 - 6b\\)\n\nFinally, divide both sides by 3 to solve for \\(a\\):\n\\(a = \\frac{{7 - 6b}}{3}\\)\n\nNow that we have the value of \\(a\\), we can substitute it into the expression \\(9a + 18b\\):\n\\(9\\left(\\frac{{7 - 6b}}{3}\\right) + 18b\\)\n\nSimplify the expression inside the parentheses by multiplying 9 with each term:\n\\(\\frac{{63 - 54b}}{3} + 18b\\)\n\nCombine like terms by finding a common denominator:\n\\(\\frac{{63 - 54b + 54b}}{3}\\)\n\nThe \\(54b\\) terms cancel out, leaving us with:\n\\(\\frac{{63}}{3}\\)\n\nSimplify the fraction:\n\\(21\\)\n\nTherefore, the value of \\(9a + 18b\\) is 21.\n\nThe answer is A) 21." } }, { "answer":"B", "choices":{ "A":"\\(a + b = 0\\)", "B":"\\(\\frac { a } { b } < 0\\)", "C":"\\(a + b < 0\\)", "D":"\\(a + b > 0\\)", "E":"\\(ab > 0\\)" }, "id":10986, "question":"If \\(a > 0\\) and \\(b < 0\\), which of the following must be true?", "explanations":{ "correct":"To determine which of the given options must be true when \\(a > 0\\) and \\(b < 0\\), let's analyze each option step-by-step:\n\nA) \\(a + b = 0\\):\nSince \\(a > 0\\) and \\(b < 0\\), their sum cannot be zero. When we add a positive number to a negative number, the result is always negative. Therefore, option A is not true.\n\nB) \\(\\frac { a } { b } < 0\\):\nWhen we divide a positive number by a negative number, the result is always negative. Therefore, option B is true.\n\nC) \\(a + b < 0\\):\nSince \\(a > 0\\) and \\(b < 0\\), their sum will be negative. Therefore, option C is true.\n\nD) \\(a + b > 0\\):\nAs mentioned earlier, when we add a positive number to a negative number, the result is always negative. Therefore, option D is not true.\n\nE) \\(ab > 0\\):\nWhen we multiply a positive number by a negative number, the result is always negative. Therefore, option E is not true.\n\nBased on the analysis above, the options that must be true are B) \\(\\frac { a } { b } < 0\\) and C) \\(a + b < 0\\).\n\nThe answer is B and C." } }, { "answer":"E", "choices":{ "A":"\\(15z\\)", "B":"\\(2z\\)", "C":"\\(1.5z\\)", "D":"\\(z\\)", "E":"\\(0.15z\\)" }, "id":10988, "question":"How much greater than 15% of z is 30% of \\(z\\) ?", "explanations":{ "correct":"To find out how much greater 30% of \\(z\\) is than 15% of \\(z\\), we need to subtract 15% of \\(z\\) from 30% of \\(z\\).\n\nStep 1: Calculate 15% of \\(z\\).\n15% of \\(z\\) can be found by multiplying \\(z\\) by 0.15.\nSo, 15% of \\(z\\) is \\(0.15z\\).\n\nStep 2: Calculate 30% of \\(z\\).\n30% of \\(z\\) can be found by multiplying \\(z\\) by 0.30.\nSo, 30% of \\(z\\) is \\(0.30z\\).\n\nStep 3: Find the difference.\nTo find the difference between 30% of \\(z\\) and 15% of \\(z\\), we subtract \\(0.15z\\) from \\(0.30z\\).\n\\(0.30z - 0.15z = 0.15z\\).\n\nThe answer is E) \\(0.15z\\)." } }, { "answer":"D", "choices":{ "A":"40", "B":"24", "C":"16", "D":"8", "E":"1" }, "id":10989, "question":"Jeweler A can set an average round-cut diamond in 20 minutes. Jeweler B requires 15 minutes to set the same type of diamond. In 8 hours, how many more diamonds can be set by Jeweler B than by Jeweler A?", "explanations":{ "correct":"To find out how many more diamonds Jeweler B can set than Jeweler A in 8 hours, we need to calculate the number of diamonds each jeweler can set in that time frame.\n\nJeweler A can set 1 diamond in 20 minutes, which means in 1 hour (60 minutes), Jeweler A can set 60/20 = 3 diamonds.\n\\(\\newline\\)In 8 hours, Jeweler A can set 8 * 3 = 24 diamonds.\n\nJeweler B can set 1 diamond in 15 minutes, which means in 1 hour (60 minutes), Jeweler B can set 60/15 = 4 diamonds.\n\\(\\newline\\)In 8 hours, Jeweler B can set 8 * 4 = 32 diamonds.\n\nTherefore, Jeweler B can set 32 - 24 = 8 more diamonds than Jeweler A in 8 hours.\n\nThe answer is D) 8." } }, { "answer":"C", "choices":{ "A":"-5", "B":"-3", "C":"-1", "D":"1", "E":"3" }, "id":10993, "question":"If \\(7^ { x+3 } + 7^2 = 98\\), what is \\(x \\)?", "explanations":{ "correct":"To solve the equation \\(7^{x+3} + 7^2 = 98\\), we need to isolate the variable \\(x\\).\n\nFirst, let's simplify the equation by evaluating the exponents:\n\\(7^{x+3} + 49 = 98\\)\n\nNext, subtract 49 from both sides of the equation:\n\\(7^{x+3} = 49\\)\n\nSince \\(7^2 = 49\\), we can rewrite the equation as:\n\\(7^{x+3} = 7^2\\)\n\nNow, we can equate the exponents:\n\\(x+3 = 2\\)\n\nTo isolate \\(x\\), subtract 3 from both sides of the equation:\n\\(x = 2 - 3\\)\n\nSimplifying further, we get:\n\\(x = -1\\)\n\nTherefore, the answer is C) -1." } }, { "answer":"A", "choices":{ "A":"83", "B":"84", "C":"85", "D":"86", "E":"87" }, "id":10999, "question":"At the beginning of the school year, Lisa's goal was to earn an average of 80% of her 50 quizzes for the year. She earned and average of 78% of the first 30 quizzes. If she wants to achieve her goal, what must be her average of the remaining quizzes?", "explanations":{ "correct":"To find out what Lisa's average must be on the remaining quizzes in order to achieve her goal, we can use the concept of weighted averages.\n\nFirst, let's calculate the total score Lisa needs to achieve her goal. Since she wants to earn an average of 80% on all 50 quizzes, her total score should be 80% of 50, which is 0.8 * 50 = 40.\n\nNext, let's calculate the total score Lisa has already earned on the first 30 quizzes. She earned an average of 78%, so her total score is 78% of 30, which is 0.78 * 30 = 23.4.\n\nNow, let's find out how many points Lisa needs to earn on the remaining quizzes. The difference between the total score she needs and the total score she has already earned is 40 - 23.4 = 16.6.\n\nSince there are 50 quizzes in total and Lisa has already completed 30 quizzes, there are 50 - 30 = 20 quizzes remaining.\n\nTo find out what average Lisa needs on the remaining quizzes, we divide the total points she needs to earn (16.6) by the number of remaining quizzes (20). This gives us 16.6 / 20 = 0.83.\n\nTherefore, Lisa must earn an average of 83% on the remaining quizzes to achieve her goal.\n\nThe answer is A) 83." } }, { "answer":"D", "choices":{ "A":"\\(p - t\\)", "B":"\\(p + t\\)", "C":"\\(\\frac { p - t } { 2 } \\)", "D":"\\(\\frac { p + t } { 2 } \\)", "E":"\\(2p - t\\)" }, "id":11002, "question":"A father is \\(t\\) inches taller than his son. If their total height is \\(p\\), in terms of \\(p\\) and \\(t\\), what is the father's height, in inches?", "explanations":{ "correct":"Let's analyze the information given in the question. We know that the father is \\(t\\) inches taller than his son. This means that if we add \\(t\\) inches to the son's height, we will get the father's height.\n\nNow, the total height of the father and son is \\(p\\). So, if we add the son's height to the father's height, it should equal \\(p\\).\n\nLet's represent the son's height as \\(s\\). Since the father is \\(t\\) inches taller, the father's height can be represented as \\(s + t\\).\n\nNow, we can set up an equation to represent the total height:\n\n\\(s + (s + t) = p\\)\n\nSimplifying the equation, we get:\n\n\\(2s + t = p\\)\n\nTo find the father's height, we need to substitute \\(s + t\\) for the father's height. So, the father's height is \\(s + t\\), which is equal to \\(\\frac{p}{2}\\) after dividing both sides of the equation by 2.\n\nTherefore, the answer is D) \\(\\frac{p + t}{2}\\)." } }, { "answer":"D", "choices":{ "A":"\\(6z^2 = (z + 6)^2\\)", "B":"\\((6z)^2 = z^2 + 6^2\\)", "C":"\\(6^2 z = z^2 - 6^2\\)", "D":"\\((6z)^2 = (z - 6)^2\\)", "E":"\\(6z^2 = z^2 + 6^2\\)" }, "id":11003, "question":"Squaring the product of z and 6 gives the same result as squaring the difference of z and 6. Which of the following equations could be used to find all the possible values of z?", "explanations":{ "correct":"To find the equation that could be used to find all the possible values of z, we need to translate the given information into an equation.\n\nThe problem states that squaring the product of z and 6 gives the same result as squaring the difference of z and 6. Let's break this down step-by-step:\n\n1. Squaring the product of z and 6: This can be written as (6z)^2.\n\n2. Squaring the difference of z and 6: This can be written as (z - 6)^2.\n\nNow, we need to find the equation that represents the given information. Comparing the two expressions we derived:\n\n(6z)^2 = (z - 6)^2\n\nWe can see that the equation that represents the given information is D) (6z)^2 = (z - 6)^2.\n\nTherefore, the answer is D." } }, { "answer":"C", "choices":{ "A":"4", "B":"6", "C":"8", "D":"\\(x + 4\\)", "E":"\\(x + 8\\)" }, "id":11005, "question":"The value of \\(x + 6\\) is how much greater than the value of \\(x - 2\\)?", "explanations":{ "correct":"To find the difference between the values of \\(x + 6\\) and \\(x - 2\\), we need to subtract the value of \\(x - 2\\) from the value of \\(x + 6\\).\n\nStep 1: Simplify \\(x + 6\\) and \\(x - 2\\).\n\\(x + 6\\) is already simplified.\nTo simplify \\(x - 2\\), we combine the like terms, which gives us \\(x - 2\\).\n\nStep 2: Subtract \\(x - 2\\) from \\(x + 6\\).\nSubtracting \\(x - 2\\) from \\(x + 6\\) can be done by subtracting the coefficients of \\(x\\) and the constant terms separately.\n\\(x - x = 0\\) (the \\(x\\) terms cancel out)\n\\(6 - (-2) = 6 + 2 = 8\\)\n\nTherefore, the value of \\(x + 6\\) is 8 greater than the value of \\(x - 2\\).\n\nThe answer is C) 8." } }, { "answer":"D", "choices":{ "A":"\\(-5\\)", "B":"\\(-\\frac{19}{6}\\)", "C":"\\(-3\\)", "D":"5", "E":"15" }, "id":11006, "question":"If \\(|12r - 2| = 58\\) and \\(|3r + 2| = 17\\), then what is the value of r?", "explanations":{ "correct":"To find the value of \\(r\\), we need to solve the given equations step-by-step.\n\nFirst, let's solve the equation \\(|12r - 2| = 58\\):\n\n1. If \\(12r - 2\\) is positive, then \\(|12r - 2| = 12r - 2\\).\n So, we have \\(12r - 2 = 58\\).\n Solving this equation, we get \\(12r = 60\\) and \\(r = 5\\).\n\n2. If \\(12r - 2\\) is negative, then \\(|12r - 2| = -(12r - 2)\\).\n So, we have \\(-(12r - 2) = 58\\).\n Expanding the negative sign, we get \\(-12r + 2 = 58\\).\n Solving this equation, we get \\(-12r = 56\\) and \\(r = -\\frac{7}{3}\\).\n\nNext, let's solve the equation \\(|3r + 2| = 17\\):\n\n1. If \\(3r + 2\\) is positive, then \\(|3r + 2| = 3r + 2\\).\n So, we have \\(3r + 2 = 17\\).\n Solving this equation, we get \\(3r = 15\\) and \\(r = 5\\).\n\n2. If \\(3r + 2\\) is negative, then \\(|3r + 2| = -(3r + 2)\\).\n So, we have \\(-(3r + 2) = 17\\).\n Expanding the negative sign, we get \\(-3r - 2 = 17\\).\n Solving this equation, we get \\(-3r = 19\\) and \\(r = -\\frac{19}{3}\\).\n\nNow, let's compare the values of \\(r\\) we obtained from both equations:\n\n- \\(r = 5\\) is a common solution.\n- \\(r = -\\frac{7}{3}\\) and \\(r = -\\frac{19}{3}\\) are different solutions.\n\nSince we are looking for the value of \\(r\\) that satisfies both equations, the only common solution is \\(r = 5\\).\n\nTherefore, the answer is D) 5." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 1 } { 9 } \\)", "B":"\\(\\frac { 1 } { 5 } \\)", "C":"\\(\\frac { 1 } { 3 } \\)", "D":"\\(\\frac { 3 } { 8 } \\)", "E":"\\(\\frac { 1 } { 2 } \\)" }, "id":11008, "question":"If a mixture of nuts consists of 3 pounds of peanuts, I pound of walnuts, and 5 pounds of cashews, by weight, what fraction of the mixture is peanuts?", "explanations":{ "correct":"To find the fraction of the mixture that is peanuts, we need to determine the total weight of the mixture and the weight of the peanuts.\n\nGiven that the mixture consists of 3 pounds of peanuts, 1 pound of walnuts, and 5 pounds of cashews, the total weight of the mixture is 3 + 1 + 5 = 9 pounds.\n\nTo find the fraction of the mixture that is peanuts, we divide the weight of the peanuts by the total weight of the mixture: \\(\\frac { 3 } { 9 } \\).\n\nSimplifying the fraction, we get \\(\\frac { 1 } { 3 } \\).\n\nTherefore, the answer is C) \\(\\frac { 1 } { 3 } \\)." } }, { "answer":"B", "choices":{ "A":"35", "B":"45", "C":"55", "D":"75", "E":"95" }, "id":11010, "question":"If \\(2x - 1 = 9\\), what is \\(10x - 5\\) ?", "explanations":{ "correct":"To find the value of \\(10x - 5\\), we first need to solve the equation \\(2x - 1 = 9\\) to find the value of \\(x\\).\n\nStep 1: Add 1 to both sides of the equation:\n\\(2x - 1 + 1 = 9 + 1\\)\n\\(2x = 10\\)\n\nStep 2: Divide both sides of the equation by 2:\n\\(\\frac{2x}{2} = \\frac{10}{2}\\)\n\\(x = 5\\)\n\nNow that we have found the value of \\(x\\), we can substitute it into the expression \\(10x - 5\\):\n\n\\(10(5) - 5 = 50 - 5 = 45\\)\n\nTherefore, the answer is B) 45." } }, { "answer":"A", "choices":{ "A":"-6", "B":"-2", "C":"\\(\\frac { 1 } { 4 } \\)", "D":"\\(\\frac { 2 } { 3 } \\)", "E":"6" }, "id":11015, "question":"What is the value of \\(b\\) if \\(1 - \\frac { 4 } { b } = \\frac { 5 } { 3 } \\) ?", "explanations":{ "correct":"To find the value of \\(b\\), we can start by isolating \\(b\\) in the given equation.\n\nWe have the equation \\(1 - \\frac{4}{b} = \\frac{5}{3}\\).\n\nTo isolate \\(b\\), we can start by subtracting 1 from both sides of the equation:\n\n\\(-\\frac{4}{b} = \\frac{5}{3} - 1\\).\n\nSimplifying the right side of the equation:\n\n\\(-\\frac{4}{b} = \\frac{5}{3} - \\frac{3}{3}\\).\n\n\\(-\\frac{4}{b} = \\frac{5-3}{3}\\).\n\n\\(-\\frac{4}{b} = \\frac{2}{3}\\).\n\nNext, we can cross-multiply to eliminate the fraction:\n\n\\(3 \\cdot (-4) = b \\cdot 2\\).\n\n\\(-12 = 2b\\).\n\nTo solve for \\(b\\), we divide both sides of the equation by 2:\n\n\\(\\frac{-12}{2} = \\frac{2b}{2}\\).\n\n\\(-6 = b\\).\n\nTherefore, the value of \\(b\\) is -6.\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"2.5", "B":"20", "C":"35", "D":"40", "E":"70" }, "id":11016, "question":"If \\(q = \\frac{5}{4}st\\), what is the value of t when \\(s = 2\\) and \\(q = 50\\)?", "explanations":{ "correct":"To find the value of t when s = 2 and q = 50, we can substitute these values into the equation q = (5/4)st and solve for t.\n\nGiven:\nq = 50\ns = 2\n\nSubstituting these values into the equation:\n50 = (5/4)(2)t\n\nTo solve for t, we can start by multiplying both sides of the equation by 4/5 to isolate t:\n(4/5)(50) = (4/5)(5/4)(2)t\n\nSimplifying:\n40 = 2t\n\nNext, we can divide both sides of the equation by 2 to solve for t:\n40/2 = 2t/2\n\nSimplifying:\n20 = t\n\nTherefore, the value of t when s = 2 and q = 50 is t = 20.\n\nThe answer is B) 20." } }, { "answer":"B", "choices":{ "A":"2", "B":"\\(\\frac { 2(x + 2 } { x - 2 } \\)", "C":"\\(\\frac { 2(x + 4) } { x - 4 } \\)", "D":"\\(\\frac { 2x + 2 } { x - 2 } \\)", "E":"\\(\\frac { 6 } { 4x - 4 } \\)" }, "id":11019, "question":"Which of the following is equivalent to \\(\\frac { 2x^2 - 8 } { x^2 - 4x + 4 } \\)?", "explanations":{ "correct":"To find the equivalent expression, we need to simplify the given expression. \n\nThe expression is \\(\\frac { 2x^2 - 8 } { x^2 - 4x + 4 } \\).\n\nFirst, let's factor the numerator and denominator:\n\nNumerator: \\(2x^2 - 8\\)\nWe can factor out a 2: \\(2(x^2 - 4)\\)\nThen, we can factor the difference of squares: \\(2(x + 2)(x - 2)\\)\n\nDenominator: \\(x^2 - 4x + 4\\)\nThis is a perfect square trinomial: \\((x - 2)^2\\)\n\nNow, we can rewrite the expression with the factored forms:\n\n\\(\\frac { 2(x + 2)(x - 2) } { (x - 2)^2 }\\)\n\nNext, we can cancel out the common factor of \\(x - 2\\) in the numerator and denominator:\n\n\\(\\frac { 2(x + 2) } { x - 2 }\\)\n\nTherefore, the equivalent expression is \\(\\frac { 2(x + 2) } { x - 2 }\\).\n\nThe answer is B." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{3}{7}\\)", "B":"\\(\\frac{3}{10}\\)", "C":"\\(\\frac{3}{11}\\)", "D":"\\(\\frac{6}{19}\\)", "E":"\\(\\frac{1}{4}\\)" }, "id":11021, "question":"Sharon has exactly 6 quarters, 5 dimes, and 10 nickels in her pocket. She pulls out a coin at random and puts it aside since the coin is not a quarter. If she pulls out a second coin at random from her pocket, what is the probability that the second coin is a quarter?", "explanations":{ "correct":"To find the probability that the second coin pulled out is a quarter, we need to consider the total number of coins remaining in Sharon's pocket after she puts aside a non-quarter coin.\n\\(\\newline\\)Initially, Sharon has a total of 6 quarters, 5 dimes, and 10 nickels, which sums up to 21 coins in total.\n\nSince Sharon puts aside a non-quarter coin, the total number of coins remaining in her pocket is 21 - 1 = 20.\n\nOut of the remaining 20 coins, there are still 6 quarters left.\n\nTherefore, the probability of pulling out a quarter as the second coin is 6/20.\n\nSimplifying this fraction, we get 3/10.\n\nSo, the answer is B) \\(\\frac{3}{10}\\)." } }, { "answer":"B", "choices":{ "A":"0.032", "B":"0.32", "C":"3.2", "D":"32", "E":"320" }, "id":11023, "question":"If \\((0.008)x = 0.032\\), then what is the value of \\(0.08x\\)?", "explanations":{ "correct":"To find the value of \\(0.08x\\), we need to first solve the equation \\(0.008x = 0.032\\) for \\(x\\).\n\nTo do this, we can divide both sides of the equation by \\(0.008\\):\n\n\\[\\frac{{0.008x}}{{0.008}} = \\frac{{0.032}}{{0.008}}\\]\n\nThis simplifies to:\n\n\\[x = 4\\]\n\nNow that we know the value of \\(x\\ is 4, we can substitute it into the expression \\(0.08x\\):\n\n\\[0.08 \\times 4 = 0.32\\]\n\nTherefore, the value of \\(0.08x\\) is 0.32.\n\nThe answer is B) 0.32." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 2 } { 3 } \\)", "B":"\\(\\frac { 4 } { 3 } \\)", "C":"2", "D":"3", "E":"\\(\\frac { 9 } { 2 } \\)" }, "id":11024, "question":"Let the function \\(g(x)\\) be defined by \\(g(x) = 2f(x) + k\\) , where \\(f(x)\\) is a linear function. If \\(g(2) = 10\\) and \\(g(5) = 18\\), what is the slope of function \\(f\\)?", "explanations":{ "correct":"To find the slope of function \\(f\\), we need to use the given information about \\(g(x)\\) and the values of \\(g(2)\\) and \\(g(5)\\).\n\nWe know that \\(g(x) = 2f(x) + k\\), where \\(f(x)\\) is a linear function. Let's substitute the values of \\(g(2)\\) and \\(g(5)\\) into the equation to create two equations:\n\nFor \\(x = 2\\):\n\\(g(2) = 2f(2) + k = 10\\)\n\nFor \\(x = 5\\):\n\\(g(5) = 2f(5) + k = 18\\)\n\nNow, we have a system of two equations with two unknowns: \\(f(2)\\) and \\(f(5)\\). To solve for the slope of function \\(f\\), we need to eliminate the constant term \\(k\\).\n\nSubtracting the first equation from the second equation, we get:\n\\(g(5) - g(2) = 2f(5) + k - (2f(2) + k)\\)\nSimplifying, we have:\n\\(18 - 10 = 2f(5) - 2f(2)\\)\n\\(8 = 2(f(5) - f(2))\\)\n\\(4 = f(5) - f(2)\\)\n\nSince \\(f(x)\\) is a linear function, the slope of \\(f\\) is equal to the difference in the function values divided by the difference in the input values. In this case, the difference in input values is \\(5 - 2 = 3\\), and the difference in function values is \\(f(5) - f(2) = 4\\).\n\nTherefore, the slope of function \\(f\\) is \\(\\frac{4}{3}\\).\n\nThe answer is B) \\(\\frac{4}{3}\\)." } }, { "answer":"E", "choices":{ "A":"\\(\\frac{1}{2}\\)", "B":"2", "C":"4", "D":"16", "E":"256" }, "id":11029, "question":"If \\(x^{\\frac{1}{2}} = 4\\), then what is the value of \\(x^2\\)?", "explanations":{ "correct":"To find the value of \\(x^2\\), we need to first solve for \\(x\\) using the given equation \\(x^{\\frac{1}{2}} = 4\\).\n\nTo eliminate the exponent of \\(\\frac{1}{2}\\), we can raise both sides of the equation to the power of 2:\n\n\\((x^{\\frac{1}{2}})^2 = 4^2\\)\n\nThis simplifies to:\n\n\\(x = 16\\)\n\nNow that we have found the value of \\(x\\), we can substitute it into the expression \\(x^2\\) to find the final answer:\n\n\\(x^2 = 16^2 = 256\\)\n\nTherefore, the value of \\(x^2\\) is 256.\n\nThe answer is E." } }, { "answer":"E", "choices":{ "A":"45", "B":"90", "C":"450", "D":"500", "E":"900" }, "id":11033, "question":"What is the sum of 15 consecutive integers if the middle integer is 60?", "explanations":{ "correct":"To find the sum of 15 consecutive integers, we need to determine the first and last integers in the sequence. Since the middle integer is given as 60, we can deduce that the first integer is 60 - 7 = 53 and the last integer is 60 + 7 = 67.\n\nTo find the sum of the sequence, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(a + l), where Sn is the sum, n is the number of terms, a is the first term, and l is the last term.\n\nPlugging in the values, we have Sn = (15/2)(53 + 67) = (15/2)(120) = 900.\n\nTherefore, the sum of the 15 consecutive integers is 900.\n\nThe answer is E." } }, { "answer":"E", "choices":{ "A":"The average of the scores", "B":"The highest score", "C":"The number of scores", "D":"The number of possible scores", "E":"The sum of the scores" }, "id":11034, "question":"Let X represent the average (arithmetic mean) of a list of test scores. What is the result of multiplying X by the number of scores?", "explanations":{ "correct":"To find the result of multiplying the average (X) by the number of scores, we need to understand the concept of average and how it relates to the total sum of the scores.\n\nThe average (X) is calculated by summing up all the scores and dividing it by the number of scores. So, if we multiply the average (X) by the number of scores, it would be equivalent to multiplying the sum of the scores by the number of scores.\n\nTherefore, the result of multiplying X by the number of scores would give us the sum of the scores.\n\nHence, the answer is E) The sum of the scores." } }, { "answer":"E", "choices":{ "A":"\\(a = b = c\\)", "B":"\\(a > b > c\\)", "C":"\\(b > c > a\\)", "D":"\\(c > b > a\\)", "E":"\\(b > a > c\\)" }, "id":11036, "question":"The various averages (arithmetic mean) of two of the three numbers a, b, and c are calculated, and are arranged as follows: \\(P\\) = The average of \\(a\\) and \\(b\\) \\(Q\\) = The average of \\(b\\) and \\(c\\) \\(R\\) = The average of \\(c\\) and \\(a\\) If \\(P > Q > R\\), then which of the following is true?", "explanations":{ "correct":"To determine which statement is true, let's analyze the given information step-by-step.\n\nWe are given that \\(P > Q > R\\), where:\n\\(P\\) is the average of \\(a\\) and \\(b\\),\n\\(Q\\) is the average of \\(b\\) and \\(c\\), and\n\\(R\\) is the average of \\(c\\) and \\(a\\).\n\nLet's compare \\(P\\) and \\(Q\\):\n\\(P > Q\\) implies that the average of \\(a\\) and \\(b\\) is greater than the average of \\(b\\) and \\(c\\).\n\nNow, let's compare \\(Q\\) and \\(R\\):\n\\(Q > R\\) implies that the average of \\(b\\) and \\(c\\) is greater than the average of \\(c\\) and \\(a\\).\n\nCombining these two inequalities, we have:\n\\(P > Q > R\\), which can be rewritten as \\(P > Q\\) and \\(Q > R\\).\n\nFrom the inequality \\(P > Q\\), we can conclude that the average of \\(a\\) and \\(b\\) is greater than the average of \\(b\\) and \\(c\\). This suggests that \\(a\\) is greater than \\(c\\).\n\nFrom the inequality \\(Q > R\\), we can conclude that the average of \\(b\\) and \\(c\\) is greater than the average of \\(c\\) and \\(a\\). This suggests that \\(b\\) is greater than \\(a\\).\n\nCombining these conclusions, we have \\(a < c\\) and \\(b > a\\). Therefore, the correct statement is \\(b > a > c\\).\n\nHence, the answer is E) \\(b > a > c\\)." } }, { "answer":"B", "choices":{ "A":"2", "B":"4", "C":"8", "D":"12", "E":"16" }, "id":11038, "question":"If \\(a + b - c = d + 6\\), \\(c - b = 8\\), and \\(3a = 2 - d\\), what is the value of \\(a\\)?", "explanations":{ "correct":"To find the value of \\(a\\), we need to solve the given system of equations step-by-step.\n\nFirst, let's rearrange the equation \\(c - b = 8\\) to solve for \\(c\\):\n\\[c = b + 8\\]\n\nNext, substitute this value of \\(c\\) into the equation \\(a + b - c = d + 6\\):\n\\[a + b - (b + 8) = d + 6\\]\n\nSimplifying the equation, we get:\n\\[a - 8 = d + 6\\]\n\nNow, let's rearrange the equation \\(3a = 2 - d\\) to solve for \\(d\\):\n\\[d = 2 - 3a\\]\n\nSubstitute this value of \\(d\\) into the previous equation:\n\\[a - 8 = (2 - 3a) + 6\\]\n\nSimplifying the equation, we get:\n\\[a - 8 = 2 - 3a + 6\\]\n\\[a - 8 = 8 - 3a\\]\n\nCombine like terms:\n\\[4a = 16\\]\n\nDivide both sides by 4:\n\\[a = 4\\]\n\nTherefore, the value of \\(a\\) is 4.\n\nThe answer is B) 4." } }, { "answer":"E", "choices":{ "A":"0", "B":"\\(\\frac { 1 } { 3 } \\)", "C":"2", "D":"\\(\\frac { 7 } { 2 } \\)", "E":"4" }, "id":11041, "question":"If \\(x^3 = 1\\), what is the value of \\(\\frac { x^2 + 3 } { x } \\) ?", "explanations":{ "correct":"To find the value of \\(\\frac { x^2 + 3 } { x }\\), we need to substitute the value of \\(x\\) from the given equation \\(x^3 = 1\\).\n\nTaking the cube root of both sides of the equation, we get \\(x = 1\\).\n\nNow, substituting \\(x = 1\\) into the expression \\(\\frac { x^2 + 3 } { x }\\), we have:\n\n\\(\\frac { 1^2 + 3 } { 1 } = \\frac { 1 + 3 } { 1 } = \\frac { 4 } { 1 } = 4\\)\n\nTherefore, the value of \\(\\frac { x^2 + 3 } { x }\\) is 4.\n\nThe answer is E." } }, { "answer":"A", "choices":{ "A":"5", "B":"6", "C":"7", "D":"8", "E":"9" }, "id":11042, "question":"The first term of a sequence is \\(a\\). Every term after the first is equal to \\(d\\) plus the preceding term. If the seventh term is 20 and the fourth term is 11, what is the value of \\(a + d\\)?", "explanations":{ "correct":"To find the value of \\(a + d\\), we need to determine the values of \\(a\\) and \\(d\\) separately.\n\nWe are given that the fourth term of the sequence is 11. Since every term after the first is equal to \\(d\\) plus the preceding term, we can write the equation:\n\\[a + 3d = 11\\] (since the fourth term is the first term plus three times \\(d\\))\n\nSimilarly, we are given that the seventh term of the sequence is 20. Using the same logic, we can write the equation:\n\\[a + 6d = 20\\] (since the seventh term is the first term plus six times \\(d\\))\n\nNow we have a system of two equations:\n\\[\\begin{cases} a + 3d = 11 \\\\ a + 6d = 20 \\end{cases}\\]\n\nTo solve this system, we can subtract the first equation from the second equation to eliminate \\(a\\):\n\\[(a + 6d) - (a + 3d) = 20 - 11\\]\nSimplifying, we get:\n\\[3d = 9\\]\nDividing both sides by 3, we find:\n\\[d = 3\\]\n\nNow that we know the value of \\(d\\), we can substitute it back into one of the original equations to find the value of \\(a\\). Let's use the first equation:\n\\[a + 3(3) = 11\\]\nSimplifying, we get:\n\\[a + 9 = 11\\]\nSubtracting 9 from both sides, we find:\n\\[a = 2\\]\n\nFinally, we can find the value of \\(a + d\\) by adding \\(a\\) and \\(d\\):\n\\[2 + 3 = 5\\]\n\nTherefore, the value of \\(a + d\\) is 5.\n\nThe answer is A) 5." } }, { "answer":"D", "choices":{ "A":"13", "B":"6", "C":"3", "D":"-3", "E":"-6" }, "id":11043, "question":"If \\(a\\) is 4 greater than \\(b\\) and \\(a^2 + b^2 = 10\\), what is the value of \\(ab\\) ?", "explanations":{ "correct":"To find the value of \\(ab\\), we need to first determine the values of \\(a\\) and \\(b\\) using the given information.\n\nLet's start by setting up the equations based on the given information:\n1) \\(a\\) is 4 greater than \\(b\\), so we can write \\(a = b + 4\\).\n2) \\(a^2 + b^2 = 10\\).\n\nNow, substitute the value of \\(a\\) from equation 1 into equation 2:\n\\((b + 4)^2 + b^2 = 10\\).\n\nExpanding the equation:\n\\(b^2 + 8b + 16 + b^2 = 10\\).\n\nCombining like terms:\n\\(2b^2 + 8b + 16 = 10\\).\n\nRearranging the equation:\n\\(2b^2 + 8b + 6 = 0\\).\n\nDividing the equation by 2 to simplify:\n\\(b^2 + 4b + 3 = 0\\).\n\nFactoring the quadratic equation:\n\\((b + 1)(b + 3) = 0\\).\n\nSetting each factor equal to zero:\n\\(b + 1 = 0\\) or \\(b + 3 = 0\\).\n\nSolving for \\(b\\):\n\\(b = -1\\) or \\(b = -3\\).\n\nNow, substitute the values of \\(b\\) back into equation 1 to find the corresponding values of \\(a\\):\nFor \\(b = -1\\), \\(a = -1 + 4 = 3\\).\nFor \\(b = -3\\), \\(a = -3 + 4 = 1\\).\n\nFinally, calculate the value of \\(ab\\) for each case:\nFor \\(b = -1\\), \\(ab = 3 \\times -1 = -3\\).\nFor \\(b = -3\\), \\(ab = 1 \\times -3 = -3\\).\n\nSince both cases yield the same value of \\(ab = -3\\), the answer is D) -3." } }, { "answer":"E", "choices":{ "A":"4", "B":"5", "C":"6", "D":"8", "E":"9" }, "id":11044, "question":"A circular grass field has a circumference of \\(120\\sqrt { \\pi } \\) meters. If Eric can mow 400 square meters of grass per hour, how many hours will he take to mow the entire field?", "explanations":{ "correct":"To find the number of hours it will take for Eric to mow the entire field, we need to determine the area of the circular grass field.\n\nThe formula for the circumference of a circle is \\(C = 2\\pi r\\), where \\(C\\) is the circumference and \\(r\\) is the radius. In this case, the circumference is given as \\(120\\sqrt{\\pi}\\) meters. So we can set up the equation as:\n\n\\(120\\sqrt{\\pi} = 2\\pi r\\)\n\nTo solve for \\(r\\), we divide both sides of the equation by \\(2\\pi\\):\n\n\\(r = \\frac{120\\sqrt{\\pi}}{2\\pi}\\)\n\nSimplifying, we get:\n\n\\(r = \\frac{60\\sqrt{\\pi}}{\\pi}\\)\n\nThe formula for the area of a circle is \\(A = \\pi r^2\\). Plugging in the value of \\(r\\), we can calculate the area of the circular grass field:\n\n\\(A = \\pi \\left(\\frac{60\\sqrt{\\pi}}{\\pi}\\right)^2\\)\n\nSimplifying further:\n\n\\(A = \\pi \\left(\\frac{60\\sqrt{\\pi}}{\\pi}\\right)^2 = \\pi \\left(\\frac{60^2 \\cdot \\pi}{\\pi^2}\\right) = \\pi \\left(\\frac{3600 \\cdot \\pi}{\\pi^2}\\right) = \\frac{3600\\pi}{\\pi} = 3600\\)\n\nSo, the area of the circular grass field is 3600 square meters.\n\nSince Eric can mow 400 square meters of grass per hour, we can divide the total area of the field by the rate at which he mows to find the number of hours it will take:\n\n\\(\\frac{3600}{400} = 9\\)\n\nTherefore, it will take Eric 9 hours to mow the entire field.\n\nThe answer is E." } }, { "answer":"C", "choices":{ "A":"\\(6b^2 - 48b + 96\\)", "B":"\\(6b^2 - 57b + 132\\)", "C":"\\(6b^2 - 57b + 144\\)", "D":"\\(6b^2 - 9b - 132\\)", "E":"\\(6b^2 - 9b + 144\\)" }, "id":11047, "question":"If \\(a = 3(2c^2 + 3c + 4)\\) and \\(b = -c + 4\\), what is a in terms of b?", "explanations":{ "correct":"To find the value of \\(a\\) in terms of \\(b\\), we need to substitute the expression for \\(b\\) into the equation for \\(a\\).\n\nGiven:\n\\(a = 3(2c^2 + 3c + 4)\\)\n\\(b = -c + 4\\)\n\nSubstituting \\(b\\) into the equation for \\(a\\):\n\\(a = 3(2c^2 + 3c + 4)\\)\n\\(a = 3(2c^2 + 3c + 4)\\)\n\\(a = 6c^2 + 9c + 12\\)\n\nNow, we need to express \\(a\\) in terms of \\(b\\). To do this, we need to express \\(c\\) in terms of \\(b\\). From the equation for \\(b\\), we have:\n\\(b = -c + 4\\)\n\\(c = -b + 4\\)\n\nSubstituting \\(c\\) into the equation for \\(a\\):\n\\(a = 6(-b + 4)^2 + 9(-b + 4) + 12\\)\n\\(a = 6(b^2 - 8b + 16) - 9b + 36 + 12\\)\n\\(a = 6b^2 - 48b + 96 - 9b + 36 + 12\\)\n\\(a = 6b^2 - 57b + 144\\)\n\nTherefore, the answer is C) \\(6b^2 - 57b + 144\\)." } }, { "answer":"C", "choices":{ "A":"-2", "B":"-1", "C":"0", "D":"1", "E":"2" }, "id":11048, "question":"If \\(3^ { x^2 + y^2 } = \\frac { 1 } { 9^xy } \\), what is the value of \\(x + y\\) ?", "explanations":{ "correct":"To find the value of \\(x + y\\), we need to solve the given equation \\(3^{x^2 + y^2} = \\frac{1}{9^{xy}}\\).\n\nFirst, let's simplify the right side of the equation. We know that \\(9 = 3^2\\), so we can rewrite \\(9^{xy}\\) as \\((3^2)^{xy}\\), which simplifies to \\(3^{2xy}\\). Therefore, the equation becomes \\(3^{x^2 + y^2} = \\frac{1}{3^{2xy}}\\).\n\nTo solve this equation, we can equate the exponents on both sides. We have \\(x^2 + y^2 = -2xy\\).\n\nNow, let's rearrange the equation to get all terms on one side: \\(x^2 + 2xy + y^2 = 0\\).\n\nThis equation can be factored as \\((x + y)^2 = 0\\).\n\nFor the equation \\((x + y)^2 = 0\\) to be true, the only possible value for \\(x + y\\) is 0. This is because any number squared is equal to 0 only when the number itself is 0.\n\nTherefore, the value of \\(x + y\\) is 0.\n\nThe answer is C) 0." } }, { "answer":"E", "choices":{ "A":"F is doubled", "B":"F is tripled", "C":"F is multiplied by 8", "D":"F is multiplied by 12", "E":"F is multiplied by 18" }, "id":11051, "question":"If \\(F = k \\frac { v^2 } { r } \\), where \\(k\\) is a constant, what happens to the value of F when \\(v\\) is tripled and \\(r\\) is halved?", "explanations":{ "correct":"To determine what happens to the value of F when v is tripled and r is halved, we can substitute the new values into the equation and compare the result to the original value of F.\n\nGiven: F = k * (v^2 / r)\n\nLet's consider the changes separately:\n\n1. When v is tripled: \nThe new value of v is 3v. Substituting this into the equation, we get:\nF' = k * ((3v)^2 / r)\nF' = k * (9v^2 / r)\nF' = 9 * (k * (v^2 / r))\nF' = 9F\n\n2. When r is halved:\nThe new value of r is r/2. Substituting this into the equation, we get:\nF'' = k * (v^2 / (r/2))\nF'' = k * (v^2 * (2/r))\nF'' = 2 * (k * (v^2 / r))\nF'' = 2F\n\nCombining the two changes, we have:\nF''' = F' * F''\nF''' = (9F) * (2F)\nF''' = 18F^2\n\nTherefore, the value of F is multiplied by 18 when v is tripled and r is halved.\n\nThe answer is E) F is multiplied by 18." } }, { "answer":"B", "choices":{ "A":"Eight", "B":"Seven", "C":"Six", "D":"Five", "E":"Four" }, "id":11053, "question":"In a certain polygon, all of the angles are equal and all of the sides are of equal length. Point Q is a vertex of the polygon. If four diagonals can be drawn from point Q to the other vertices of the polygon, how many sides does the polygon have?", "explanations":{ "correct":"To determine the number of sides of the polygon, we need to analyze the number of diagonals that can be drawn from point Q to the other vertices.\n\\(\\newline\\)In a polygon, the number of diagonals that can be drawn from a single vertex is given by the formula (n - 3), where n represents the number of sides of the polygon.\n\nSince four diagonals can be drawn from point Q, we can set up the equation (n - 3) = 4.\n\nSolving for n, we add 3 to both sides of the equation: n = 4 + 3.\n\nTherefore, the polygon has 7 sides.\n\nThe answer is B) Seven." } }, { "answer":"D", "choices":{ "A":"\\(pq\\)", "B":"\\(p + q\\)", "C":"\\(p - q\\)", "D":"\\(\\frac { p } { q } \\)", "E":"\\(\\frac { q } { p } \\)" }, "id":11054, "question":"If \\(p < q < 0\\), which of the following MUST be greater than 1?", "explanations":{ "correct":"To determine which of the given options must be greater than 1 when \\(p < q < 0\\), let's evaluate each option step-by-step:\n\nA) \\(pq\\): Since both \\(p\\) and \\(q\\) are negative numbers, multiplying them together will result in a positive number. Therefore, \\(pq\\) will be greater than 1. \n\nB) \\(p + q\\): Since both \\(p\\) and \\(q\\) are negative numbers, adding them together will result in a negative number. Therefore, \\(p + q\\) will be less than 1.\n\nC) \\(p - q\\): Subtracting a negative number from another negative number will result in a negative number. Therefore, \\(p - q\\) will be less than 1.\n\nD) \\(\\frac { p } { q }\\): Dividing a negative number (\\(p\\)) by a negative number (\\(q\\)) will result in a positive number. Therefore, \\(\\frac { p } { q }\\) will be greater than 1.\n\nE) \\(\\frac { q } { p }\\): Dividing a negative number (\\(q\\)) by a negative number (\\(p\\)) will result in a positive number. Therefore, \\(\\frac { q } { p }\\) will be greater than 1.\n\nBased on the reasoning above, the only option that MUST be greater than 1 is \\(\\frac { p } { q }\\). \n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"128", "B":"80", "C":"64", "D":"32", "E":"16" }, "id":11056, "question":"There are a total of 256 people at a party. If after every 5 minutes, half of the people present leave, how many people will still remain after 20 minutes?", "explanations":{ "correct":"To find the number of people who will still remain after 20 minutes, we need to determine how many times the process of half the people leaving occurs in 20 minutes.\n\nSince half of the people leave every 5 minutes, we can divide 20 by 5 to find the number of times this process occurs. \n\n20 ÷ 5 = 4\n\nTherefore, the process of half the people leaving occurs 4 times in 20 minutes.\n\nNow, let's calculate the number of people who will remain after each occurrence:\n\nAfter the first occurrence: 256 ÷ 2 = 128 people remain\nAfter the second occurrence: 128 ÷ 2 = 64 people remain\nAfter the third occurrence: 64 ÷ 2 = 32 people remain\nAfter the fourth occurrence: 32 ÷ 2 = 16 people remain\n\nSo, after 20 minutes, there will be 16 people remaining.\n\nThe answer is E) 16." } }, { "answer":"C", "choices":{ "A":"1", "B":"2", "C":"3", "D":"4", "E":"5" }, "id":11057, "question":"If \\(4^{4x + 6} = 64^{2x}\\), what is the value of x?", "explanations":{ "correct":"To solve the equation \\(4^{4x + 6} = 64^{2x}\\), we need to simplify both sides of the equation and then solve for \\(x\\).\n\nFirst, let's simplify the right side of the equation. We know that \\(64 = 4^3\\), so we can rewrite \\(64^{2x}\\) as \\((4^3)^{2x}\\). Using the property of exponents, we can simplify this to \\(4^{6x}\\).\n\nNow, our equation becomes \\(4^{4x + 6} = 4^{6x}\\). Since the bases are the same, we can equate the exponents:\n\n\\(4x + 6 = 6x\\)\n\nNext, let's isolate \\(x\\) by subtracting \\(4x\\) from both sides:\n\n\\(6 = 2x\\)\n\nFinally, divide both sides by 2 to solve for \\(x\\):\n\n\\(3 = x\\)\n\nTherefore, the value of \\(x\\) is 3.\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"70%", "B":"75%", "C":"80%", "D":"85%", "E":"90%" }, "id":11059, "question":"Eva and Magnus took a road trip and shared the driving. Eva drove four times as many miles as Magnus drove. What percent of the total miles of the trip did Eva drive?", "explanations":{ "correct":"Let's assume that Magnus drove x miles. According to the question, Eva drove four times as many miles as Magnus, so Eva drove 4x miles.\n\nTo find the percentage of the total miles that Eva drove, we need to calculate the ratio of Eva's miles to the total miles of the trip.\n\nThe total miles of the trip is the sum of Magnus's miles and Eva's miles, which is x + 4x = 5x.\n\nTo find the percentage, we divide Eva's miles (4x) by the total miles (5x) and multiply by 100:\n\n(4x / 5x) * 100 = (4/5) * 100 = 80%\n\nTherefore, Eva drove 80% of the total miles of the trip.\n\nThe answer is C) 80%." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 1 } { 24 } \\)", "B":"\\(\\frac { 1 } { 12 } \\)", "C":"\\(\\frac { 7 } { 12 } \\)", "D":"\\(\\frac { 3 } { 4 } \\)", "E":"It depends on the number of pens in the box." }, "id":11063, "question":"A certain box contains only black, blue, and red pens. If a pen is chosen at random from the box, the probability of choosing a black pen is \\(\\frac { 1 } { 4 } \\) and the probability of choosing a blue pen is \\(\\frac { 1 } { 6 } \\). What is the probability of choosing a red pen?", "explanations":{ "correct":"To find the probability of choosing a red pen, we need to consider that the sum of the probabilities of choosing all possible outcomes must equal 1.\n\nGiven that the probability of choosing a black pen is \\(\\frac { 1 } { 4 }\\) and the probability of choosing a blue pen is \\(\\frac { 1 } { 6 }\\), we can calculate the probability of choosing a red pen as follows:\n\nLet \\(P(R)\\) represent the probability of choosing a red pen.\n\nSince there are only three colors of pens in the box (black, blue, and red), the sum of the probabilities of choosing each color must equal 1:\n\n\\(\\frac { 1 } { 4 } + \\frac { 1 } { 6 } + P(R) = 1\\)\n\nTo find \\(P(R)\\), we can rearrange the equation:\n\n\\(P(R) = 1 - \\left( \\frac { 1 } { 4 } + \\frac { 1 } { 6 } \\right)\\)\n\nSimplifying the expression inside the parentheses:\n\n\\(P(R) = 1 - \\left( \\frac { 3 } { 12 } + \\frac { 2 } { 12 } \\right)\\)\n\nCombining the fractions:\n\n\\(P(R) = 1 - \\frac { 5 } { 12 }\\)\n\nSubtracting the fractions:\n\n\\(P(R) = \\frac { 12 } { 12 } - \\frac { 5 } { 12 }\\)\n\nSimplifying:\n\n\\(P(R) = \\frac { 7 } { 12 }\\)\n\nTherefore, the probability of choosing a red pen is \\(\\frac { 7 } { 12 }\\).\n\nThe answer is C) \\(\\frac { 7 } { 12 }\\)." } }, { "answer":"B", "choices":{ "A":"None", "B":"1", "C":"More than 1 but less than 142", "D":"142", "E":"More than 142" }, "id":11069, "question":"How many primes less than 1000 are divisible by 7?", "explanations":{ "correct":"To determine how many primes less than 1000 are divisible by 7, we need to analyze the divisibility rule for 7. According to the rule, a number is divisible by 7 if and only if the difference between twice the units digit and the remaining digits is divisible by 7.\n\nLet's consider the primes less than 1000 and check if they are divisible by 7:\n- 2 is not divisible by 7.\n- 3 is not divisible by 7.\n- 5 is not divisible by 7.\n- 7 is divisible by 7.\n- 11 is not divisible by 7.\n- 13 is not divisible by 7.\n- 17 is not divisible by 7.\n- 19 is not divisible by 7.\n- 23 is not divisible by 7.\n- 29 is not divisible by 7.\n- 31 is not divisible by 7.\n- 37 is divisible by 7.\n- 41 is not divisible by 7.\n- 43 is not divisible by 7.\n- 47 is not divisible by 7.\n- 53 is not divisible by 7.\n- 59 is not divisible by 7.\n- 61 is not divisible by 7.\n- 67 is not divisible by 7.\n- 71 is not divisible by 7.\n- 73 is not divisible by 7.\n- 79 is not divisible by 7.\n- 83 is not divisible by 7.\n- 89 is not divisible by 7.\n- 97 is not divisible by 7.\n- ...\n\nFrom this analysis, we can see that only two primes less than 1000, namely 7 and 37, are divisible by 7. Therefore, the answer is B) 1.\n\nThe answer is B." } }, { "answer":"C", "choices":{ "A":"4", "B":"8", "C":"12", "D":"16", "E":"32" }, "id":11073, "question":"If a positive number a is 150 percent of \\(4p\\), and if \\(p\\) is 25 percent of \\(8b\\), then what is the value of \\(\\frac { a } { b } \\)?", "explanations":{ "correct":"To find the value of \\\\(\\frac { a } { b } \\\\), we need to first determine the values of \\\\(a\\\\) and \\\\(b\\\\) using the given information.\n\nLet's start with the equation \"a is 150 percent of \\\\(4p\\\\)\". This can be written as:\n\n\\\\(a = 1.5 \\times 4p\\\\)\n\nNext, we are told that \\\\(p\\\\) is 25 percent of \\\\(8b\\\\). This can be written as:\n\n\\\\(p = 0.25 \\times 8b\\\\)\n\nNow, let's substitute the value of \\\\(p\\\\) in the equation for \\\\(a\\\\):\n\n\\\\(a = 1.5 \\times 4 \\times (0.25 \\times 8b)\\\\)\n\nSimplifying this equation, we get:\n\n\\\\(a = 1.5 \\times 4 \\times 2b\\\\)\n\n\\\\(a = 12b\\\\)\n\nNow, we can substitute this value of \\\\(a\\\\) in the expression \\\\(\\frac { a } { b }\\\\):\n\n\\\\(\\frac { a } { b } = \\frac { 12b } { b }\\\\)\n\nSimplifying this expression, we get:\n\n\\\\(\\frac { a } { b } = 12\\\\)\n\nTherefore, the value of \\\\(\\frac { a } { b }\\\\) is 12.\n\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"\\(w(\\frac { r } { 100 } )(\\frac { s } { 100 } )\\)", "B":"\\(w(1 + \\frac { r } { 100 } )(1 -\\frac { s } { 100 } )\\)", "C":"\\(w(\\frac { rs } { 100 } )\\)", "D":"\\(w + \\frac { wr } { 100 } - \\frac { ws } { 100 } \\)", "E":"\\(w(\\frac { wr } { 100 } - \\frac { ws } { 100 } )\\)" }, "id":11081, "question":"The wholesale price of a car is w dollars. The retail price of this car is r percent greater than the wholesale price. During a special promotion, the retail price of the car is then discounted by s percent. Which of the following expressions represents the price, in dollars, of this car during the special promotion?", "explanations":{ "correct":"To find the price of the car during the special promotion, we need to calculate the retail price after the discount is applied.\n\nFirst, we know that the retail price is r percent greater than the wholesale price. This means the retail price is \\(w + \\frac{r}{100}w\\).\n\nNext, we apply the discount to the retail price. The discount is s percent, so the discounted price is \\((w + \\frac{r}{100}w) - \\frac{s}{100}(w + \\frac{r}{100}w)\\).\n\nSimplifying this expression, we get \\(w(1 + \\frac{r}{100})(1 - \\frac{s}{100})\\).\n\nTherefore, the correct answer is B) \\(w(1 + \\frac{r}{100})(1 - \\frac{s}{100})\\).\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"\\(m + 3\\)", "B":"\\(m^2 - 3\\)", "C":"\\(\\frac { m + 1 } { 9 } \\)", "D":"\\(\\frac { m - 9 } { 9 } \\)", "E":"\\(\\frac { 1 } { 3 } m\\)" }, "id":11086, "question":"When \\(n\\) is a positive integer, \\(9 + 3^ { n + 2 } = m\\), what is the value of \\(3^n\\) in terms of \\(m\\)?", "explanations":{ "correct":"To find the value of \\(3^n\\) in terms of \\(m\\), we need to isolate \\(3^n\\) in the given equation \\(9 + 3^{n + 2} = m\\).\n\nFirst, let's simplify the equation by subtracting 9 from both sides:\n\\[3^{n + 2} = m - 9\\]\n\nNext, we can rewrite \\(3^{n + 2}\\) as \\(3^n \\cdot 3^2\\), using the property of exponents:\n\\[3^n \\cdot 3^2 = m - 9\\]\n\nSimplifying further, we have:\n\\[9 \\cdot 3^n = m - 9\\]\n\nNow, divide both sides of the equation by 9 to isolate \\(3^n\\):\n\\[3^n = \\frac{m - 9}{9}\\]\n\nTherefore, the value of \\(3^n\\) in terms of \\(m\\) is \\(\\frac{m - 9}{9}\\).\n\nThe answer is D) \\(\\frac{m - 9}{9}\\)." } }, { "answer":"B", "choices":{ "A":"84", "B":"77", "C":"49", "D":"42", "E":"35" }, "id":11087, "question":"If the average (arithmetic mean) of 7 numbers is greater than 7 and less than 12, which of the following could be the sum of the 7 numbers?", "explanations":{ "correct":"To find the sum of the 7 numbers, we need to consider the range of possible values for the average. \n\nGiven that the average is greater than 7 and less than 12, we can determine the minimum and maximum possible sums.\n\nThe minimum possible sum occurs when the average is 7.1 (slightly greater than 7). In this case, the sum would be 7.1 * 7 = 49.7.\n\nThe maximum possible sum occurs when the average is 11.9 (slightly less than 12). In this case, the sum would be 11.9 * 7 = 83.3.\n\nTherefore, the sum of the 7 numbers could be any value between 49.7 and 83.3, inclusive.\n\nLooking at the answer choices:\nA) 84 is greater than the maximum possible sum of 83.3, so it is not a valid answer.\nB) 77 is within the range of possible sums, so it could be the sum of the 7 numbers.\nC) 49 is the minimum possible sum, so it could be the sum of the 7 numbers.\nD) 42 is less than the minimum possible sum of 49.7, so it is not a valid answer.\nE) 35 is less than the minimum possible sum of 49.7, so it is not a valid answer.\n\nTherefore, the only valid answer is B) 77.\n\nThe answer is B)." } }, { "answer":"B", "choices":{ "A":"5", "B":"6", "C":"7", "D":"8", "E":"9" }, "id":11091, "question":"A triangle has a perimeter of 16 and one side of length 4. If the lengths of the other two sides are equal, what is the length of each of the other two sides?", "explanations":{ "correct":"To find the lengths of the other two sides of the triangle, we can use the fact that the perimeter of a triangle is the sum of the lengths of all three sides.\n\nLet's call the length of the first side 4, and the lengths of the other two sides x. Since the lengths of the other two sides are equal, we can write the equation:\n\n4 + x + x = 16\n\nSimplifying the equation, we have:\n\n4 + 2x = 16\n\nSubtracting 4 from both sides, we get:\n\n2x = 12\n\nDividing both sides by 2, we find:\n\nx = 6\n\nTherefore, the lengths of the other two sides are both 6.\n\nThe answer is B) 6." } }, { "answer":"B", "choices":{ "A":"2", "B":"\\(2\\frac { 2 } { 9 } \\)", "C":"3", "D":"\\(3\\frac { 1 } { 3 } \\)", "E":"4" }, "id":11092, "question":"Mr. Lopez can clean the house in 5 hours. His son, Carl, can clean the house in 10 hours. How long will it take them to clean \\(\\frac { 2 } { 3 } \\) of the house, in hours, if they work together?", "explanations":{ "correct":"To solve this problem, we can use the concept of work rates. \n\nLet's first find the work rate of Mr. Lopez. We know that he can clean the house in 5 hours, so his work rate is 1 house cleaned per 5 hours, which can be written as \\(\\frac{1}{5}\\) house per hour.\n\nSimilarly, Carl's work rate is 1 house cleaned per 10 hours, or \\(\\frac{1}{10}\\) house per hour.\n\nWhen they work together, their work rates add up. So the combined work rate is \\(\\frac{1}{5} + \\frac{1}{10} = \\frac{3}{10}\\) house per hour.\n\nNow, we need to find how long it will take them to clean \\(\\frac{2}{3}\\) of the house. Let's represent the time it takes as \\(t\\) hours.\n\nSince their combined work rate is \\(\\frac{3}{10}\\) house per hour, we can set up the equation:\n\n\\(\\frac{3}{10} \\cdot t = \\frac{2}{3}\\)\n\nTo solve for \\(t\\), we can multiply both sides of the equation by \\(\\frac{10}{3}\\):\n\n\\(t = \\frac{2}{3} \\cdot \\frac{10}{3} = \\frac{20}{9} = 2\\frac{2}{9}\\)\n\nTherefore, it will take them \\(2\\frac{2}{9}\\) hours to clean \\(\\frac{2}{3}\\) of the house.\n\nThe answer is B) \\(2\\frac{2}{9}\\)." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{1}{3}, \\frac{1}{6}, \\frac{1}{9}\\)", "B":"\\(\\frac{2}{5}, \\frac{10}{25}, \\frac{12}{30}\\)", "C":"\\(\\frac{1}{2}, \\frac{2}{4}, \\frac{4}{6}\\)", "D":"\\(\\frac{2}{3}, \\frac{2}{6}, \\frac{2}{9}\\)", "E":"\\(\\frac{3}{4}, \\frac{10}{12}, \\frac{75}{100}\\)" }, "id":11093, "question":"Which of the following groups contains three fractions that are equal?", "explanations":{ "correct":"To determine which group contains three fractions that are equal, we need to compare the fractions in each group.\n\nA) \\(\\frac{1}{3}, \\frac{1}{6}, \\frac{1}{9}\\)\nThe fractions in this group have different denominators, so they are not equal.\n\nB) \\(\\frac{2}{5}, \\frac{10}{25}, \\frac{12}{30}\\)\nTo compare these fractions, we need to simplify them to their simplest form.\n\\(\\frac{2}{5}\\) is already in simplest form.\n\\(\\frac{10}{25}\\) can be simplified by dividing both the numerator and denominator by 5, resulting in \\(\\frac{2}{5}\\).\n\\(\\frac{12}{30}\\) can be simplified by dividing both the numerator and denominator by 6, resulting in \\(\\frac{2}{5}\\).\nTherefore, all three fractions in this group are equal.\n\nC) \\(\\frac{1}{2}, \\frac{2}{4}, \\frac{4}{6}\\)\nTo compare these fractions, we need to simplify them to their simplest form.\n\\(\\frac{1}{2}\\) is already in simplest form.\n\\(\\frac{2}{4}\\) can be simplified by dividing both the numerator and denominator by 2, resulting in \\(\\frac{1}{2}\\).\n\\(\\frac{4}{6}\\) can be simplified by dividing both the numerator and denominator by 2, resulting in \\(\\frac{2}{3}\\).\nTherefore, not all three fractions in this group are equal.\n\nD) \\(\\frac{2}{3}, \\frac{2}{6}, \\frac{2}{9}\\)\nTo compare these fractions, we need to simplify them to their simplest form.\n\\(\\frac{2}{3}\\) is already in simplest form.\n\\(\\frac{2}{6}\\) can be simplified by dividing both the numerator and denominator by 2, resulting in \\(\\frac{1}{3}\\).\n\\(\\frac{2}{9}\\) is already in simplest form.\nTherefore, not all three fractions in this group are equal.\n\nE) \\(\\frac{3}{4}, \\frac{10}{12}, \\frac{75}{100}\\)\nTo compare these fractions, we need to simplify them to their simplest form.\n\\(\\frac{3}{4}\\) is already in simplest form.\n\\(\\frac{10}{12}\\) can be simplified by dividing both the numerator and denominator by 2, resulting in \\(\\frac{5}{6}\\).\n\\(\\frac{75}{100}\\) can be simplified by dividing both the numerator and denominator by 25, resulting in \\(\\frac{3}{4}\\).\nTherefore, not all three fractions in this group are equal.\n\nBased on our analysis, the group that contains three fractions that are equal is B) \\(\\frac{2}{5}, \\frac{10}{25}, \\frac{12}{30}\\).\n\nThe answer is B." } }, { "answer":"C", "choices":{ "A":"1.2y", "B":"10y", "C":"12y", "D":"15y", "E":"120y" }, "id":11097, "question":"If \\(y > 0\\), what is 60 percent of 20y?", "explanations":{ "correct":"To find 60 percent of 20y, we need to multiply 20y by 0.60. \n\nStep 1: Multiply 20y by 0.60:\n20y * 0.60 = 12y\n\nTherefore, 60 percent of 20y is 12y.\n\nThe answer is C) 12y." } }, { "answer":"C", "choices":{ "A":"\\(3\\pi\\)", "B":"\\(6\\pi\\)", "C":"\\(9\\pi\\)", "D":"\\(12\\pi\\)", "E":"\\(36\\pi\\)" }, "id":11099, "question":"If A is point (-4, 1) and B is point (2, 1) what is the area of the circle that has \\(\\overline { AB } \\) as a diameter?", "explanations":{ "correct":"To find the area of a circle, we need to know its radius. The radius of a circle is half the length of its diameter. \n\nGiven that point A is (-4, 1) and point B is (2, 1), we can find the length of the diameter by using the distance formula. The distance formula is given by:\n\n\\(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\)\n\nSubstituting the coordinates of A and B into the formula, we have:\n\n\\(d = \\sqrt{(2 - (-4))^2 + (1 - 1)^2}\\)\n\\(d = \\sqrt{(6)^2 + (0)^2}\\)\n\\(d = \\sqrt{36}\\)\n\\(d = 6\\)\n\nSo, the diameter of the circle is 6 units. \n\nThe radius of the circle is half the length of the diameter, which is \\(6/2 = 3\\) units. \n\nNow, we can find the area of the circle using the formula for the area of a circle:\n\n\\(A = \\pi r^2\\)\n\nSubstituting the value of the radius, we have:\n\n\\(A = \\pi (3)^2\\)\n\\(A = 9\\pi\\)\n\nTherefore, the area of the circle that has \\(\\overline{AB}\\) as a diameter is \\(9\\pi\\).\n\nThe answer is C) \\(9\\pi\\)." } }, { "answer":"D", "choices":{ "A":"4", "B":"5", "C":"8", "D":"10", "E":"15" }, "id":11101, "question":"Patty uses 2 gallons of paint to cover 875 square feet of surface. At this rate, how many gallons will she need to cover 4,375 square feet of surface?", "explanations":{ "correct":"To find out how many gallons Patty will need to cover 4,375 square feet of surface, we can set up a proportion using the given information.\n\nWe know that Patty uses 2 gallons of paint to cover 875 square feet of surface. Let's set up the proportion:\n\n2 gallons / 875 square feet = x gallons / 4,375 square feet\n\nTo solve for x, we can cross-multiply:\n\n2 * 4,375 = 875 * x\n\n8,750 = 875x\n\nTo isolate x, we divide both sides of the equation by 875:\n\n8,750 / 875 = x\n\nx = 10\n\nTherefore, Patty will need 10 gallons of paint to cover 4,375 square feet of surface.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"2.5", "B":"20", "C":"35", "D":"40", "E":"70" }, "id":11105, "question":"If \\(q = \\frac{5}{4}st\\), what is the value of t when \\(s = 2\\) and \\(q = 50\\)?", "explanations":{ "correct":"To find the value of t when s = 2 and q = 50, we can substitute these values into the equation q = (5/4)st and solve for t.\n\nGiven:\nq = 50\ns = 2\n\nSubstituting these values into the equation:\n50 = (5/4)(2)t\n\nTo solve for t, we can start by multiplying both sides of the equation by 4/5 to isolate t:\n(4/5)(50) = (4/5)(5/4)(2)t\n\nSimplifying:\n40 = 2t\n\nNext, we can divide both sides of the equation by 2 to solve for t:\n40/2 = 2t/2\n\nSimplifying:\n20 = t\n\nTherefore, the value of t when s = 2 and q = 50 is t = 20.\n\nThe answer is B) 20." } }, { "answer":"A", "choices":{ "A":"\\(22\\frac { 2 } { 9 } %\\)", "B":"\\(28\\frac { 4 } { 7 } %\\)", "C":"\\(33\\frac { 1 } { 3 } %\\)", "D":"\\(44\\frac { 4 } { 9 } %\\)", "E":"It cannot be determined from the information given." }, "id":11106, "question":"Phil's Phone Shop sells three models of cellular phones, priced at \\$ 100, \\$ 125, and \\$ 225. In January, Phil sold exactly the same number of each model. What percent of the total income from the sales of cellular phones was attributable to sales of the cheapest model?", "explanations":{ "correct":"To find the percent of the total income from the sales of the cheapest model, we need to determine the proportion of the income that comes from selling the cheapest model.\n\nLet's assume that Phil sold \\(x\\) units of each model in January. Since he sold the same number of each model, the total number of units sold is \\(3x\\).\n\nThe income from selling the cheapest model is \\(100 \\times x\\), as each unit is priced at \\$ 100. The total income from all three models is \\(100 \\times x + 125 \\times x + 225 \\times x\\).\n\nTo find the percent of the total income from the sales of the cheapest model, we divide the income from selling the cheapest model by the total income and multiply by 100:\n\n\\(\\frac{{100 \\times x}}{{100 \\times x + 125 \\times x + 225 \\times x}} \\times 100\\)\n\nSimplifying the expression:\n\n\\(\\frac{{100x}}{{450x}} \\times 100 = \\frac{{100}}{{450}} \\times 100 = \\frac{{10000}}{{450}}\\)\n\nTo simplify the fraction, we can divide both the numerator and denominator by 50:\n\n\\(\\frac{{200}}{{9}}\\)\n\nTherefore, the percent of the total income from the sales of the cheapest model is \\(22\\frac{2}{9}%\\).\n\nThe answer is A) \\(22\\frac{2}{9}%\\)." } }, { "answer":"B", "choices":{ "A":"-2", "B":"-1", "C":"0", "D":"2", "E":"4" }, "id":11107, "question":"If \\(|1 - 2k| = k + 4\\), which of the following could be the value of \\(k\\)?", "explanations":{ "correct":"To find the possible values of \\(k\\), we need to solve the equation \\(|1 - 2k| = k + 4\\).\n\nStep 1: Split the equation into two cases based on the absolute value.\n\nCase 1: \\(1 - 2k = k + 4\\)\\(\\newline\\)In this case, we don't need to consider the absolute value since \\(1 - 2k\\) is already positive.\nSimplifying the equation:\n\\(1 - 2k = k + 4\\)\n\\(1 = 3k + 4\\)\n\\(3k = -3\\)\n\\(k = -1\\)\n\nCase 2: \\(1 - 2k = -(k + 4)\\)\\(\\newline\\)In this case, we need to consider the absolute value since \\(1 - 2k\\) is negative.\nSimplifying the equation:\n\\(1 - 2k = -k - 4\\)\n\\(1 = -k - 4 + 2k\\)\n\\(1 = -3k - 4\\)\n\\(3 = -3k\\)\n\\(k = -1\\)\n\nStep 2: Check if the values of \\(k\\) satisfy the original equation.\n\nFor \\(k = -1\\):\n\\(1 - 2(-1) = -1 + 4\\)\n\\(1 + 2 = 3\\)\n\\(3 = 3\\)\n\nSince both values of \\(k\\) satisfy the original equation, the possible values of \\(k\\) are -1.\n\nTherefore, the answer is B) -1." } }, { "answer":"C", "choices":{ "A":"1", "B":"2", "C":"\\(2^26\\)", "D":"\\(2^27\\)", "E":"\\(2^32\\)" }, "id":11109, "question":"The first term of sequence I is 2, and each subsequent term is 2 more than the preceding term. The first term of sequence II is 2 and each subsequent term is 2 times the preceding term. What is the ratio of the 32nd term of sequence II to the 32nd term of sequence I?", "explanations":{ "correct":"To find the ratio of the 32nd term of sequence II to the 32nd term of sequence I, we need to determine the value of each term in both sequences.\n\nFor sequence I, we know that the first term is 2, and each subsequent term is 2 more than the preceding term. So, the second term is 2 + 2 = 4, the third term is 4 + 2 = 6, and so on. We can see that the nth term of sequence I can be represented as 2n.\n\nFor sequence II, we know that the first term is 2, and each subsequent term is 2 times the preceding term. So, the second term is 2 * 2 = 4, the third term is 4 * 2 = 8, and so on. We can see that the nth term of sequence II can be represented as 2^n.\n\nTo find the ratio of the 32nd term of sequence II to the 32nd term of sequence I, we substitute n = 32 into the formulas for each sequence.\n\nFor sequence I, the 32nd term is 2 * 32 = 64.\n\nFor sequence II, the 32nd term is 2^32.\n\nTherefore, the ratio of the 32nd term of sequence II to the 32nd term of sequence I is (2^32) / 64.\n\nSimplifying this expression, we have (2^(32-6)) = 2^26.\n\nTherefore, the answer is C) \\(2^26\\)." } }, { "answer":"B", "choices":{ "A":"2.5%", "B":"13%", "C":"20%", "D":"35%", "E":"52%" }, "id":11117, "question":"While away at school, Eileen receives an allowance of \\$ 400 each month, 35 percent of which she uses to pay her bills. If she budgets 30 percent of the remainder for shopping, allots \\$ 130 for entertainment, and saves the rest of the money, what percentage of her allowance is she able to save each month?", "explanations":{ "correct":"To find the percentage of Eileen's allowance that she is able to save each month, we need to follow these steps:\n\n1. Calculate the amount of money Eileen uses to pay her bills: \\$ 400 * 35% = \\$ 140.\n\n2. Calculate the remaining amount after paying the bills: \\$ 400 - \\$ 140 = \\$ 260.\n\n3. Calculate the amount of money Eileen budgets for shopping: \\$ 260 * 30% = \\$ 78.\n\n4. Subtract the amount budgeted for shopping from the remaining amount: \\$ 260 - \\$ 78 = \\$ 182.\n\n5. Subtract the amount allotted for entertainment from the remaining amount: \\$ 182 - \\$ 130 = \\$ 52.\n\n6. Calculate the percentage of the remaining amount that Eileen is able to save: ($ 52 / \\$ 400) * 100% = 13%.\n\nTherefore, the answer is B) 13%." } }, { "answer":"A", "choices":{ "A":"The second graph will be 9 units above the first.", "B":"The second graph will be 3 units above the first.", "C":"The second graph will be 5 units bellow the first.", "D":"The second graph will be 9 units bellow the first.", "E":"The second graph will have a greater slope than the first." }, "id":11120, "question":"The graph of \\(y = x - 2\\) is drawn on a set of coordinate axes. If the graph of \\(y = x + 7\\) is then drawn on the same axes, how does the second graph compare to the first?", "explanations":{ "correct":"To compare the graphs of \\(y = x - 2\\) and \\(y = x + 7\\), we need to analyze their slopes and y-intercepts.\n\nThe equation \\(y = x - 2\\) is in slope-intercept form, where the coefficient of \\(x\\) represents the slope and the constant term represents the y-intercept. In this case, the slope is 1 and the y-intercept is -2.\n\nSimilarly, the equation \\(y = x + 7\\) also has a slope of 1, but the y-intercept is 7.\n\nSince both equations have the same slope, the second graph will have the same steepness as the first graph. However, the y-intercept of the second graph is higher than the y-intercept of the first graph. This means that the second graph will be shifted vertically upwards compared to the first graph.\n\nTo determine the vertical shift, we can subtract the y-intercept of the first graph from the y-intercept of the second graph: \\(7 - (-2) = 9\\).\n\nTherefore, the second graph will be 9 units above the first graph.\n\nThe answer is A) The second graph will be 9 units above the first." } }, { "answer":"D", "choices":{ "A":"\\(2S\\)", "B":"\\(S + 50\\)", "C":"\\(S + 100\\)", "D":"\\(S + 2500\\)", "E":"\\(S + 10000\\)" }, "id":11126, "question":"S = 1 + 2 + 3 + 4 + ... + 49 + 50 T = 51 + 52 + 53 + ... + 99 + 100 If the sum of the positive integers from 1 to 50 is \\(S\\), and the sum of the positive integers from 51 to 100 is \\(T\\), what is \\(T\\) in terms of \\(S\\)?", "explanations":{ "correct":"To find the sum of the positive integers from 51 to 100, we can use the formula for the sum of an arithmetic series. The formula is given by:\n\n\\(S_n = \\frac{n}{2}(a_1 + a_n)\\)\n\nwhere \\(S_n\\) is the sum of the series, \\(n\\) is the number of terms, \\(a_1\\) is the first term, and \\(a_n\\) is the last term.\n\\(\\newline\\)In this case, the first term is 51 and the last term is 100. So, we have:\n\n\\(T = \\frac{50}{2}(51 + 100)\\)\n\nSimplifying this expression, we get:\n\n\\(T = 25(151)\\)\n\n\\(T = 3775\\)\n\nNow, we need to express \\(T\\) in terms of \\(S\\). We know that \\(S\\) is the sum of the positive integers from 1 to 50. Using the formula for the sum of an arithmetic series again, we have:\n\n\\(S = \\frac{50}{2}(1 + 50)\\)\n\nSimplifying this expression, we get:\n\n\\(S = 25(51)\\)\n\n\\(S = 1275\\)\n\nTo express \\(T\\) in terms of \\(S\\), we can substitute the value of \\(S\\) into the expression for \\(T\\):\n\n\\(T = 3775 = S + 2500\\)\n\nTherefore, the answer is D) \\(S + 2500\\)." } }, { "answer":"B", "choices":{ "A":"4", "B":"5", "C":"6", "D":"7", "E":"8" }, "id":11135, "question":"Which of the following is the distance between the points (7, 3) and (11, 0) in the xy-coordinate plane?", "explanations":{ "correct":"To find the distance between two points in the xy-coordinate plane, we can use the distance formula. The distance formula is given by:\n\nd = √((x2 - x1)^2 + (y2 - y1)^2)\n\\(\\newline\\)In this case, the coordinates of the first point are (7, 3) and the coordinates of the second point are (11, 0). Let's substitute these values into the distance formula:\n\nd = √((11 - 7)^2 + (0 - 3)^2)\n = √(4^2 + (-3)^2)\n = √(16 + 9)\n = √25\n = 5\n\nTherefore, the distance between the points (7, 3) and (11, 0) in the xy-coordinate plane is 5. \n\nThe answer is B) 5." } }, { "answer":"C", "choices":{ "A":"3", "B":"4", "C":"5", "D":"6", "E":"7" }, "id":11137, "question":"The Suzukis are having a picnic at their house where they will serve pasta salad to all of the guests. One pound of pasta salad either serves 6 adults or 10 children. If the Suzukis have 34 guests, 10 of whom are children, how many pounds of pasta salad will they need to serve all of the guests?", "explanations":{ "correct":"To determine how many pounds of pasta salad the Suzukis will need to serve all of the guests, we need to calculate the number of pounds required for the adults and the number of pounds required for the children separately.\n\nGiven that 1 pound of pasta salad serves 6 adults, we can calculate the number of pounds needed for the adults by dividing the total number of adult guests by 6. Since there are 34 guests in total and 10 of them are children, the number of adult guests is 34 - 10 = 24. Therefore, the number of pounds needed for the adults is 24 / 6 = 4 pounds.\n\nSimilarly, given that 1 pound of pasta salad serves 10 children, we can calculate the number of pounds needed for the children by dividing the total number of children by 10. In this case, there are 10 children, so the number of pounds needed for the children is 10 / 10 = 1 pound.\n\nTo find the total number of pounds needed, we add the pounds needed for the adults and the pounds needed for the children: 4 pounds + 1 pound = 5 pounds.\n\nTherefore, the Suzukis will need 5 pounds of pasta salad to serve all of the guests.\n\nThe answer is C) 5." } }, { "answer":"E", "choices":{ "A":"48", "B":"72", "C":"96", "D":"120", "E":"144" }, "id":11139, "question":"At a certain party, an executive committee provided one soda for 8 people, one apple for 4 people and one cake for 6 people. If the total number of sodas, apples, and cakes was 78, how many people were at the party?", "explanations":{ "correct":"Let's solve this problem step-by-step.\n\nFirst, let's find the number of sodas, apples, and cakes provided at the party. We are given that one soda is provided for 8 people, one apple is provided for 4 people, and one cake is provided for 6 people.\n\nLet's assume there are x people at the party. Therefore, the number of sodas provided would be x/8, the number of apples provided would be x/4, and the number of cakes provided would be x/6.\n\nAccording to the problem, the total number of sodas, apples, and cakes is 78. So we can write the equation:\n\nx/8 + x/4 + x/6 = 78\n\nTo solve this equation, we need to find a common denominator. The least common multiple of 8, 4, and 6 is 24. Multiplying each term by 24, we get:\n\n3x + 6x + 4x = 78 * 24\n13x = 78 * 24\n13x = 1872\n\nDividing both sides of the equation by 13, we find:\n\nx = 1872 / 13\nx ≈ 144\n\nTherefore, there were approximately 144 people at the party.\n\nThe answer is E) 144." } }, { "answer":"A", "choices":{ "A":"\\(8a + 16\\)", "B":"\\(4a + 4\\)", "C":"\\(8a\\)", "D":"\\(4a + 16\\)", "E":"\\(4a\\)" }, "id":11141, "question":"If a, b, and c are consecutive even integers, such that \\(0 < a < b < c\\), what is \\(c^2 - a^2\\) in terms of \\(a\\) ?", "explanations":{ "correct":"To find the value of \\(c^2 - a^2\\) in terms of \\(a\\), we need to first determine the values of \\(a\\), \\(b\\), and \\(c\\) in terms of \\(a\\).\n\nGiven that \\(a\\), \\(b\\), and \\(c\\) are consecutive even integers, we can express them as \\(a\\), \\(a + 2\\), and \\(a + 4\\) respectively.\n\nNow, let's substitute these values into the expression \\(c^2 - a^2\\):\n\n\\((a + 4)^2 - a^2\\)\n\nExpanding the square of \\(a + 4\\), we get:\n\n\\(a^2 + 8a + 16 - a^2\\)\n\nSimplifying the expression, we can cancel out the \\(a^2\\) terms:\n\n\\(8a + 16\\)\n\nTherefore, \\(c^2 - a^2\\) in terms of \\(a\\) is \\(8a + 16\\).\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"290", "B":"522", "C":"870", "D":"1,450", "E":"2,610" }, "id":11142, "question":"If 3 kilograms of rose petals are needed to produce 5 grams of perfume, how many kilograms of rose petals are needed to produce 870 grams of perfume?", "explanations":{ "correct":"To find out how many kilograms of rose petals are needed to produce 870 grams of perfume, we can set up a proportion using the given information.\n\nWe know that 3 kilograms of rose petals produce 5 grams of perfume. Let's set up the proportion:\n\n3 kilograms / 5 grams = x kilograms / 870 grams\n\nTo solve for x, we can cross multiply:\n\n3 * 870 = 5 * x\n\n2,610 = 5x\n\nDividing both sides of the equation by 5:\n\n2,610 / 5 = x\n\nx = 522\n\nTherefore, the answer is B) 522." } }, { "answer":"C", "choices":{ "A":"(3,1)", "B":"(3,-3)", "C":"(5,-1)", "D":"(5,3)", "E":"(7,-5)" }, "id":11144, "question":"Square ABCD in the coordinate plane has points A(1,3), B(5,7) and C(9,3). Which of the following could be the coordinates of point D?", "explanations":{ "correct":"To find the coordinates of point D, we need to understand the properties of a square. In a square, opposite sides are parallel and equal in length. \n\nGiven that point A is (1,3) and point B is (5,7), we can find the length of one side of the square using the distance formula. \n\nThe distance formula is: \nd = √((x2 - x1)^2 + (y2 - y1)^2)\n\nUsing the distance formula, we can find the length of AB: \ndAB = √((5 - 1)^2 + (7 - 3)^2) = √(4^2 + 4^2) = √(16 + 16) = √32\n\nSince opposite sides of a square are equal in length, the length of side AB is equal to the length of side CD. Therefore, the length of CD is also √32.\n\nNow, let's consider the possible coordinates for point D:\n\nA) (3,1): The distance from A to D would be √((3 - 1)^2 + (1 - 3)^2) = √(2^2 + (-2)^2) = √8, which is not equal to √32. So, (3,1) cannot be the coordinates of point D.\n\nB) (3,-3): The distance from A to D would be √((3 - 1)^2 + (-3 - 3)^2) = √(2^2 + (-6)^2) = √40, which is not equal to √32. So, (3,-3) cannot be the coordinates of point D.\n\nC) (5,-1): The distance from A to D would be √((5 - 1)^2 + (-1 - 3)^2) = √(4^2 + (-4)^2) = √32. The length of side AD matches the length of side AB, so (5,-1) could be the coordinates of point D.\n\nD) (5,3): The distance from A to D would be √((5 - 1)^2 + (3 - 3)^2) = √(4^2 + 0^2) = √16, which is not equal to √32. So, (5,3) cannot be the coordinates of point D.\n\nE) (7,-5): The distance from A to D would be √((7 - 1)^2 + (-5 - 3)^2) = √(6^2 + (-8)^2) = √100, which is not equal to √32. So, (7,-5) cannot be the coordinates of point D.\n\nBased on the calculations, the only option that matches the length of side AB and could be the coordinates of point D is C) (5,-1).\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { 49 } { 121 } \\)", "B":"\\(\\frac { 7 } { 11 } \\)", "C":"1", "D":"\\(\\frac { 11 } { 7 } \\)", "E":"\\(\\frac { 121 } { 49 } \\)" }, "id":11147, "question":"If \\(a(\\frac { 7 } { 11 } ) = (\\frac { 7 } { 11 } )b\\), then \\(\\frac { a } { b } \\) =", "explanations":{ "correct":"To find the value of \\(\\frac{a}{b}\\), we can start by multiplying both sides of the equation \\(a(\\frac{7}{11}) = (\\frac{7}{11})b\\) by \\(\\frac{11}{7}\\) to eliminate the fractions.\n\n\\(\\frac{11}{7} \\cdot a(\\frac{7}{11}) = \\frac{11}{7} \\cdot (\\frac{7}{11})b\\)\n\nSimplifying, we have:\n\n\\(a = b\\)\n\nSince \\(a\\) is equal to \\(b\\), the value of \\(\\frac{a}{b}\\) is equal to 1.\n\nTherefore, the answer is C) 1." } }, { "answer":"A", "choices":{ "A":"250", "B":"500", "C":"750", "D":"1,000", "E":"2,000" }, "id":11153, "question":"The populations of two species of bacteria in a petri dish vary inversely. If there are 1,000 bacteria of species A when there are 500 bacteria of species B, how many bacteria of species B are there when there are 2,000 bacteria of species A ?", "explanations":{ "correct":"To solve this problem, we can use the inverse variation formula, which states that the product of the two variables remains constant. Let's denote the number of bacteria of species A as A and the number of bacteria of species B as B.\n\nAccording to the problem, the populations of the two species of bacteria vary inversely. This means that A * B = k, where k is a constant.\n\nWe are given that when there are 1,000 bacteria of species A, there are 500 bacteria of species B. So, we can set up the equation as follows:\n\n1,000 * 500 = k\n\nSimplifying this equation, we find that k = 500,000.\n\nNow, we need to find the number of bacteria of species B when there are 2,000 bacteria of species A. Let's denote this as B'.\n\nUsing the inverse variation formula, we have:\n\n2,000 * B' = 500,000\n\nDividing both sides of the equation by 2,000, we find:\n\nB' = 500,000 / 2,000\n\nSimplifying this expression, we get:\n\nB' = 250\n\nTherefore, when there are 2,000 bacteria of species A, there are 250 bacteria of species B.\n\nThe answer is A) 250." } }, { "answer":"E", "choices":{ "A":"the set of all positive integers", "B":"the set of all positive real numbers", "C":"the set of positive multiples of 3", "D":"the set of positive multiples of 5", "E":"the set of positive multiples of 6" }, "id":11155, "question":"If X is the set of positive multiples of 2 and Y is the set of positive multiples of 3, then the intersection of X and Y is", "explanations":{ "correct":"To find the intersection of two sets, we need to identify the elements that are common to both sets. \n\\(\\newline\\)In this case, X represents the set of positive multiples of 2, which includes numbers like 2, 4, 6, 8, and so on. \nY represents the set of positive multiples of 3, which includes numbers like 3, 6, 9, 12, and so on. \n\nTo find the intersection, we need to identify the numbers that are present in both sets. Looking at the lists, we can see that the number 6 appears in both sets. \n\nTherefore, the intersection of X and Y is the set of positive multiples of 6. \n\nThe answer is E) the set of positive multiples of 6." } }, { "answer":"B", "choices":{ "A":"6.2", "B":"8", "C":"9", "D":"16", "E":"62" }, "id":11156, "question":"1. Choose a number between 0.5 and 0.99. 2. Multiply the number from the previous step by 20. 3. Determine the largest integer less than or equal to the number obtained from the previous step. 4. Subtract 4 from the number obtained from the previous step. 5. Write down the resulting number. If 0.62 is the number chosen in step 1, what is the number written in step 5?", "explanations":{ "correct":"To find the number written in step 5, we need to follow the given steps:\n\n1. Choose a number between 0.5 and 0.99.\nThe number chosen in step 1 is 0.62.\n\n2. Multiply the number from the previous step by 20.\n0.62 * 20 = 12.4\n\n3. Determine the largest integer less than or equal to the number obtained from the previous step.\nThe largest integer less than or equal to 12.4 is 12.\n\n4. Subtract 4 from the number obtained from the previous step.\n12 - 4 = 8\n\n5. Write down the resulting number.\nThe resulting number is 8.\n\nTherefore, the number written in step 5 is 8.\n\nThe answer is B) 8." } }, { "answer":"D", "choices":{ "A":"0", "B":"3", "C":"6", "D":"7", "E":"35" }, "id":11158, "question":"If \\(f(x) = \\frac{x^3 - 6}{x^2 - 2x + 6}\\), then what is \\(f(6)\\)?", "explanations":{ "correct":"To find the value of \\(f(6)\\), we need to substitute \\(x = 6\\) into the given function \\(f(x) = \\frac{x^3 - 6}{x^2 - 2x + 6}\\).\n\nStep 1: Substitute \\(x = 6\\) into the numerator:\n\\(6^3 - 6 = 216 - 6 = 210\\).\n\nStep 2: Substitute \\(x = 6\\) into the denominator:\n\\(6^2 - 2(6) + 6 = 36 - 12 + 6 = 30\\).\n\nStep 3: Divide the numerator by the denominator:\n\\(\\frac{210}{30} = 7\\).\n\nTherefore, \\(f(6) = 7\\).\n\nThe answer is D) 7." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { p + q } { 2 } \\)", "B":"\\(\\frac { 10p + 20q } { 30 } \\)", "C":"\\(\\frac { 10p + 20q } { p + q } \\)", "D":"\\(\\frac { 10p + 20q } { 100 } \\)", "E":"\\(\\frac { 100p + 200q } { p + q } \\)" }, "id":11162, "question":"If \\(p\\) gallons of 10 percent antifreeze solution is added to \\(q\\) gallons of 20 percent antifreeze solution, what is the percent antifreeze of the resulting solution in terms of \\(p\\) and \\(q\\)?", "explanations":{ "correct":"To find the percent antifreeze of the resulting solution, we need to consider the amount of antifreeze in each solution and the total volume of the resulting solution.\n\nLet's start by calculating the amount of antifreeze in the first solution. Since it is a 10 percent antifreeze solution, we have \\(0.10p\\) gallons of antifreeze.\n\nNext, let's calculate the amount of antifreeze in the second solution. Since it is a 20 percent antifreeze solution, we have \\(0.20q\\) gallons of antifreeze.\n\nTo find the total amount of antifreeze in the resulting solution, we add the amounts from the two solutions: \\(0.10p + 0.20q\\).\n\nNow, let's find the total volume of the resulting solution. We add the volumes of the two solutions: \\(p + q\\).\n\nTo calculate the percent antifreeze of the resulting solution, we divide the total amount of antifreeze by the total volume of the resulting solution and multiply by 100:\n\n\\(\\frac{{0.10p + 0.20q}}{{p + q}} \\times 100\\)\n\nSimplifying this expression, we get:\n\n\\(\\frac{{10p + 20q}}{{p + q}}\\)\n\nTherefore, the correct answer is C) \\(\\frac{{10p + 20q}}{{p + q}}\\).\n\nThe answer is C." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { 1 } { 80 } \\)", "B":"\\(\\frac { 1 } { 24 } \\)", "C":"\\(\\frac { 1 } { 8 } \\)", "D":"\\(\\frac { 1 } { 6 } \\)", "E":"\\(\\frac { 1 } { 5 } \\)" }, "id":11168, "question":"If Jonas has already run 4 miles in a 20-mile race, what fraction of the race has he already run?", "explanations":{ "correct":"To find the fraction of the race that Jonas has already run, we need to divide the distance he has already run (4 miles) by the total distance of the race (20 miles).\n\nSo, the fraction can be calculated as \\(\\frac { 4 } { 20 }\\).\n\nTo simplify this fraction, we can divide both the numerator and denominator by their greatest common divisor, which is 4.\n\n\\(\\frac { 4 } { 20 } = \\frac { 1 } { 5 }\\).\n\nTherefore, Jonas has already run \\(\\frac { 1 } { 5 }\\) of the race.\n\nThe answer is E) \\(\\frac { 1 } { 5 }\\)." } }, { "answer":"C", "choices":{ "A":"39", "B":"41", "C":"42", "D":"45", "E":"81" }, "id":11174, "question":"If \\(\\frac{y}{y - 3} = \\frac{42}{39}\\), then what does y equal?", "explanations":{ "correct":"To find the value of y, we can start by cross-multiplying the given equation. \n\nCross-multiplying means multiplying the numerator of the first fraction with the denominator of the second fraction, and vice versa. \n\nSo, we have:\n\n\\(y \\cdot 39 = 42 \\cdot (y - 3)\\)\n\nExpanding the equation, we get:\n\n\\(39y = 42y - 126\\)\n\nNext, we can simplify the equation by subtracting 42y from both sides:\n\n\\(39y - 42y = -126\\)\n\nSimplifying further, we have:\n\n\\(-3y = -126\\)\n\nTo isolate y, we can divide both sides of the equation by -3:\n\n\\(\\frac{-3y}{-3} = \\frac{-126}{-3}\\)\n\nThis simplifies to:\n\n\\(y = 42\\)\n\nTherefore, the answer is C) 42." } }, { "answer":"C", "choices":{ "A":"10.25", "B":"14.5", "C":"17", "D":"18.25", "E":"19" }, "id":11180, "question":"If the average (arithmetic mean) of 14 consecutive integers is 20.5, what is the average of the first seven integers?", "explanations":{ "correct":"To find the average of a set of consecutive integers, we can use the formula: average = (first term + last term) / 2.\n\\(\\newline\\)In this case, we are given that the average of 14 consecutive integers is 20.5. Let's assume the first term of these 14 integers is x. Since they are consecutive, the last term would be x + 13.\n\nUsing the formula, we can write the equation as:\n20.5 = (x + x + 13) / 2\n\nSimplifying the equation, we get:\n41 = 2x + 13\n\nSubtracting 13 from both sides, we have:\n28 = 2x\n\nDividing both sides by 2, we find:\nx = 14\n\nSo, the first term of the 14 consecutive integers is 14.\n\nNow, we need to find the average of the first seven integers. The first seven integers would be 14, 15, 16, 17, 18, 19, and 20.\n\nTo find the average, we add up these seven integers and divide by 7:\n(14 + 15 + 16 + 17 + 18 + 19 + 20) / 7 = 119 / 7 = 17\n\nTherefore, the average of the first seven integers is 17.\n\nThe answer is C) 17." } }, { "answer":"A", "choices":{ "A":"7", "B":"14", "C":"15", "D":"21", "E":"It cannot be determined from the information given" }, "id":11184, "question":"\\(x + y = 10\\) \\(y + z = 15\\) \\(x + z = 17\\) What is the average (arithmetic mean) of \\(x\\), \\(y\\), and \\(z\\)?", "explanations":{ "correct":"To find the average of \\(x\\), \\(y\\), and \\(z\\), we need to first determine the values of \\(x\\), \\(y\\), and \\(z\\). \n\nGiven the system of equations:\n\\(x + y = 10\\) ...(Equation 1)\n\\(y + z = 15\\) ...(Equation 2)\n\\(x + z = 17\\) ...(Equation 3)\n\nWe can solve this system of equations using various methods, such as substitution or elimination. Let's use the elimination method:\n\nAdding Equation 1 and Equation 3, we get:\n\\((x + y) + (x + z) = 10 + 17\\)\n\\(2x + y + z = 27\\) ...(Equation 4)\n\nSubtracting Equation 2 from Equation 4, we get:\n\\(2x + y + z - (y + z) = 27 - 15\\)\n\\(2x = 12\\)\n\\(x = 6\\)\n\nSubstituting the value of \\(x\\) into Equation 1, we get:\n\\(6 + y = 10\\)\n\\(y = 4\\)\n\nSubstituting the values of \\(x\\) and \\(y\\) into Equation 2, we get:\n\\(4 + z = 15\\)\n\\(z = 11\\)\n\nNow that we have the values of \\(x\\), \\(y\\), and \\(z\\), we can find their average by adding them up and dividing by 3:\n\\(\\text{Average} = \\frac{x + y + z}{3} = \\frac{6 + 4 + 11}{3} = \\frac{21}{3} = 7\\)\n\nTherefore, the average (arithmetic mean) of \\(x\\), \\(y\\), and \\(z\\) is 7.\n\nThe answer is A) 7." } }, { "answer":"B", "choices":{ "A":"44", "B":"96", "C":"128", "D":"144", "E":"168" }, "id":11185, "question":"A frame 2 inches wide is placed around a rectangular picture with dimensions 8 inches by 12 inches. What is the area of the frame in square inches?", "explanations":{ "correct":"To find the area of the frame, we need to subtract the area of the inner rectangle (picture) from the area of the outer rectangle (frame).\n\nThe dimensions of the outer rectangle (frame) can be calculated by adding 2 inches to each side of the inner rectangle (picture). \n\nThe length of the outer rectangle is 8 inches + 2 inches + 2 inches = 12 inches.\nThe width of the outer rectangle is 12 inches + 2 inches + 2 inches = 16 inches.\n\nThe area of the outer rectangle (frame) is the product of its length and width:\nArea of the outer rectangle = 12 inches * 16 inches = 192 square inches.\n\nThe dimensions of the inner rectangle (picture) are given as 8 inches by 12 inches.\n\nThe area of the inner rectangle (picture) is the product of its length and width:\nArea of the inner rectangle = 8 inches * 12 inches = 96 square inches.\n\nTo find the area of the frame, we subtract the area of the inner rectangle (picture) from the area of the outer rectangle (frame):\nArea of the frame = Area of the outer rectangle - Area of the inner rectangle\nArea of the frame = 192 square inches - 96 square inches = 96 square inches.\n\nTherefore, the area of the frame is 96 square inches.\n\nThe answer is B." } }, { "answer":"E", "choices":{ "A":"xyz", "B":"\\(\\frac{z}{x}\\)", "C":"\\(\\frac{z^2}{x^2}\\)", "D":"\\(\\frac{z^3}{x^3}\\)", "E":"\\(\\frac{z^4}{x^4}\\)" }, "id":11191, "question":"If \\(xyz \\neq 0\\), then \\(\\frac{x^2 y^3 z^6}{x^6 y^3 z^2} =\\)", "explanations":{ "correct":"To simplify the expression \\(\\frac{x^2 y^3 z^6}{x^6 y^3 z^2}\\), we can use the properties of exponents. \n\nFirst, let's simplify the numerator: \\(x^2 y^3 z^6\\). Since the exponents are being multiplied, we can add the exponents of the same base. Therefore, \\(x^2 y^3 z^6\\) simplifies to \\(x^{2+0} y^{3+0} z^{6+0}\\), which further simplifies to \\(x^2 y^3 z^6\\).\n\nNext, let's simplify the denominator: \\(x^6 y^3 z^2\\). Again, using the property of exponents, we add the exponents of the same base. Therefore, \\(x^6 y^3 z^2\\) simplifies to \\(x^{6+0} y^{3+0} z^{2+0}\\), which further simplifies to \\(x^6 y^3 z^2\\).\n\nNow, we can substitute the simplified numerator and denominator back into the original expression: \\(\\frac{x^2 y^3 z^6}{x^6 y^3 z^2}\\).\n\nSince the numerator and denominator have the same bases (x, y, and z), we can cancel out the common factors. Cancelling out \\(y^3\\) from the numerator and denominator, we are left with \\(\\frac{x^2 z^6}{x^6 z^2}\\).\n\nNext, we can simplify the expression further by subtracting the exponents of the same base. Therefore, \\(\\frac{x^2 z^6}{x^6 z^2}\\) simplifies to \\(x^{2-6} z^{6-2}\\), which further simplifies to \\(x^{-4} z^4\\).\n\nFinally, we can rewrite \\(x^{-4}\\) as \\(\\frac{1}{x^4}\\). Therefore, the expression \\(\\frac{x^2 z^6}{x^6 z^2}\\) simplifies to \\(\\frac{1}{x^4} z^4\\).\n\nTherefore, the answer is E) \\(\\frac{z^4}{x^4}\\)." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"I and II only", "D":"I and III only", "E":"I, II, and III" }, "id":11192, "question":"If \\(a < b < c < d\\) and the average (arithmetic mean) of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is 10, which of the following MUST be true? \\(\\newline\\)I. \\(a + d - b + c\\) \\(\\newline\\)II. \\(a < 10 and d > 10\\) \\(\\newline\\)III. \\(b < 10 and c > 10\\)", "explanations":{ "correct":"To find the average of \\(a\\), \\(b\\), \\(c\\), and \\(d\\), we add them together and divide by 4. Since the average is 10, we have:\n\n\\(\\frac{a + b + c + d}{4} = 10\\)\n\nMultiplying both sides by 4, we get:\n\n\\(a + b + c + d = 40\\)\n\nNow let's analyze each statement:\n\\(\\newline\\)I. \\(a + d - b + c\\)\n\nRearranging the terms, we have:\n\n\\(a - b + c + d\\)\n\nSince \\(a < b < c < d\\), we can see that \\(a - b\\) will be negative, and \\(c + d\\) will be positive. Therefore, \\(a - b + c + d\\) can be any value, positive or negative, depending on the specific values of \\(a\\), \\(b\\), \\(c\\), and \\(d\\). This statement is not necessarily true.\n\\(\\newline\\)II. \\(a < 10\\) and \\(d > 10\\)\n\nSince the average of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is 10, it is possible for \\(a\\) to be less than 10 and \\(d\\) to be greater than 10. For example, if \\(a = 5\\), \\(b = 8\\), \\(c = 12\\), and \\(d = 15\\), both conditions are satisfied. This statement can be true.\n\\(\\newline\\)III. \\(b < 10\\) and \\(c > 10\\)\n\nSince the average of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is 10, it is not possible for both \\(b\\) to be less than 10 and \\(c\\) to be greater than 10. The average would be greater than 10 in this case. This statement is not true.\n\nBased on our analysis, statement II is the only one that must be true. Therefore, the answer is B) II only." } }, { "answer":"D", "choices":{ "A":"16", "B":"22", "C":"23", "D":"32", "E":"36" }, "id":11194, "question":"The percent increase from 6 to 16 is equal to the percent increase from 12 to what number?", "explanations":{ "correct":"To find the percent increase from 6 to 16, we can use the formula:\n\nPercent Increase = (New Value - Original Value) / Original Value * 100\n\nLet's calculate the percent increase from 6 to 16:\n\nPercent Increase = (16 - 6) / 6 * 100\nPercent Increase = 10 / 6 * 100\nPercent Increase = 1.6667 * 100\nPercent Increase = 166.67%\n\nNow, we need to find the number that has the same percent increase when going from 12 to that number.\n\nLet's set up the equation:\n\n166.67% = (New Value - 12) / 12 * 100\n\nTo solve for the new value, we can cross multiply:\n\n166.67 * 12 = New Value - 12\n\n2000.04 = New Value - 12\n\nNew Value = 2000.04 + 12\nNew Value = 2012.04\n\nTherefore, the number that has the same percent increase from 12 to that number as the percent increase from 6 to 16 is 2012.04.\n\nThe answer is D) 32." } }, { "answer":"D", "choices":{ "A":"60", "B":"78", "C":"138", "D":"150", "E":"168" }, "id":11195, "question":"A bag contains 228 marbles, and one marble is selected at random. The probability of selecting a black marble is \\(\\frac { 5 } { 19 } \\), and the probability of selecting a gold marble is \\(\\frac { 3 } { 38 } \\). How many marbles in the bag are neither black nor gold?", "explanations":{ "correct":"Let's start by finding the total probability of selecting a black or gold marble. The probability of selecting a black marble is given as \\( \\frac{5}{19} \\) and the probability of selecting a gold marble is given as \\( \\frac{3}{38} \\).\n\nTo find the total probability, we add the probabilities of selecting a black marble and a gold marble:\n\n\\( \\frac{5}{19} + \\frac{3}{38} = \\frac{10}{38} + \\frac{3}{38} = \\frac{13}{38} \\)\n\nThis means that the probability of selecting a marble that is either black or gold is \\( \\frac{13}{38} \\).\n\nNow, let's find the probability of selecting a marble that is neither black nor gold. We can do this by subtracting the probability of selecting a black or gold marble from 1 (since the sum of all probabilities must equal 1):\n\n\\( 1 - \\frac{13}{38} = \\frac{38}{38} - \\frac{13}{38} = \\frac{25}{38} \\)\n\nSo, the probability of selecting a marble that is neither black nor gold is \\( \\frac{25}{38} \\).\n\nTo find the number of marbles in the bag that are neither black nor gold, we multiply the probability by the total number of marbles:\n\n\\( \\frac{25}{38} \\times 228 = \\frac{5700}{38} = 150 \\)\n\nTherefore, the answer is D) 150." } }, { "answer":"B", "choices":{ "A":"95", "B":"93", "C":"90", "D":"89", "E":"88" }, "id":11197, "question":"The average (arithmetic mean) of a student's grades on four history tests is 86. If the student received two grades of 83 and one of 85, what grade did the student receive on fourth test?", "explanations":{ "correct":"To find the grade the student received on the fourth test, we need to use the concept of the average (arithmetic mean). \n\nThe average of the student's grades on four history tests is given as 86. This means that the sum of the grades on all four tests is 86 multiplied by 4, which is 344.\n\nWe are given that the student received two grades of 83 and one grade of 85. To find the grade on the fourth test, we can subtract the sum of these three grades from the total sum of all four grades.\n\nSum of the three known grades: 83 + 83 + 85 = 251\n\nGrade on the fourth test = Total sum of all four grades - Sum of the three known grades\nGrade on the fourth test = 344 - 251 = 93\n\nTherefore, the student received a grade of 93 on the fourth test.\n\nThe answer is B) 93." } }, { "answer":"C", "choices":{ "A":"20", "B":"18", "C":"8", "D":"6", "E":"4" }, "id":11203, "question":"If \\(2 \\leq p \\leq 6\\) and \\(1 \\leq q \\leq 12\\), what is the smallest value of \\((p + q)^2 - (p - q)^2\\)?", "explanations":{ "correct":"To find the smallest value of \\((p + q)^2 - (p - q)^2\\), we need to simplify the expression step-by-step.\n\nFirst, let's expand the expression:\n\\((p + q)^2 - (p - q)^2 = (p^2 + 2pq + q^2) - (p^2 - 2pq + q^2)\\)\n\nNext, let's simplify the expression by removing the parentheses:\n\\(p^2 + 2pq + q^2 - p^2 + 2pq - q^2\\)\n\nNow, let's combine like terms:\n\\(4pq\\)\n\nSince we want to find the smallest value of \\(4pq\\), we need to find the smallest possible values for \\(p\\) and \\(q\\).\n\nGiven that \\(2 \\leq p \\leq 6\\) and \\(1 \\leq q \\leq 12\\), the smallest possible values for \\(p\\) and \\(q\\) are \\(2\\) and \\(1\\) respectively.\n\nSubstituting these values into \\(4pq\\), we get:\n\\(4 \\cdot 2 \\cdot 1 = 8\\)\n\nTherefore, the smallest value of \\((p + q)^2 - (p - q)^2\\) is 8.\n\nThe answer is C) 8." } }, { "answer":"E", "choices":{ "A":"4", "B":"8", "C":"9", "D":"12", "E":"13" }, "id":11211, "question":"There are 5 red, 5 green, 5 blue, and 5 yellow hats packaged in 20 identical, unmarked boxes, with 1 hat per box. What is the least number of boxes that must be selected in order to be sure that among the boxes selected, 4 or more contain hats of the same color?", "explanations":{ "correct":"To determine the least number of boxes that must be selected in order to be sure that among the boxes selected, 4 or more contain hats of the same color, we need to consider the worst-case scenario.\n\\(\\newline\\)In the worst-case scenario, we would select 3 boxes of each color, which would give us a total of 3 x 4 = 12 boxes. However, this would not guarantee that we have 4 or more boxes of the same color.\n\nTo ensure that we have 4 or more boxes of the same color, we need to select an additional box of any color. This additional box will guarantee that we have at least 4 boxes of one color.\n\nTherefore, the least number of boxes that must be selected is 12 + 1 = 13.\n\nThe answer is E." } }, { "answer":"C", "choices":{ "A":"\\(-1\\)", "B":"1", "C":"2", "D":"3", "E":"5" }, "id":11215, "question":"If a and b are integers and \\(2a + 5b = 15\\), which of the following CANNOT be a value of b?", "explanations":{ "correct":"To determine which value of b cannot be a solution, we need to find the possible values of b that satisfy the equation \\(2a + 5b = 15\\).\n\nFirst, let's rearrange the equation to solve for a:\n\\(2a = 15 - 5b\\)\n\\(a = \\frac{15 - 5b}{2}\\)\n\nSince a and b are integers, the numerator of the fraction must be divisible by 2. Therefore, 15 - 5b must be an even number.\n\nNow, let's analyze each option:\n\nA) If b = -1, then 15 - 5(-1) = 20, which is an even number. This is a possible value for b.\n\nB) If b = 1, then 15 - 5(1) = 10, which is an even number. This is a possible value for b.\n\nC) If b = 2, then 15 - 5(2) = 5, which is an odd number. This is not a possible value for b.\n\nD) If b = 3, then 15 - 5(3) = 0, which is an even number. This is a possible value for b.\n\nE) If b = 5, then 15 - 5(5) = -10, which is an even number. This is a possible value for b.\n\nTherefore, the only value that cannot be a solution for b is C) 2.\n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"\\(\\frac{25}{2}\\)", "B":"\\(\\frac{19}{2}\\)", "C":"11", "D":"5", "E":"50" }, "id":11216, "question":"If \\(4(n + 6) = 44\\), what is the value of n?", "explanations":{ "correct":"To find the value of \\(n\\), we need to solve the equation \\(4(n + 6) = 44\\).\n\nFirst, we can simplify the equation by distributing the 4 to both terms inside the parentheses:\n\\(4n + 24 = 44\\).\n\nNext, we can isolate the variable \\(n\\) by subtracting 24 from both sides of the equation:\n\\(4n = 44 - 24\\),\n\\(4n = 20\\).\n\nTo solve for \\(n\\), we divide both sides of the equation by 4:\n\\(n = \\frac{20}{4}\\),\n\\(n = 5\\).\n\nTherefore, the value of \\(n\\) is 5.\n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"54 degrees", "B":"240 degrees", "C":"720 degrees", "D":"2,160 degrees", "E":"2,400 degrees" }, "id":11219, "question":"Carmel is spinning a basketball on his finger so that it turns around completely every 1.5 seconds. How many degrees does the logo on the ball turn in 10 seconds, assuming it is not on the spinning axis?", "explanations":{ "correct":"To find the number of degrees the logo on the ball turns in 10 seconds, we need to determine how many complete rotations the ball makes in that time.\n\nWe know that the ball completes one full rotation every 1.5 seconds. Therefore, in 10 seconds, the ball will complete 10 / 1.5 = 6.67 rotations.\n\nSince each rotation is 360 degrees, we can find the total number of degrees the logo turns by multiplying the number of rotations by 360.\n\n6.67 rotations * 360 degrees/rotation = 2,401.2 degrees\n\nHowever, the question specifies that the logo is not on the spinning axis. This means that the logo will not complete a full rotation for every rotation of the ball. Instead, it will turn slightly less than 360 degrees for each rotation.\n\nTo find the exact number of degrees the logo turns, we need to multiply the number of rotations by the fraction of a full rotation the logo completes.\n\nSince the logo completes one full rotation for every 1.5 seconds, it completes 360 degrees in 1.5 seconds. Therefore, the logo turns 360 degrees / 1.5 seconds = 240 degrees per second.\n\nTo find the number of degrees the logo turns in 10 seconds, we multiply the number of seconds by the rate at which the logo turns.\n\n10 seconds * 240 degrees/second = 2,400 degrees\n\nTherefore, the answer is E) 2,400 degrees." } }, { "answer":"A", "choices":{ "A":"Green", "B":"Red", "C":"Blue", "D":"Yellow", "E":"Purple" }, "id":11222, "question":"A line of plastic ducks moves across a conveyor belt at a shooting gallery. The color of each duck follows the repeating pattern orange, green, red, blue, yellow, purple, continuing indefinitely. If the first duck is orange, what is the color of the 50th duck?", "explanations":{ "correct":"To determine the color of the 50th duck, we need to analyze the pattern of colors. The pattern repeats every 6 ducks: orange, green, red, blue, yellow, purple. \n\nTo find the color of the 50th duck, we can divide 50 by 6 and find the remainder. \n\n50 divided by 6 is 8 with a remainder of 2. \n\nSince the pattern repeats every 6 ducks, the color of the 50th duck will be the same as the color of the 2nd duck in the pattern, which is green. \n\nTherefore, the answer is A) Green." } }, { "answer":"A", "choices":{ "A":"8", "B":"6", "C":"4", "D":"3", "E":"2" }, "id":11225, "question":"A square and a circle drawn in a plane can intersect in at most how many points?", "explanations":{ "correct":"To determine the maximum number of points at which a square and a circle can intersect, we need to consider the possible scenarios.\n\n1. The square is completely inside the circle: In this case, the square and the circle do not intersect at any point.\n\n2. The circle is completely inside the square: In this case, the circle and the square can intersect at a maximum of 4 points, where the circle touches the sides of the square.\n\n3. The square and the circle overlap partially: In this case, the square and the circle can intersect at a maximum of 8 points. This occurs when the circle intersects the square at 4 points and also touches the sides of the square at 4 points.\n\nTherefore, the square and the circle can intersect in at most 8 points.\n\nThe answer is A) 8." } }, { "answer":"B", "choices":{ "A":"4", "B":"2", "C":"\\(\\frac{7}{6}\\)", "D":"\\(\\frac{1}{2}\\)", "E":"\\(\\frac{1}{4}\\)" }, "id":11226, "question":"Flour, sugar, and baking soda are mixed by weight in the ratio of 6:4:2, respectively, to produce a certain type of cookie. In order to make 6 pounds of this dough, what weight of sugar, in pounds, is required?", "explanations":{ "correct":"To find the weight of sugar required to make 6 pounds of dough, we need to determine the weight of sugar in the given ratio.\n\nThe ratio of flour to sugar to baking soda is 6:4:2. This means that for every 6 parts of flour, we have 4 parts of sugar and 2 parts of baking soda.\n\nTo find the weight of sugar, we need to determine the fraction of the total ratio that represents sugar. We can do this by dividing the weight of sugar by the sum of the weights of all the ingredients.\n\nThe total ratio is 6 + 4 + 2 = 12.\n\nTo find the weight of sugar, we divide the weight of sugar by the total ratio and multiply it by the total weight of the dough.\n\nWeight of sugar = (4/12) * 6 = 2 pounds.\n\nTherefore, the weight of sugar required to make 6 pounds of dough is 2 pounds.\n\nThe answer is B) 2." } }, { "answer":"A", "choices":{ "A":"10", "B":"20", "C":"50", "D":"100", "E":"200" }, "id":11227, "question":"If \\(k = 10\\), what is the value of \\(\\frac { 2k(k + 1) - 2k(k - 1) } { 4 } \\) ?", "explanations":{ "correct":"To find the value of the expression \\\\(\\frac { 2k(k + 1) - 2k(k - 1) } { 4 }\\\\), we need to substitute the value of \\\\(k\\\\) and simplify the expression step-by-step.\n\nGiven that \\\\(k = 10\\\\), we can substitute this value into the expression:\n\n\\\\(\\frac { 2(10)(10 + 1) - 2(10)(10 - 1) } { 4 }\\\\)\n\nNext, we simplify the expression within the parentheses:\n\n\\\\(\\frac { 2(10)(11) - 2(10)(9) } { 4 }\\\\)\n\nNow, we can simplify the multiplication:\n\n\\\\(\\frac { 220 - 180 } { 4 }\\\\)\n\nSubtracting \\\\(220 - 180\\\\) gives us:\n\n\\\\(\\frac { 40 } { 4 }\\\\)\n\nFinally, we divide \\\\(40\\\\) by \\\\(4\\\\) to get the final answer:\n\n\\\\(\\frac { 40 } { 4 } = 10\\\\)\n\nTherefore, the value of the expression is \\\\(10\\\\).\n\nThe answer is A) 10." } }, { "answer":"C", "choices":{ "A":"\\(-4\\)", "B":"\\(-2\\)", "C":"0", "D":"2", "E":"4" }, "id":11229, "question":"If \\(\\frac{4}{\\sqrt{r + 4}} = 2\\), what is the value of r?", "explanations":{ "correct":"To find the value of \\(r\\), we can start by isolating the variable \\(r\\) in the given equation.\n\nGiven: \\(\\frac{4}{\\sqrt{r + 4}} = 2\\)\n\nFirst, we can multiply both sides of the equation by \\(\\sqrt{r + 4}\\) to eliminate the fraction:\n\n\\(4 = 2\\sqrt{r + 4}\\)\n\nNext, we can divide both sides of the equation by 2 to solve for \\(\\sqrt{r + 4}\\):\n\n\\(2 = \\sqrt{r + 4}\\)\n\nTo eliminate the square root, we can square both sides of the equation:\n\n\\(2^2 = (\\sqrt{r + 4})^2\\)\n\nSimplifying further:\n\n\\(4 = r + 4\\)\n\nNow, we can isolate \\(r\\) by subtracting 4 from both sides of the equation:\n\n\\(4 - 4 = r + 4 - 4\\)\n\n\\(0 = r\\)\n\nTherefore, the value of \\(r\\) is 0.\n\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 1 } { 4 } \\)", "B":"\\(\\frac { 1 } { 2 } \\)", "C":"1", "D":"2", "E":"4" }, "id":11230, "question":"The volume of a cylinder whose height is 4 and whose radius is 2 is how many times the volume of a cylinder whose height is 2 and whose radius is 4?", "explanations":{ "correct":"To find the volume of a cylinder, we use the formula V = πr^2h, where r is the radius and h is the height.\n\nLet's calculate the volume of the first cylinder with a height of 4 and a radius of 2:\nV1 = π(2^2)(4)\nV1 = 4π(4)\nV1 = 16π\n\nNow, let's calculate the volume of the second cylinder with a height of 2 and a radius of 4:\nV2 = π(4^2)(2)\nV2 = 16π(2)\nV2 = 32π\n\nTo find how many times the volume of the first cylinder is compared to the second cylinder, we divide V1 by V2:\nV1/V2 = (16π)/(32π)\nV1/V2 = 1/2\n\nTherefore, the volume of the first cylinder is half the volume of the second cylinder.\n\nThe answer is B) \\(\\frac { 1 } { 2 } \\)." } }, { "answer":"B", "choices":{ "A":"\\(\\{-2,-1,0,1,2\\}\\)", "B":"\\(\\{-2,-1, 0,1, 3\\}\\)", "C":"\\(\\{-2,0,0,0, 2\\}\\)", "D":"\\(\\{-3,-1,0,1,2\\}\\)", "E":"\\(\\{-3,-1, 0,1, 3\\}\\)" }, "id":11231, "question":"For which of the following sets of numbers is the average (arithmetic mean) greater than the median?", "explanations":{ "correct":"To determine which set of numbers has an average greater than the median, we need to compare the average and the median for each set.\n\nLet's start with set A: {-2, -1, 0, 1, 2}.\nThe median is 0, which is the middle number when the numbers are arranged in ascending order. The average is ( -2 + -1 + 0 + 1 + 2 ) / 5 = 0 / 5 = 0.\nSince the average (0) is equal to the median (0), set A does not have an average greater than the median.\n\nNext, let's consider set B: {-2, -1, 0, 1, 3}.\nThe median is 0, which is the middle number when the numbers are arranged in ascending order. The average is ( -2 + -1 + 0 + 1 + 3 ) / 5 = 1 / 5 = 0.2.\nSince the average (0.2) is greater than the median (0), set B does have an average greater than the median.\n\nMoving on to set C: {-2, 0, 0, 0, 2}.\nThe median is 0, which is the middle number when the numbers are arranged in ascending order. The average is ( -2 + 0 + 0 + 0 + 2 ) / 5 = 0 / 5 = 0.\nSince the average (0) is equal to the median (0), set C does not have an average greater than the median.\n\nNow let's analyze set D: {-3, -1, 0, 1, 2}.\nThe median is 0, which is the middle number when the numbers are arranged in ascending order. The average is ( -3 + -1 + 0 + 1 + 2 ) / 5 = -1 / 5 = -0.2.\nSince the average (-0.2) is less than the median (0), set D does not have an average greater than the median.\n\nFinally, let's examine set E: {-3, -1, 0, 1, 3}.\nThe median is 0, which is the middle number when the numbers are arranged in ascending order. The average is ( -3 + -1 + 0 + 1 + 3 ) / 5 = 0 / 5 = 0.\nSince the average (0) is equal to the median (0), set E does not have an average greater than the median.\n\nBased on our analysis, the only set that has an average greater than the median is set B: {-2, -1, 0, 1, 3}.\nTherefore, the answer is B." } }, { "answer":"B", "choices":{ "A":"100", "B":"121", "C":"144", "D":"222", "E":"1,011" }, "id":11240, "question":"At a certain school, if the ratio of teachers to students is 1 to 10, which of the following could be the total number of teachers and students?", "explanations":{ "correct":"To find the total number of teachers and students, we need to consider the given ratio of teachers to students, which is 1 to 10. This means that for every 1 teacher, there are 10 students.\n\nLet's analyze each option:\n\nA) 100: If there are 100 people in total, the number of teachers would be 100/11 = 9.09, which is not a whole number. Therefore, option A is not possible.\n\nB) 121: If there are 121 people in total, the number of teachers would be 121/11 = 11, which is a whole number. The number of students would be 121 - 11 = 110. This satisfies the given ratio of 1 teacher to 10 students. Therefore, option B is possible.\n\nC) 144: If there are 144 people in total, the number of teachers would be 144/11 = 13.09, which is not a whole number. Therefore, option C is not possible.\n\nD) 222: If there are 222 people in total, the number of teachers would be 222/11 = 20.18, which is not a whole number. Therefore, option D is not possible.\n\nE) 1,011: If there are 1,011 people in total, the number of teachers would be 1,011/11 = 91.91, which is not a whole number. Therefore, option E is not possible.\n\nBased on the analysis, the only option that could be the total number of teachers and students is B) 121.\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"1", "B":"2", "C":"4", "D":"6", "E":"15" }, "id":11242, "question":"Ninety people work in a certain office. If the ratio of men to women is 7:8, how many more women than men work in the office?", "explanations":{ "correct":"To find the number of men and women working in the office, we need to divide the total number of people by the sum of the parts of the ratio (7 + 8 = 15). \n\nFirst, we find the number of men by multiplying the total number of people (90) by the fraction representing the ratio of men to the total ratio (7/15):\nNumber of men = 90 * (7/15) = 42\n\nNext, we find the number of women by multiplying the total number of people (90) by the fraction representing the ratio of women to the total ratio (8/15):\nNumber of women = 90 * (8/15) = 48\n\nTo find the difference between the number of women and men, we subtract the number of men from the number of women:\nDifference = Number of women - Number of men = 48 - 42 = 6\n\nTherefore, there are 6 more women than men working in the office.\n\nThe answer is D) 6." } }, { "answer":"E", "choices":{ "A":"\\(j^2 + 9\\)", "B":"\\(j^2 + 6j + 9\\)", "C":"\\((j^2 + 9) \\sqrt { 2 } \\)", "D":"\\(\\frac { j^2 + 9 } { 2 } \\)", "E":"\\(\\frac { j^2 + 6j + 9 } { 2 } \\)" }, "id":11243, "question":"The length of the diagonal of a square is \\(j + 3\\). Which of the following represents the area of the square?", "explanations":{ "correct":"To find the area of a square, we need to know the length of one of its sides. However, in this question, we are given the length of the diagonal of the square, which is \\(j + 3\\). \n\nTo find the length of one side of the square, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the lengths of the other two sides.\n\\(\\newline\\)In this case, the diagonal of the square is the hypotenuse, and the two sides of the square are equal in length (since it is a square). Let's call the length of one side of the square \\(s\\).\n\nUsing the Pythagorean theorem, we have:\n\n\\((j + 3)^2 = s^2 + s^2\\)\n\nSimplifying this equation, we get:\n\n\\(j^2 + 6j + 9 = 2s^2\\)\n\nTo find the area of the square, we need to square the length of one of its sides. From the equation above, we can solve for \\(s^2\\) by dividing both sides by 2:\n\n\\(\\frac{j^2 + 6j + 9}{2} = s^2\\)\n\nTherefore, the area of the square is \\(\\frac{j^2 + 6j + 9}{2}\\).\n\nThe answer is E) \\(\\frac{j^2 + 6j + 9}{2}\\)." } }, { "answer":"C", "choices":{ "A":"\\(x < x^2 < x^3\\)", "B":"\\(x^2 < x < x^3\\)", "C":"\\(x < x^3 < x^2\\)", "D":"\\(x^3 < x^2 < x\\)", "E":"\\(x^3 < x < x^2\\)" }, "id":11244, "question":"If \\(-1 < x < 0\\), which of the following is true?", "explanations":{ "correct":"To determine which of the given options is true when \\(-1 < x < 0\\), we need to evaluate each option step-by-step.\n\nOption A: \\(x < x^2 < x^3\\)\nLet's substitute \\(x = -\\frac{1}{2}\\) into this option:\n\\(-\\frac{1}{2} < \\left(-\\frac{1}{2}\\right)^2 < \\left(-\\frac{1}{2}\\right)^3\\)\n\\(-\\frac{1}{2} < \\frac{1}{4} < -\\frac{1}{8}\\)\nThis statement is not true since \\(\\frac{1}{4}\\) is not less than \\(-\\frac{1}{8}\\).\n\nOption B: \\(x^2 < x < x^3\\)\nLet's substitute \\(x = -\\frac{1}{2}\\) into this option:\n\\(\\left(-\\frac{1}{2}\\right)^2 < -\\frac{1}{2} < \\left(-\\frac{1}{2}\\right)^3\\)\n\\(\\frac{1}{4} < -\\frac{1}{2} < -\\frac{1}{8}\\)\nThis statement is also not true since \\(\\frac{1}{4}\\) is not less than \\(-\\frac{1}{2}\\).\n\nOption C: \\(x < x^3 < x^2\\)\nLet's substitute \\(x = -\\frac{1}{2}\\) into this option:\n\\(-\\frac{1}{2} < \\left(-\\frac{1}{2}\\right)^3 < \\left(-\\frac{1}{2}\\right)^2\\)\n\\(-\\frac{1}{2} < -\\frac{1}{8} < \\frac{1}{4}\\)\nThis statement is true since \\(-\\frac{1}{8}\\) is indeed less than \\(\\frac{1}{4}\\).\n\nOption D: \\(x^3 < x^2 < x\\)\nLet's substitute \\(x = -\\frac{1}{2}\\) into this option:\n\\(\\left(-\\frac{1}{2}\\right)^3 < \\left(-\\frac{1}{2}\\right)^2 < -\\frac{1}{2}\\)\n\\(-\\frac{1}{8} < \\frac{1}{4} < -\\frac{1}{2}\\)\nThis statement is not true since \\(\\frac{1}{4}\\) is not less than \\(-\\frac{1}{2}\\).\n\nOption E: \\(x^3 < x < x^2\\)\nLet's substitute \\(x = -\\frac{1}{2}\\) into this option:\n\\(\\left(-\\frac{1}{2}\\right)^3 < -\\frac{1}{2} < \\left(-\\frac{1}{2}\\right)^2\\)\n\\(-\\frac{1}{8} < -\\frac{1}{2} < \\frac{1}{4}\\)\nThis statement is not true since \\(-\\frac{1}{2}\\) is not less than \\(\\frac{1}{4}\\).\n\nAfter evaluating each option, we find that the only true statement is Option C: \\(x < x^3 < x^2\\).\n\nTherefore, the answer is C." } }, { "answer":"E", "choices":{ "A":"\\(k^3n^2\\)", "B":"\\(\\frac { k^3 } { n^2 } \\)", "C":"\\(\\frac { 1 } { k^3n^2 } \\)", "D":"\\(\\frac { k^6 } { n^3 } \\)", "E":"\\(\\frac { n^2 } { k^3 } \\)" }, "id":11246, "question":"If \\(k^6 = m^ { -2 } n^4\\), where \\(k\\), \\(m\\), and \\(n\\) are positive numbers, which of the following is the value of \\(m\\)?", "explanations":{ "correct":"To find the value of \\(m\\) in the equation \\(k^6 = m^{-2}n^4\\), we need to isolate \\(m\\) on one side of the equation.\n\nFirst, let's rewrite \\(m^{-2}\\) as \\(\\frac{1}{m^2}\\). Substituting this into the equation, we have \\(k^6 = \\frac{1}{m^2}n^4\\).\n\nNext, let's multiply both sides of the equation by \\(m^2\\) to get rid of the fraction. This gives us \\(k^6m^2 = n^4\\).\n\nNow, let's divide both sides of the equation by \\(k^6\\) to solve for \\(m^2\\). This gives us \\(m^2 = \\frac{n^4}{k^6}\\).\n\nFinally, to find the value of \\(m\\), we take the square root of both sides of the equation. This gives us \\(m = \\sqrt{\\frac{n^4}{k^6}}\\).\n\nSimplifying the expression under the square root, we have \\(m = \\frac{n^2}{k^3}\\).\n\nTherefore, the answer is E) \\(\\frac{n^2}{k^3}\\)." } }, { "answer":"B", "choices":{ "A":"0", "B":"1", "C":"\\(a^{-4}\\)", "D":"a", "E":"\\(a^4\\)" }, "id":11250, "question":"If a is a positive number, which of the following is equal to \\(a^2 \\times a^{-2}\\)?", "explanations":{ "correct":"To simplify the expression \\(a^2 \\times a^{-2}\\), we can use the rule of exponents that states \\(a^m \\times a^n = a^{m+n}\\).\n\\(\\newline\\)In this case, we have \\(m = 2\\) and \\(n = -2\\). Applying the rule, we get:\n\n\\(a^2 \\times a^{-2} = a^{2 + (-2)} = a^0\\)\n\nAccording to another rule of exponents, any number raised to the power of 0 is equal to 1. Therefore, \\(a^0 = 1\\).\n\nSo, the expression \\(a^2 \\times a^{-2}\\) simplifies to 1.\n\nThe answer is B) 1." } }, { "answer":"E", "choices":{ "A":"(12, 10)", "B":"(11, 10)", "C":"(6, 5)", "D":"(2, 3)", "E":"(4, 7)" }, "id":11252, "question":"Which of the following points \\((x, y)\\) does NOT satisfy \\(|5 - x| > y - 5\\)?", "explanations":{ "correct":"To determine which point does not satisfy the inequality \\\\(|5 - x| > y - 5\\\\), we need to substitute the coordinates of each point into the inequality and check if the inequality holds true.\n\nLet's start with point A) (12, 10):\nSubstituting the coordinates into the inequality, we get \\\\(|5 - 12| > 10 - 5\\\\, which simplifies to \\\\(|-7| > 5\\\\). This is true since the absolute value of -7 is greater than 5.\n\nNext, let's check point B) (11, 10):\nSubstituting the coordinates into the inequality, we get \\\\(|5 - 11| > 10 - 5\\\\, which simplifies to \\\\(|-6| > 5\\\\). This is also true since the absolute value of -6 is greater than 5.\n\nNow, let's check point C) (6, 5):\nSubstituting the coordinates into the inequality, we get \\\\(|5 - 6| > 5 - 5\\\\, which simplifies to \\\\(|-1| > 0\\\\). This is true since the absolute value of -1 is greater than 0.\n\nNext, let's check point D) (2, 3):\nSubstituting the coordinates into the inequality, we get \\\\(|5 - 2| > 3 - 5\\\\, which simplifies to \\\\(|3| > -2\\\\). This is also true since the absolute value of 3 is greater than -2.\n\nFinally, let's check point E) (4, 7):\nSubstituting the coordinates into the inequality, we get \\\\(|5 - 4| > 7 - 5\\\\, which simplifies to \\\\(|1| > 2\\\\). This is not true since the absolute value of 1 is not greater than 2.\n\nTherefore, the point that does NOT satisfy the inequality is E) (4, 7).\n\nThe answer is E." } }, { "answer":"D", "choices":{ "A":"11:30 AM", "B":"12:00 PM", "C":"12:30 PM", "D":"12:50 PM", "E":"1:00 PM" }, "id":11257, "question":"Alexis programmed her VCR to record for exactly 225 minutes. If it began recording at 9:05 A.M., at what time did it stop recording?", "explanations":{ "correct":"To find the time when the VCR stopped recording, we need to add 225 minutes to the starting time of 9:05 A.M.\n\nFirst, let's convert the starting time to minutes. Since there are 60 minutes in an hour, we can multiply 9 by 60 and add 5 to get the total number of minutes: 9 * 60 + 5 = 545 minutes.\n\nNext, we add 225 minutes to the starting time: 545 + 225 = 770 minutes.\n\nNow, we need to convert the total minutes back to hours and minutes. Since there are 60 minutes in an hour, we divide 770 by 60 to get the number of hours: 770 / 60 = 12 hours.\n\nWe also need to find the remaining minutes. We can find this by taking the remainder when dividing 770 by 60: 770 % 60 = 50 minutes.\n\nTherefore, the VCR stopped recording at 12:50 PM.\n\nThe answer is D) 12:50 PM." } }, { "answer":"D", "choices":{ "A":"\\(6\\frac { 2 } { 3 } \\)", "B":"\\(4\\frac { 2 } { 3 } \\)", "C":"\\(4\\frac { 1 } { 3 } \\)", "D":"\\(4\\frac { 1 } { 6 } \\)", "E":"\\(3\\frac { 1 } { 3 } \\)" }, "id":11259, "question":"Gilda drove 650 miles at an average speed of 50 miles per hour. How many miles per hour faster would she have had to drive in order for the trip to have taken I hour less?", "explanations":{ "correct":"To find the answer, we need to determine the current travel time and the desired travel time. \n\nFirst, we calculate the current travel time by dividing the total distance (650 miles) by the average speed (50 miles per hour):\n\nCurrent travel time = 650 miles / 50 miles per hour = 13 hours\n\nNext, we need to determine the desired travel time, which is 1 hour less than the current travel time:\n\nDesired travel time = Current travel time - 1 hour = 13 hours - 1 hour = 12 hours\n\nNow, we can calculate the speed needed to achieve the desired travel time. We divide the total distance (650 miles) by the desired travel time (12 hours):\n\nSpeed needed = 650 miles / 12 hours = 54\\frac { 1 } { 6 } miles per hour\n\nTo find how many miles per hour faster Gilda would have to drive, we subtract her current speed (50 miles per hour) from the speed needed:\n\nSpeed difference = Speed needed - Current speed = 54\\frac { 1 } { 6 } miles per hour - 50 miles per hour = 4\\frac { 1 } { 6 } miles per hour\n\nTherefore, the answer is D) \\(4\\frac { 1 } { 6 } \\)." } }, { "answer":"D", "choices":{ "A":"The 15th", "B":"The 18th", "C":"The 25th", "D":"The 26th", "E":"The 27th" }, "id":11262, "question":"The terms in a sequence are given by \\(a_n = a + (n -1)d\\), where \\(n\\) is the number of terms, \\(a\\) is the first term, and each term after the first is found by adding 8 to the term immediately preceding it. If the first term is 10, which term in the sequence is equal to 210?", "explanations":{ "correct":"To find the term in the sequence that is equal to 210, we need to use the formula given: \\(a_n = a + (n-1)d\\), where \\(a\\) is the first term, \\(n\\) is the number of terms, and \\(d\\) is the common difference.\n\nGiven that the first term, \\(a\\), is 10, and each term after the first is found by adding 8 to the term immediately preceding it, we can determine the common difference, \\(d\\), to be 8.\n\nNow, we can substitute the values into the formula and solve for \\(n\\):\n\n\\(210 = 10 + (n-1) \\cdot 8\\)\n\nSimplifying the equation:\n\n\\(210 = 10 + 8n - 8\\)\n\nCombining like terms:\n\n\\(210 = 2 + 8n\\)\n\nSubtracting 2 from both sides:\n\n\\(208 = 8n\\)\n\nDividing both sides by 8:\n\n\\(26 = n\\)\n\nTherefore, the term in the sequence that is equal to 210 is the 26th term.\n\nThe answer is D) The 26th." } }, { "answer":"C", "choices":{ "A":"1.2y", "B":"10y", "C":"12y", "D":"15y", "E":"120y" }, "id":11272, "question":"If \\(y > 0\\), what is 60 percent of 20y?", "explanations":{ "correct":"To find 60 percent of 20y, we need to multiply 20y by 0.60. \n\nStep 1: Multiply 20y by 0.60:\n20y * 0.60 = 12y\n\nTherefore, 60 percent of 20y is 12y.\n\nThe answer is C) 12y." } }, { "answer":"D", "choices":{ "A":"10%", "B":"12%", "C":"25%", "D":"28%", "E":"30%" }, "id":11273, "question":"Laurie inherited 40% of her father's estate. After paying a tax equal to 30% of her inheritance, what percent of her father's estate did she own?", "explanations":{ "correct":"To find the percent of her father's estate that Laurie owns after inheriting 40% and paying a tax of 30% on her inheritance, we can follow these steps:\n\n1. Start with the initial inheritance of 40% of her father's estate.\n2. Calculate the amount of the inheritance after paying the tax. To do this, multiply the inheritance by (100% - 30%) or 0.7. This is because the tax is equal to 30% of the inheritance, so the remaining amount is 100% - 30% = 70% or 0.7.\n3. Multiply the initial inheritance by the amount after paying the tax to find the final percent of her father's estate that Laurie owns. This is equal to 40% * 0.7 = 28%.\n\nTherefore, the answer is D) 28%." } }, { "answer":"D", "choices":{ "A":"\\(-2\\) only", "B":"0 only", "C":"2 only", "D":"\\(-2\\) and 2 only", "E":"\\(-2\\), 0, and 2" }, "id":11281, "question":"If \\(g(t) = t^2 - 4\\), then the graph of \\(g(t)\\) crosses the x-axis when t equals", "explanations":{ "correct":"To find when the graph of \\(g(t) = t^2 - 4\\) crosses the x-axis, we need to determine the values of \\(t\\) that make \\(g(t) = 0\\). \n\nSetting \\(g(t)\\) equal to zero, we have:\n\n\\(t^2 - 4 = 0\\)\n\nTo solve this equation, we can factor it as a difference of squares:\n\n\\((t - 2)(t + 2) = 0\\)\n\nNow, we can use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. \n\nSo, we have two possibilities:\n\n\\(t - 2 = 0\\) or \\(t + 2 = 0\\)\n\nSolving each equation separately, we find:\n\n\\(t - 2 = 0\\) implies \\(t = 2\\)\n\n\\(t + 2 = 0\\) implies \\(t = -2\\)\n\nTherefore, the graph of \\(g(t)\\) crosses the x-axis when \\(t\\) equals -2 and 2.\n\nThe answer is D) \\(-2\\) and 2 only." } }, { "answer":"A", "choices":{ "A":"0", "B":"1", "C":"\\(p\\)", "D":"\\(p^2\\)", "E":"\\(p^3\\)" }, "id":11288, "question":"If \\(p > 0\\), then \\(p^3(\\frac { 1 } { p } -\\frac { 1 } { p } )\\) =", "explanations":{ "correct":"To simplify the expression \\(p^3(\\frac{1}{p} - \\frac{1}{p})\\), we can start by simplifying the subtraction inside the parentheses.\n\n\\(\\frac{1}{p} - \\frac{1}{p}\\) can be simplified to \\(0\\), since the numerator of both fractions is the same and subtracting them will result in zero.\n\nSo, the expression becomes \\(p^3(0)\\).\n\nAny number multiplied by zero is always equal to zero.\n\nTherefore, \\(p^3(0) = 0\\).\n\nThe answer is A) 0." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { 4 } { 3 } \\)", "B":"\\(\\frac { 16 } { 9 } \\)", "C":"\\(\\frac { \\sqrt { 3 } } { 4 } \\)", "D":"\\(\\frac { 4\\sqrt { 3 } } { 3 } \\)", "E":"\\(\\frac { 16\\sqrt { 3 } } { 9 } \\)" }, "id":11290, "question":"A square and an equilateral triangle each have sides of length 5. What is the ratio of the area of the square to the area of the triangle?", "explanations":{ "correct":"To find the ratio of the area of the square to the area of the equilateral triangle, we need to calculate the areas of both shapes.\n\nThe area of a square is given by the formula \\(A_{\\text{square}} = s^2\\), where \\(s\\) is the length of a side. In this case, the side length of the square is 5, so the area of the square is \\(A_{\\text{square}} = 5^2 = 25\\).\n\nThe area of an equilateral triangle is given by the formula \\(A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4} \\cdot s^2\\), where \\(s\\) is the length of a side. In this case, the side length of the equilateral triangle is also 5, so the area of the triangle is \\(A_{\\text{triangle}} = \\frac{\\sqrt{3}}{4} \\cdot 5^2 = \\frac{\\sqrt{3}}{4} \\cdot 25 = \\frac{25\\sqrt{3}}{4}\\).\n\nNow, we can calculate the ratio of the area of the square to the area of the triangle by dividing the area of the square by the area of the triangle:\n\n\\(\\text{Ratio} = \\frac{A_{\\text{square}}}{A_{\\text{triangle}}} = \\frac{25}{\\frac{25\\sqrt{3}}{4}} = \\frac{25}{1} \\cdot \\frac{4}{25\\sqrt{3}} = \\frac{4}{\\sqrt{3}} = \\frac{4\\sqrt{3}}{3}\\).\n\nTherefore, the answer is D) \\(\\frac{4\\sqrt{3}}{3}\\)." } }, { "answer":"B", "choices":{ "A":"-32", "B":"-27", "C":"-22", "D":"32", "E":"37" }, "id":11294, "question":"In a certain sequence the difference between any two consecutive terms is 5. If the 20th term is 63, what is the 2nd term?", "explanations":{ "correct":"To find the 2nd term in the sequence, we need to determine the common difference between consecutive terms. We are given that the difference between any two consecutive terms is 5.\n\nLet's assume the first term in the sequence is represented by 'a'. Since the difference between any two consecutive terms is 5, the second term can be represented as 'a + 5'.\n\nWe are also given that the 20th term is 63. Using this information, we can set up an equation to solve for 'a':\n\na + 19 * 5 = 63\n\nSimplifying the equation:\n\na + 95 = 63\n\nSubtracting 95 from both sides:\n\na = 63 - 95\n\na = -32\n\nTherefore, the first term in the sequence is -32. The second term can be found by adding the common difference of 5 to the first term:\n\n-32 + 5 = -27\n\nSo, the 2nd term in the sequence is -27.\n\nThe answer is B) -27." } }, { "answer":"C", "choices":{ "A":"5", "B":"18", "C":"25", "D":"36", "E":"40" }, "id":11296, "question":"The nth term of a sequence is defined to be \\(5n + 2\\). The 35th term is how much greater than the 30th term?", "explanations":{ "correct":"To find the difference between the 35th term and the 30th term of the sequence, we need to substitute the values of n into the given expression and calculate the difference.\n\nThe 35th term is given by \\(5n + 2\\), where n = 35. Substituting this value into the expression, we get:\n\\(5(35) + 2 = 175 + 2 = 177\\)\n\nSimilarly, the 30th term is given by \\(5n + 2\\), where n = 30. Substituting this value into the expression, we get:\n\\(5(30) + 2 = 150 + 2 = 152\\)\n\nTo find the difference between the 35th term and the 30th term, we subtract the 30th term from the 35th term:\n\\(177 - 152 = 25\\)\n\nTherefore, the 35th term is 25 greater than the 30th term.\n\nThe answer is C) 25." } }, { "answer":"A", "choices":{ "A":"-9", "B":"-8", "C":"-3", "D":"1", "E":"2" }, "id":11298, "question":"If \\(|x + 3| > 5\\), which of the following is a possible value of \\(x\\) ?", "explanations":{ "correct":"To solve the inequality \\(|x + 3| > 5\\), we need to consider two cases: when \\(x + 3\\) is positive and when \\(x + 3\\) is negative.\n\nCase 1: \\(x + 3\\) is positive\\(\\newline\\)If \\(x + 3\\) is positive, then the absolute value of \\(x + 3\\) is equal to \\(x + 3\\). So we have the inequality \\(x + 3 > 5\\). Subtracting 3 from both sides, we get \\(x > 2\\).\n\nCase 2: \\(x + 3\\) is negative\\(\\newline\\)If \\(x + 3\\) is negative, then the absolute value of \\(x + 3\\) is equal to \\(-(x + 3)\\). So we have the inequality \\(-(x + 3) > 5\\). Multiplying both sides by -1, we get \\(x + 3 < -5\\). Subtracting 3 from both sides, we get \\(x < -8\\).\n\nCombining the results from both cases, we have \\(x > 2\\) or \\(x < -8\\). Therefore, the possible values of \\(x\\) are -9, -8, -3, 1, and 2.\n\nHowever, we need to check if these values satisfy the original inequality. Let's substitute each value into the inequality \\(|x + 3| > 5\\) and see if it holds true.\n\nFor \\(x = -9\\), we have \\(|-9 + 3| > 5\\), which simplifies to \\(|-6| > 5\\). Since \\(|-6| = 6\\) and \\(6 > 5\\), this value satisfies the inequality.\n\nFor \\(x = -8\\), we have \\(|-8 + 3| > 5\\), which simplifies to \\(|-5| > 5\\). Since \\(|-5| = 5\\) and \\(5 > 5\\) is not true, this value does not satisfy the inequality.\n\nFor \\(x = -3\\), we have \\(|-3 + 3| > 5\\), which simplifies to \\(|0| > 5\\). Since \\(|0| = 0\\) and \\(0 > 5\\) is not true, this value does not satisfy the inequality.\n\nFor \\(x = 1\\), we have \\(|1 + 3| > 5\\), which simplifies to \\(|4| > 5\\). Since \\(|4| = 4\\) and \\(4 > 5\\) is not true, this value does not satisfy the inequality.\n\nFor \\(x = 2\\), we have \\(|2 + 3| > 5\\), which simplifies to \\(|5| > 5\\). Since \\(|5| = 5\\) and \\(5 > 5\\) is not true, this value does not satisfy the inequality.\n\nTherefore, the only possible value of \\(x\\) that satisfies the inequality \\(|x + 3| > 5\\) is -9.\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"\\(x^4 + x^2 + 4\\)", "B":"\\(x^5 -1\\)", "C":"\\(x^6 - 1\\)", "D":"\\(x^6 + x^2 + 1\\)", "E":"\\(x^2 + 10\\)" }, "id":11299, "question":"Which of the following expressions must be negative if \\(x < 0\\)?", "explanations":{ "correct":"To determine which of the given expressions must be negative when \\(x < 0\\), we can substitute a negative value for \\(x\\) and evaluate each expression. \n\nLet's start with option A: \\(x^4 + x^2 + 4\\). If we substitute \\(x = -1\\), we get \\((-1)^4 + (-1)^2 + 4 = 1 + 1 + 4 = 6\\), which is positive. Therefore, option A is not the correct answer.\n\nMoving on to option B: \\(x^5 - 1\\). Substituting \\(x = -1\\), we have \\((-1)^5 - 1 = -1 - 1 = -2\\), which is negative. Thus, option B is a possible answer.\n\nNext, let's consider option C: \\(x^6 - 1\\). Substituting \\(x = -1\\), we get \\((-1)^6 - 1 = 1 - 1 = 0\\), which is not negative. Therefore, option C is not the correct answer.\n\nNow, let's examine option D: \\(x^6 + x^2 + 1\\). Substituting \\(x = -1\\), we have \\((-1)^6 + (-1)^2 + 1 = 1 + 1 + 1 = 3\\), which is positive. Hence, option D is not the correct answer.\n\nLastly, let's analyze option E: \\(x^2 + 10\\). Substituting \\(x = -1\\), we get \\((-1)^2 + 10 = 1 + 10 = 11\\), which is positive. Therefore, option E is not the correct answer.\n\nAfter evaluating each expression, we find that the only expression that is guaranteed to be negative when \\(x < 0\\) is option B: \\(x^5 - 1\\).\n\nTherefore, the answer is B." } }, { "answer":"D", "choices":{ "A":"100", "B":"150", "C":"200", "D":"260", "E":"300" }, "id":11303, "question":"The profit \\(p\\), in dollars, from a car wash is given by the function \\(p(x) = \\frac { 50x - 200 } { x } + k\\), where x is the number of cars washed and k is a constant. If 20 cars were washed today for a total profit of \\$ 300, what is the value of \\(k\\)?", "explanations":{ "correct":"To find the value of \\(k\\), we can use the given information that 20 cars were washed today for a total profit of \\$ 300.\n\nWe are given the profit function \\(p(x) = \\frac{50x - 200}{x} + k\\), where \\(x\\) is the number of cars washed and \\(k\\) is a constant.\n\nSubstituting the values into the equation, we have:\n\\[300 = \\frac{50(20) - 200}{20} + k\\]\n\nSimplifying the equation, we get:\n\\[300 = \\frac{1000 - 200}{20} + k\\]\n\\[300 = \\frac{800}{20} + k\\]\n\\[300 = 40 + k\\]\n\nTo isolate \\(k\\), we subtract 40 from both sides of the equation:\n\\[300 - 40 = k\\]\n\\[260 = k\\]\n\nTherefore, the value of \\(k\\) is 260.\n\nThe answer is D) 260." } }, { "answer":"E", "choices":{ "A":"3", "B":"4", "C":"12", "D":"24", "E":"36" }, "id":11307, "question":"If \\(s = 3q\\) and \\(q = 4\\), what is the value of 3s?", "explanations":{ "correct":"To find the value of 3s, we need to substitute the value of q into the equation s = 3q. Given that q = 4, we can substitute this value into the equation:\n\ns = 3q\ns = 3(4)\ns = 12\n\nNow, we need to find the value of 3s. We can multiply 3 by the value of s:\n\n3s = 3(12)\n3s = 36\n\nTherefore, the value of 3s is 36.\n\nThe answer is E) 36." } }, { "answer":"B", "choices":{ "A":"3", "B":"5", "C":"7", "D":"9", "E":"11" }, "id":11308, "question":"The hypotenuse of an isosceles triangle has endpoints (4, 3) and (9, 8). What is the length of one of the legs of the triangle?", "explanations":{ "correct":"To find the length of one of the legs of the isosceles triangle, we need to determine the length of the hypotenuse first. \n\nUsing the distance formula, we can find the length of the hypotenuse by calculating the distance between the two given endpoints. The distance formula is given by:\n\nd = √((x2 - x1)^2 + (y2 - y1)^2)\n\nLet's calculate the distance between the two endpoints (4, 3) and (9, 8):\n\nd = √((9 - 4)^2 + (8 - 3)^2)\n = √(5^2 + 5^2)\n = √(25 + 25)\n = √50\n = 5√2\n\nNow that we know the length of the hypotenuse is 5√2, we can determine the length of one of the legs of the triangle. In an isosceles triangle, the length of the legs is equal.\n\nSince the length of one leg is equal to the length of the other leg, the length of one of the legs is also 5√2.\n\nTherefore, the answer is B) 5.\n\nNote: It's important to double-check our reasoning and calculations to avoid mistakes." } }, { "answer":"E", "choices":{ "A":"\\(2x + 7\\) is odd", "B":"\\(3x^2 + 5\\) is odd", "C":"\\(x^3 - x^2 + x - 1\\) is odd", "D":"\\(3x + 4\\) is even", "E":"\\((3x - 5)(5x - 3) is even\\)" }, "id":11317, "question":"If \\(x\\) is an even number, then each of the following must be true EXCEPT", "explanations":{ "correct":"To determine which statement is NOT true when \\(x\\) is an even number, we need to evaluate each statement one by one.\n\nA) \\(2x + 7\\) is odd: \nWhen we substitute an even number for \\(x\\), we get \\(2 \\times \\text{{even number}} + 7\\). Multiplying an even number by 2 gives us an even number, and adding 7 to an even number gives us an odd number. Therefore, statement A is true.\n\nB) \\(3x^2 + 5\\) is odd: \nWhen we substitute an even number for \\(x\\), we get \\(3 \\times \\text{{even number}}^2 + 5\\). Squaring an even number gives us an even number, and multiplying an even number by 3 gives us an even number. Adding 5 to an even number gives us an odd number. Therefore, statement B is true.\n\nC) \\(x^3 - x^2 + x - 1\\) is odd: \nWhen we substitute an even number for \\(x\\), we get \\(\\text{{even number}}^3 - \\text{{even number}}^2 + \\text{{even number}} - 1\\). Cubing an even number gives us an even number, squaring an even number gives us an even number, and subtracting an even number from an even number gives us an even number. Subtracting 1 from an even number gives us an odd number. Therefore, statement C is true.\n\nD) \\(3x + 4\\) is even: \nWhen we substitute an even number for \\(x\\), we get \\(3 \\times \\text{{even number}} + 4\\). Multiplying an even number by 3 gives us an even number, and adding 4 to an even number gives us an even number. Therefore, statement D is true.\n\nE) \\((3x - 5)(5x - 3)\\) is even: \nWhen we substitute an even number for \\(x\\), we get \\((3 \\times \\text{{even number}} - 5)(5 \\times \\text{{even number}} - 3)\\). Multiplying an even number by 3 gives us an even number, subtracting an odd number from an even number gives us an odd number, multiplying an even number by 5 gives us an even number, and subtracting an odd number from an even number gives us an odd number. Therefore, statement E is false.\n\nBased on the above analysis, the statement that is NOT true when \\(x\\) is an even number is E. \n\nThe answer is E." } }, { "answer":"C", "choices":{ "A":"\\$ 26.00", "B":"\\$ 30.00", "C":"\\$ 32.50", "D":"\\$ 42.50", "E":"\\$ 45.00" }, "id":11321, "question":"At a certain store, the price of a new compact disc is five times the price of a used compact disc. The difference between the two prices is ten dollars. What would be the total price of three used compact discs and two new compact discs at this store?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nLet's assume the price of a used compact disc is \"x\" dollars. According to the information given, the price of a new compact disc would be 5 times the price of a used compact disc, which is 5x dollars.\n\nThe difference between the two prices is given as ten dollars, so we can set up the equation:\n\n5x - x = 10\n\nSimplifying the equation, we have:\n\n4x = 10\n\nDividing both sides of the equation by 4, we get:\n\nx = 2.5\n\nSo, the price of a used compact disc is 2.5 dollars, and the price of a new compact disc is 5 times that, which is 12.5 dollars.\n\nTo find the total price of three used compact discs, we multiply the price of one used compact disc by 3:\n\n3 * 2.5 = 7.5\n\nTo find the total price of two new compact discs, we multiply the price of one new compact disc by 2:\n\n2 * 12.5 = 25\n\nFinally, we add the total price of three used compact discs and two new compact discs:\n\n7.5 + 25 = 32.5\n\nTherefore, the total price of three used compact discs and two new compact discs at this store would be \\$ 32.50.\n\nThe answer is C) \\$ 32.50." } }, { "answer":"E", "choices":{ "A":"6", "B":"15", "C":"21", "D":"39", "E":"56" }, "id":11322, "question":"Which of the following CANNOT be written as the sum of three consecutive integers?", "explanations":{ "correct":"To determine which of the given numbers cannot be written as the sum of three consecutive integers, we can use the fact that the sum of three consecutive integers is always divisible by 3.\n\nLet's go through each option and check if it satisfies this condition:\n\nA) 6: The sum of three consecutive integers can be written as n + (n+1) + (n+2) = 3n + 3, which is divisible by 3. Therefore, 6 can be written as the sum of three consecutive integers.\n\nB) 15: The sum of three consecutive integers can be written as n + (n+1) + (n+2) = 3n + 3, which is divisible by 3. Therefore, 15 can be written as the sum of three consecutive integers.\n\nC) 21: The sum of three consecutive integers can be written as n + (n+1) + (n+2) = 3n + 3, which is divisible by 3. Therefore, 21 can be written as the sum of three consecutive integers.\n\nD) 39: The sum of three consecutive integers can be written as n + (n+1) + (n+2) = 3n + 3, which is divisible by 3. Therefore, 39 can be written as the sum of three consecutive integers.\n\nE) 56: The sum of three consecutive integers can be written as n + (n+1) + (n+2) = 3n + 3, which is divisible by 3. However, 56 is not divisible by 3. Therefore, 56 cannot be written as the sum of three consecutive integers.\n\nBased on the reasoning above, the answer is E." } }, { "answer":"B", "choices":{ "A":"7", "B":"5", "C":"3", "D":"10", "E":"13" }, "id":11323, "question":"What is the slope of a line with the equation \\(y + 3 = 5(x - 2)\\) ?", "explanations":{ "correct":"To find the slope of a line, we need to rewrite the equation in slope-intercept form, which is in the form of \\\\(y = mx + b\\\\), where \\\\(m\\\\) represents the slope.\n\nGiven the equation \\\\(y + 3 = 5(x - 2)\\\\), let's simplify it step-by-step:\n\n1. Distribute the 5 to the terms inside the parentheses:\n\\\\(y + 3 = 5x - 10\\\\)\n\n2. Move the constant term to the other side of the equation by subtracting 3 from both sides:\n\\\\(y = 5x - 10 - 3\\\\)\n\\\\(y = 5x - 13\\\\)\n\nNow we can see that the equation is in the form \\\\(y = mx + b\\\\), where the coefficient of \\\\(x\\\\) is the slope. In this case, the coefficient of \\\\(x\\\\) is 5.\n\nTherefore, the slope of the line is 5.\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"\\(-5\\)", "B":"\\(-2\\)", "C":"1", "D":"5", "E":"8" }, "id":11326, "question":"If t is 5 more than s, and s is 3 less than r, what is t when \\(r = 3\\)?", "explanations":{ "correct":"To find the value of t when r = 3, we need to follow the given information step-by-step.\n\nFirst, we are told that s is 3 less than r. So, if r = 3, then s = 3 - 3 = 0.\n\nNext, we are told that t is 5 more than s. Since s = 0, t = 0 + 5 = 5.\n\nTherefore, when r = 3, t = 5.\n\nThe answer is D) 5." } }, { "answer":"E", "choices":{ "A":"-3", "B":"0", "C":"1", "D":"2", "E":"4" }, "id":11327, "question":"If \\(x = -3\\), then \\(||x - 2| - |2 + x||\\) =", "explanations":{ "correct":"To find the value of \\\\(||x - 2| - |2 + x||\\\\)| when \\\\(x = -3\\\\), we substitute \\\\(-3\\\\) for \\\\(x\\\\) in the expression.\n\nFirst, let's evaluate \\\\(|x - 2|\\). When \\\\(x = -3\\\\), we have \\\\(|-3 - 2| = |-5| = 5\\\\).\n\nNext, let's evaluate \\\\(|2 + x|\\). When \\\\(x = -3\\\\), we have \\\\(|2 + (-3)| = |2 - 3| = |-1| = 1\\\\).\n\nNow, we substitute the values we found back into the expression: \\\\(||x - 2| - |2 + x||\\\\) = \\\\(|5 - 1| = |4| = 4\\\\).\n\nTherefore, the answer is E) 4." } }, { "answer":"A", "choices":{ "A":"\\(e = 0\\)", "B":"\\(c = 0\\)", "C":"\\(a > 0\\)", "D":"\\(b < 0\\)", "E":"\\(b > 3\\)" }, "id":11328, "question":"If \\(abcd = 3\\) and \\(abce = 0\\), which of the following must be true?", "explanations":{ "correct":"To solve this problem, we need to analyze the given equations and determine the possible values of the variables.\n\nFrom the equation \\(abcd = 3\\), we know that the product of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is equal to 3.\n\nFrom the equation \\(abce = 0\\), we know that the product of \\(a\\), \\(b\\), \\(c\\), and \\(e\\) is equal to 0.\n\nNow let's consider the possible values of each variable:\n\n1. \\(a\\): Since the product of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is equal to 3, and the product of \\(a\\), \\(b\\), \\(c\\), and \\(e\\) is equal to 0, we can conclude that \\(a\\) must be equal to 0. This is because any number multiplied by 0 will result in 0.\n\n2. \\(b\\): Since the product of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is equal to 3, and the product of \\(a\\), \\(b\\), \\(c\\), and \\(e\\) is equal to 0, we can conclude that \\(b\\) can be any value. This is because any number multiplied by 0 will result in 0, so the value of \\(b\\) does not affect the product.\n\n3. \\(c\\): Since the product of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is equal to 3, and the product of \\(a\\), \\(b\\), \\(c\\), and \\(e\\) is equal to 0, we can conclude that \\(c\\) must be equal to 0. This is because any number multiplied by 0 will result in 0.\n\n4. \\(d\\): Since the product of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) is equal to 3, and the product of \\(a\\), \\(b\\), \\(c\\), and \\(e\\) is equal to 0, we can conclude that \\(d\\) must be a non-zero value. This is because if \\(d\\) were equal to 0, the product of \\(a\\), \\(b\\), \\(c\\), and \\(d\\) would also be 0.\n\n5. \\(e\\): Since the product of \\(a\\), \\(b\\), \\(c\\), and \\(e\\) is equal to 0, we can conclude that \\(e\\) must be equal to 0. This is because any number multiplied by 0 will result in 0.\n\nBased on our analysis, we can see that the only statement that must be true is \\(e = 0\\). Therefore, the answer is A) \\(e = 0\\)." } }, { "answer":"C", "choices":{ "A":"840", "B":"280", "C":"252", "D":"175", "E":"84" }, "id":11339, "question":"40 percent of 210 is the same as \\(33 \\frac{1}{3}\\) percent of what number?", "explanations":{ "correct":"To find the number that is equivalent to 40 percent of 210, we can set up the equation:\n\n40% of 210 = \\(\\frac{33 \\frac{1}{3}}{100}\\) of x\n\nTo simplify the equation, we convert 33 1/3 percent to a decimal:\n\n\\(\\frac{33 \\frac{1}{3}}{100} = \\frac{100}{3} \\div 100 = \\frac{1}{3}\\)\n\nNow we can rewrite the equation:\n\n40% of 210 = \\(\\frac{1}{3}\\) of x\n\nTo solve for x, we can multiply both sides of the equation by 3:\n\n3 * (40% of 210) = 3 * \\(\\frac{1}{3}\\) of x\n\n120% of 210 = x\n\nTo find 120% of 210, we can multiply 210 by 1.2:\n\nx = 210 * 1.2\n\nx = 252\n\nTherefore, the number that is equivalent to 40 percent of 210 is 252.\n\nThe answer is C." } }, { "answer":"E", "choices":{ "A":"18", "B":"36", "C":"72", "D":"144", "E":"260" }, "id":11340, "question":"If \\(f(x) = x^2 + \\sqrt { x } \\) and \\(g(x) = f(4x)\\), what is the value of \\(g(4)\\)?", "explanations":{ "correct":"To find the value of \\(g(4)\\), we need to substitute \\(x = 4\\) into the function \\(g(x) = f(4x)\\).\n\nFirst, let's find the value of \\(f(4x)\\). We substitute \\(4x\\) into the function \\(f(x) = x^2 + \\sqrt{x}\\):\n\n\\(f(4x) = (4x)^2 + \\sqrt{4x}\\)\n\nSimplifying this expression, we get:\n\n\\(f(4x) = 16x^2 + 2\\sqrt{x}\\)\n\nNow, we substitute \\(x = 4\\) into the expression we just found:\n\n\\(g(4) = 16(4)^2 + 2\\sqrt{4}\\)\n\nSimplifying further:\n\n\\(g(4) = 16(16) + 2(2)\\)\n\n\\(g(4) = 256 + 4\\)\n\n\\(g(4) = 260\\)\n\nTherefore, the value of \\(g(4)\\) is 260.\n\nThe answer is E." } }, { "answer":"B", "choices":{ "A":"7", "B":"11", "C":"\\(3x + 7\\)", "D":"\\(3x + 11\\)", "E":"\\(6x + 7\\)" }, "id":11349, "question":"The value of \\(3x + 9\\) is how much more than the value of \\(3x - 2\\)?", "explanations":{ "correct":"To find the difference between the values of \\(3x + 9\\) and \\(3x - 2\\), we need to subtract the second expression from the first expression.\n\nSubtracting \\(3x - 2\\) from \\(3x + 9\\) can be done by distributing the negative sign to each term inside the parentheses of \\(3x - 2\\). This gives us:\n\n\\(3x + 9 - (3x - 2)\\)\n\nNext, we can remove the parentheses by changing the signs of each term inside the parentheses:\n\n\\(3x + 9 - 3x + 2\\)\n\nNow, we can combine like terms:\n\n\\(3x - 3x + 9 + 2\\)\n\nSimplifying further:\n\n\\(0x + 11\\)\n\nSince any number multiplied by zero is zero, we can simplify \\(0x\\) to just \\(0\\):\n\n\\(0 + 11\\)\n\nFinally, we have:\n\n\\(11\\)\n\nTherefore, the value of \\(3x + 9\\) is 11 more than the value of \\(3x - 2\\).\n\nThe answer is B) 11." } }, { "answer":"D", "choices":{ "A":"36", "B":"38", "C":"40", "D":"42", "E":"44" }, "id":11352, "question":"If 60 percent of Jared's jigsaw puzzles have 50 pieces and 40 percent of his jigsaw puzzles have 30 pieces each, what is the average (arithmetic mean) number of pieces per puzzle?", "explanations":{ "correct":"To find the average number of pieces per puzzle, we need to calculate the weighted average based on the percentages given.\n\nFirst, we calculate the contribution of the 50-piece puzzles:\n60 percent of Jared's puzzles have 50 pieces, so the contribution of these puzzles to the average is 60/100 * 50 = 30.\n\nNext, we calculate the contribution of the 30-piece puzzles:\n40 percent of Jared's puzzles have 30 pieces, so the contribution of these puzzles to the average is 40/100 * 30 = 12.\n\nTo find the total contribution, we add the contributions from the 50-piece and 30-piece puzzles:\n30 + 12 = 42.\n\nFinally, we divide the total contribution by the total number of puzzles (100 percent):\n42 / 100 = 0.42.\n\nTherefore, the average number of pieces per puzzle is 42.\n\nThe answer is D) 42." } }, { "answer":"E", "choices":{ "A":"12", "B":"5", "C":"3", "D":"2.5", "E":"1.5" }, "id":11353, "question":"If y is inversely proportional to \\(x^2\\) and if \\(y = 6\\) and \\(x = 5\\), what is the value of \\(y\\) when \\(x = 10\\)?", "explanations":{ "correct":"To solve this problem, we need to use the concept of inverse proportionality. Inverse proportionality means that as one variable increases, the other variable decreases, and vice versa, while their product remains constant.\n\nGiven that \\(y\\) is inversely proportional to \\(x^2\\), we can write the equation as \\(y = k/x^2\\), where \\(k\\) is the constant of proportionality.\n\nTo find the value of \\(k\\), we can substitute the given values of \\(y\\) and \\(x\\) into the equation. We have \\(6 = k/5^2\\), which simplifies to \\(6 = k/25\\). To solve for \\(k\\), we can multiply both sides of the equation by 25, giving us \\(k = 150\\).\n\nNow that we have the value of \\(k\\), we can use it to find the value of \\(y\\) when \\(x = 10\\). Substituting the values into the equation, we have \\(y = 150/10^2\\), which simplifies to \\(y = 150/100\\). Evaluating this expression, we find \\(y = 1.5\\).\n\nTherefore, the value of \\(y\\) when \\(x = 10\\) is 1.5.\n\nThe answer is E) 1.5." } }, { "answer":"D", "choices":{ "A":"\\(3x\\)", "B":"\\(9x\\)", "C":"\\(12x\\)", "D":"\\(24x\\)", "E":"\\(27x\\)" }, "id":11355, "question":"In a sequence of numbers, the first term is x and the ratio of each subsequent term to the previous term is 3 to 1. Which of the following expressions represents the difference between the fourth term and the second term?", "explanations":{ "correct":"To find the difference between the fourth term and the second term in the sequence, we need to determine the values of these terms.\n\nLet's analyze the given information. The ratio of each subsequent term to the previous term is 3 to 1. This means that each term is three times greater than the previous term.\n\nLet's write out the terms of the sequence:\nFirst term: x\nSecond term: 3x (since it is three times greater than the first term)\nThird term: 9x (since it is three times greater than the second term)\nFourth term: 27x (since it is three times greater than the third term)\n\nNow, we can calculate the difference between the fourth term and the second term:\nDifference = Fourth term - Second term\nDifference = 27x - 3x\nDifference = 24x\n\nTherefore, the correct expression that represents the difference between the fourth term and the second term is 24x.\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"280", "B":"360", "C":"480", "D":"560", "E":"630" }, "id":11358, "question":"Hoover High School has 840 students, and the ratio of the number of students taking Spanish to the number not taking Spanish is 4:3. How many of the students take Spanish?", "explanations":{ "correct":"To find the number of students taking Spanish, we need to determine the ratio of students taking Spanish to the total number of students. \n\nLet's represent the number of students taking Spanish as 4x and the number of students not taking Spanish as 3x. \n\nAccording to the given information, the total number of students is 840. So, we can set up the equation: \n\n4x + 3x = 840\n\nCombining like terms, we get: \n\n7x = 840\n\nTo solve for x, we divide both sides of the equation by 7: \n\nx = 840 / 7\n\nSimplifying, we find that x = 120. \n\nNow, we can find the number of students taking Spanish by substituting x back into the equation: \n\n4x = 4 * 120 = 480\n\nTherefore, the answer is C) 480." } }, { "answer":"C", "choices":{ "A":"3", "B":"8", "C":"27", "D":"64", "E":"208" }, "id":11359, "question":"Some clay is shaped into a sphere with a radius of 2 inches. If more clay is added to the original sphere so that the radius is increased by 4 inches, then the volume of the new sphere is how many times greater than the volume of the original sphere?", "explanations":{ "correct":"To find the volume of a sphere, we use the formula V = (4/3)πr^3, where V is the volume and r is the radius.\n\nThe original sphere has a radius of 2 inches, so its volume is V1 = (4/3)π(2^3) = (4/3)π(8) = (32/3)π.\n\nWhen more clay is added to the original sphere, the radius is increased by 4 inches. Therefore, the new radius is 2 + 4 = 6 inches.\n\nThe volume of the new sphere is V2 = (4/3)π(6^3) = (4/3)π(216) = (864/3)π = 288π.\n\nTo find how many times greater the volume of the new sphere is compared to the original sphere, we divide V2 by V1:\n\n(V2 / V1) = (288π) / (32/3)π = (288/1) / (32/3) = (288/1) * (3/32) = 9.\n\nTherefore, the volume of the new sphere is 9 times greater than the volume of the original sphere.\n\nThe answer is C) 27." } }, { "answer":"E", "choices":{ "A":"\\((1, 1)\\)", "B":"\\((2, 2)\\)", "C":"\\((1, 3)\\)", "D":"\\((0, 2)\\)", "E":"\\((2, 3)\\)" }, "id":11360, "question":"Line l has a slope of 1 and contains the point \\((1,2)\\). Which of the following points is also on line l?", "explanations":{ "correct":"To determine which of the given points is also on line \\(l\\), we need to use the point-slope form of a linear equation. The point-slope form is given by \\(y - y_1 = m(x - x_1)\\), where \\(m\\) is the slope of the line and \\((x_1, y_1)\\) is a point on the line.\n\\(\\newline\\)In this case, we are given that line \\(l\\) has a slope of 1 and contains the point \\((1,2)\\). Plugging these values into the point-slope form, we have \\(y - 2 = 1(x - 1)\\).\n\nNow, let's check each of the given points to see if they satisfy this equation:\n\nA) \\((1, 1)\\):\nPlugging in \\(x = 1\\) and \\(y = 1\\) into the equation, we have \\(1 - 2 = 1(1 - 1)\\), which simplifies to \\(-1 = 0\\). This is not true, so \\((1, 1)\\) is not on line \\(l\\).\n\nB) \\((2, 2)\\):\nPlugging in \\(x = 2\\) and \\(y = 2\\) into the equation, we have \\(2 - 2 = 1(2 - 1)\\), which simplifies to \\(0 = 1\\). This is not true, so \\((2, 2)\\) is not on line \\(l\\).\n\nC) \\((1, 3)\\):\nPlugging in \\(x = 1\\) and \\(y = 3\\) into the equation, we have \\(3 - 2 = 1(1 - 1)\\), which simplifies to \\(1 = 0\\). This is not true, so \\((1, 3)\\) is not on line \\(l\\).\n\nD) \\((0, 2)\\):\nPlugging in \\(x = 0\\) and \\(y = 2\\) into the equation, we have \\(2 - 2 = 1(0 - 1)\\), which simplifies to \\(0 = -1\\). This is not true, so \\((0, 2)\\) is not on line \\(l\\).\n\nE) \\((2, 3)\\):\nPlugging in \\(x = 2\\) and \\(y = 3\\) into the equation, we have \\(3 - 2 = 1(2 - 1)\\), which simplifies to \\(1 = 1\\). This is true, so \\((2, 3)\\) is on line \\(l\\).\n\nAfter checking all the points, we can conclude that the point \\((2, 3)\\) is also on line \\(l\\).\n\nTherefore, the answer is E." } }, { "answer":"D", "choices":{ "A":"-16", "B":"-8", "C":"0", "D":"8", "E":"16" }, "id":11362, "question":"If \\(8 - (8 - m) = 8\\), then \\(m\\) =", "explanations":{ "correct":"To solve the equation \\(8 - (8 - m) = 8\\), we need to simplify the expression on the left side step by step.\n\nStep 1: Simplify the expression inside the parentheses.\n\\(8 - (8 - m)\\) can be rewritten as \\(8 - 8 + m\\) using the distributive property.\nThis simplifies to \\(0 + m\\) or simply \\(m\\).\n\nStep 2: Rewrite the equation using the simplified expression.\nThe equation \\(8 - (8 - m) = 8\\) becomes \\(m = 8\\).\n\nTherefore, the value of \\(m\\) is 8.\n\nThe answer is D) 8." } }, { "answer":"C", "choices":{ "A":"0", "B":"1", "C":"\\(\\sqrt { 2 } \\)", "D":"\\(\\sqrt { 7 } \\)", "E":"7" }, "id":11363, "question":"If \\(f(x) = x^2 - 1\\) and \\(g(x) = 1 - x^2\\), then for which of the following values of a does \\(f(2a) - 7 = g(2a) + 7\\)?", "explanations":{ "correct":"To find the value of \\(a\\) that satisfies the equation \\(f(2a) - 7 = g(2a) + 7\\), we need to substitute the given functions \\(f(x) = x^2 - 1\\) and \\(g(x) = 1 - x^2\\) into the equation.\n\nFirst, let's substitute \\(f(2a)\\) and \\(g(2a)\\) into the equation:\n\n\\(f(2a) - 7 = g(2a) + 7\\)\n\n\\((2a)^2 - 1 - 7 = 1 - (2a)^2 + 7\\)\n\nSimplifying both sides of the equation:\n\n\\(4a^2 - 1 - 7 = 1 - 4a^2 + 7\\)\n\n\\(4a^2 - 8 = -4a^2 + 8\\)\n\nAdding \\(4a^2\\) to both sides:\n\n\\(8a^2 - 8 = 8\\)\n\nAdding 8 to both sides:\n\n\\(8a^2 = 16\\)\n\nDividing both sides by 8:\n\n\\(a^2 = 2\\)\n\nTaking the square root of both sides:\n\n\\(a = \\sqrt{2}\\) or \\(a = -\\sqrt{2}\\)\n\nTherefore, the value of \\(a\\) that satisfies the equation is \\(\\sqrt{2}\\).\n\nThe answer is C." } }, { "answer":"A", "choices":{ "A":"24", "B":"28", "C":"32", "D":"48", "E":"50" }, "id":11365, "question":"In a jar of cookies, \\(\\frac { 1 } { 8 } \\) of the cookies are oatmeal raisin, \\(\\frac { 1 } { 4 } \\) are peanut butter, \\(\\frac { 1 } { 2 } \\) are chocolate chip, and the remaining 12 cookies are mint. How many peanut butter cookies are in the jar?", "explanations":{ "correct":"To find the number of peanut butter cookies in the jar, we need to determine the fraction of cookies that are peanut butter. \n\nGiven that \\\\(\\frac { 1 } { 8 }\\\\) of the cookies are oatmeal raisin, \\\\(\\frac { 1 } { 4 }\\\\) are peanut butter, and \\\\(\\frac { 1 } { 2 }\\\\) are chocolate chip, we can add these fractions together to find the total fraction of cookies that are accounted for:\n\n\\\\(\\frac { 1 } { 8 } + \\frac { 1 } { 4 } + \\frac { 1 } { 2 } = \\frac { 1 } { 8 } + \\frac { 2 } { 8 } + \\frac { 4 } { 8 } = \\frac { 7 } { 8 }\\\\)\n\nThis means that \\\\(\\frac { 7 } { 8 }\\\\) of the cookies are accounted for, and the remaining fraction of cookies is \\\\(\\frac { 1 } { 8 }\\\\).\n\nWe are given that the remaining 12 cookies are mint. Since these mint cookies represent \\\\(\\frac { 1 } { 8 }\\\\) of the total number of cookies, we can set up the following equation:\n\n\\\\(\\frac { 1 } { 8 } \\cdot \\text{Total number of cookies} = 12\\\\)\n\nTo solve for the total number of cookies, we can multiply both sides of the equation by 8:\n\n\\\\(\\text{Total number of cookies} = 12 \\cdot 8 = 96\\\\)\n\nNow that we know there are 96 cookies in total, we can find the number of peanut butter cookies by multiplying the total number of cookies by the fraction that represents the peanut butter cookies:\n\n\\\\(96 \\cdot \\frac { 1 } { 4 } = 24\\\\)\n\nTherefore, the answer is A) 24." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { 3 } { 4 } \\)", "B":"\\(\\frac { 4 } { 5 } \\)", "C":"\\(\\frac { 1 } { 1 } \\)", "D":"\\(\\frac { 5 } { 4 } \\)", "E":"\\(\\frac { 4 } { 3 } \\)" }, "id":11367, "question":"At Essex High School 100 students are taking chemistry and 80 students are taking are biology. If 20 students are taking both chemistry and biology, what is the ratio of the number of students taking only chemistry to the number taking only biology?", "explanations":{ "correct":"To find the ratio of the number of students taking only chemistry to the number taking only biology, we need to subtract the number of students taking both chemistry and biology from the total number of students taking each subject.\n\nGiven that 100 students are taking chemistry and 80 students are taking biology, and 20 students are taking both, we can calculate the number of students taking only chemistry by subtracting the number of students taking both from the total number of students taking chemistry: 100 - 20 = 80.\n\nSimilarly, we can calculate the number of students taking only biology by subtracting the number of students taking both from the total number of students taking biology: 80 - 20 = 60.\n\nTherefore, the ratio of the number of students taking only chemistry to the number taking only biology is 80:60, which simplifies to 4:3.\n\nThe answer is E) \\(\\frac { 4 } { 3 } \\)." } }, { "answer":"A", "choices":{ "A":"1", "B":"2", "C":"5", "D":"6", "E":"There are no solutions" }, "id":11369, "question":"If \\(f(x) = x + 5\\), which of the following is a solution of \\(f(3a) + 2 = f(2a) + 3\\) ?", "explanations":{ "correct":"To find the solution to the equation \\\\(f(3a) + 2 = f(2a) + 3\\\\), we need to substitute the given function \\\\(f(x) = x + 5\\\\) into the equation.\n\nFirst, let's substitute \\\\(3a\\\\) into the function \\\\(f(x)\\\\):\n\\\\(f(3a) = 3a + 5\\\\)\n\nNext, let's substitute \\\\(2a\\\\) into the function \\\\(f(x)\\\\):\n\\\\(f(2a) = 2a + 5\\\\)\n\nNow, let's substitute these values back into the original equation:\n\\\\(3a + 5 + 2 = 2a + 5 + 3\\\\)\n\nSimplifying the equation, we have:\n\\\\(3a + 7 = 2a + 8\\\\)\n\nTo solve for \\\\(a\\\\), we need to isolate the variable on one side of the equation. Let's subtract \\\\(2a\\\\) from both sides:\n\\\\(3a - 2a + 7 = 2a - 2a + 8\\\\)\n\\\\(a + 7 = 8\\\\)\n\nNext, let's subtract 7 from both sides:\n\\\\(a + 7 - 7 = 8 - 7\\\\)\n\\\\(a = 1\\\\)\n\nTherefore, the solution to the equation \\\\(f(3a) + 2 = f(2a) + 3\\\\) is \\\\(a = 1\\\\).\n\nThe answer is A) 1." } }, { "answer":"C", "choices":{ "A":"900", "B":"700", "C":"500", "D":"0.007", "E":"0.005" }, "id":11370, "question":"If 0.05 percent of n is 5, what is 5 percent of n?", "explanations":{ "correct":"To find the answer, we need to set up an equation based on the given information. We are told that 0.05 percent of n is equal to 5. \n\nFirst, let's convert 0.05 percent to decimal form by dividing it by 100: 0.05/100 = 0.0005.\n\nNext, we can set up the equation: 0.0005n = 5.\n\nTo solve for n, we divide both sides of the equation by 0.0005: n = 5/0.0005 = 10,000.\n\nNow that we know the value of n is 10,000, we can find 5 percent of n by multiplying 10,000 by 0.05: 10,000 * 0.05 = 500.\n\nTherefore, 5 percent of n is 500.\n\nThe answer is C) 500." } }, { "answer":"B", "choices":{ "A":"35", "B":"56", "C":"63", "D":"75", "E":"91" }, "id":11371, "question":"In the sequence 7, 14, 28, x, 112.. what is the value of x?", "explanations":{ "correct":"To find the value of x in the given sequence, we need to identify the pattern. Looking at the sequence, we can see that each term is obtained by multiplying the previous term by 2. \n\nStarting with 7, the next term is obtained by multiplying 7 by 2, which gives us 14. \nThen, we multiply 14 by 2 to get 28. \nNext, we multiply 28 by 2 to get x. \nFinally, we multiply x by 2 to get 112. \n\nTo find the value of x, we can divide 112 by 2. \n\n112 ÷ 2 = 56\n\nTherefore, the value of x is 56.\n\nThe answer is B) 56." } }, { "answer":"C", "choices":{ "A":"5", "B":"6", "C":"8", "D":"10", "E":"12" }, "id":11373, "question":"In a regular n-sided polygon, the measure of each interior angle d, in degrees, is given by the function \\(d = \\frac { 180(n - 2 } { n } \\). If the regular polygon has and interior angle of \\(135^o\\), what is the number of sides of the polygon?", "explanations":{ "correct":"To find the number of sides of the polygon, we can use the formula for the measure of each interior angle of a regular polygon: \n\n\\(d = \\frac{180(n-2)}{n}\\)\n\nGiven that the interior angle is \\(135^o\\), we can substitute this value into the formula:\n\n\\(135 = \\frac{180(n-2)}{n}\\)\n\nTo solve for \\(n\\), we can cross-multiply:\n\n\\(135n = 180(n-2)\\)\n\nExpanding the equation:\n\n\\(135n = 180n - 360\\)\n\nRearranging the equation:\n\n\\(180n - 135n = 360\\)\n\n\\(45n = 360\\)\n\nDividing both sides by 45:\n\n\\(n = \\frac{360}{45}\\)\n\nSimplifying:\n\n\\(n = 8\\)\n\nTherefore, the number of sides of the polygon is 8.\n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"625%", "B":"400%", "C":"80%", "D":"25%", "E":"4%" }, "id":11374, "question":"If \\(\\frac { 4 } { 5 } p\\) is two times the value of \\(\\frac { q } { 10 } \\), where \\(p\\) and \\(q\\) are both positive integers, then \\(p\\) is what percent of \\(q\\) ?", "explanations":{ "correct":"To find the percentage of \\(p\\) in relation to \\(q\\), we need to compare the values of \\(p\\) and \\(q\\) in the given equation.\n\nThe equation states that \\(\\frac{4}{5}p\\) is two times the value of \\(\\frac{q}{10}\\). We can write this as an equation:\n\n\\(\\frac{4}{5}p = 2 \\cdot \\frac{q}{10}\\)\n\nTo simplify this equation, we can cancel out the common factors:\n\n\\(\\frac{4}{5}p = \\frac{2}{1} \\cdot \\frac{q}{10}\\)\n\nSimplifying further, we have:\n\n\\(\\frac{4}{5}p = \\frac{q}{5}\\)\n\nTo isolate \\(p\\), we can multiply both sides of the equation by \\(\\frac{5}{4}\\):\n\n\\(p = \\frac{5}{4} \\cdot \\frac{q}{5}\\)\n\nSimplifying, we get:\n\n\\(p = \\frac{q}{4}\\)\n\nThis equation shows that \\(p\\) is one-fourth of \\(q\\). To find the percentage, we can express this as a fraction with a denominator of 100:\n\n\\(\\frac{p}{q} = \\frac{1}{4} = \\frac{25}{100}\\)\n\nTherefore, \\(p\\) is 25% of \\(q\\).\n\nThe answer is D) 25%." } }, { "answer":"B", "choices":{ "A":"\\(-1\\)", "B":"14", "C":"19", "D":"26", "E":"32" }, "id":11377, "question":"If \\(3x + 2 = 14\\), what is the value of \\(5x - 6\\)?", "explanations":{ "correct":"To find the value of \\(5x - 6\\), we first need to solve the equation \\(3x + 2 = 14\\) for \\(x\\).\n\nStep 1: Subtract 2 from both sides of the equation:\n\\(3x + 2 - 2 = 14 - 2\\)\n\\(3x = 12\\)\n\nStep 2: Divide both sides of the equation by 3 to isolate \\(x\\):\n\\(\\frac{3x}{3} = \\frac{12}{3}\\)\n\\(x = 4\\)\n\nNow that we have found the value of \\(x\\), we can substitute it into the expression \\(5x - 6\\) to find the final answer.\n\nStep 3: Substitute \\(x = 4\\) into \\(5x - 6\\):\n\\(5(4) - 6\\)\n\\(20 - 6\\)\n\\(14\\)\n\nTherefore, the value of \\(5x - 6\\) is 14.\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"74", "B":"85", "C":"90", "D":"94", "E":"95" }, "id":11378, "question":"Jordan has taken five math tests so far this semester. If he gets a 70 on his next test, that grade will lower his test average (arithmetic mean) by 4 points. What is his average now?", "explanations":{ "correct":"To find Jordan's current average, we need to consider the average of his first five tests and the impact of his next test score.\n\nLet's assume Jordan's current average is x.\n\nThe sum of his first five test scores is 5x.\n\\(\\newline\\)If he gets a 70 on his next test, the sum of his six test scores will be 5x + 70.\n\nTo find his new average, we divide the sum of his six test scores by 6:\n\n(5x + 70) / 6\n\nWe know that this new average will be 4 points lower than his current average, so we can set up the equation:\n\n(5x + 70) / 6 = x - 4\n\nTo solve for x, we can start by multiplying both sides of the equation by 6 to eliminate the denominator:\n\n5x + 70 = 6(x - 4)\n\nDistribute on the right side:\n\n5x + 70 = 6x - 24\n\nSubtract 5x from both sides:\n\n70 = x - 24\n\nAdd 24 to both sides:\n\n94 = x\n\nTherefore, Jordan's current average is 94.\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"15", "B":"20", "C":"30", "D":"45", "E":"60" }, "id":11382, "question":"Six lines intersect at one point to form 12 equal angles that are non-overlapping. What is the measure, in degrees, of one of these angles?", "explanations":{ "correct":"To find the measure of one of the angles formed by the intersection of six lines, we can use the fact that the sum of the angles around a point is 360 degrees.\n\nSince there are 12 equal angles formed by the intersection of the lines, we can divide 360 degrees by 12 to find the measure of one angle.\n\n360 degrees ÷ 12 = 30 degrees\n\nTherefore, the measure of one of these angles is 30 degrees.\n\nThe answer is C) 30." } }, { "answer":"B", "choices":{ "A":"1", "B":"3", "C":"5", "D":"7", "E":"9" }, "id":11385, "question":"If \\(4 < \\sqrt { 6k } < 5\\), then \\(k\\) =", "explanations":{ "correct":"To find the value of \\(k\\) given the inequality \\(4 < \\sqrt{6k} < 5\\), we need to isolate \\(k\\) by squaring both sides of the inequality.\n\nFirst, let's square the left side of the inequality:\n\\((4)^2 < (\\sqrt{6k})^2\\)\n\\(16 < 6k\\)\n\nNext, let's square the right side of the inequality:\n\\((\\sqrt{6k})^2 < (5)^2\\)\n\\(6k < 25\\)\n\nNow, we have the following inequalities:\n\\(16 < 6k < 25\\)\n\nTo solve for \\(k\\), we divide all parts of the inequality by 6:\n\\(\\frac{16}{6} < \\frac{6k}{6} < \\frac{25}{6}\\)\n\\(\\frac{8}{3} < k < \\frac{25}{6}\\)\n\nSince \\(k\\) must be an integer, we need to find the integer value of \\(k\\) that satisfies the inequality. The only integer value of \\(k\\) that falls within the range \\(\\frac{8}{3} < k < \\frac{25}{6}\\) is 3.\n\nTherefore, the answer is B) 3." } }, { "answer":"C", "choices":{ "A":"\\(a\\sqrt { 2 } \\)", "B":"\\(a\\sqrt { 3 } \\)", "C":"\\(a\\sqrt { 5 } \\)", "D":"\\(3a\\)", "E":"\\(5a\\)" }, "id":11386, "question":"A rectangle is twice as long as it is wide. If the width is a, what is the length of a diagonal?", "explanations":{ "correct":"To find the length of the diagonal of a rectangle, we can use the Pythagorean theorem. \n\nLet's assume the width of the rectangle is \\(a\\). Since the rectangle is twice as long as it is wide, the length of the rectangle would be \\(2a\\).\n\nNow, let's draw a diagonal in the rectangle, which forms a right triangle with the width and length of the rectangle. The diagonal is the hypotenuse of this right triangle.\n\nUsing the Pythagorean theorem, we have:\n\n\\((\\text{Hypotenuse})^2 = (\\text{Width})^2 + (\\text{Length})^2\\)\n\n\\((\\text{Diagonal})^2 = a^2 + (2a)^2\\)\n\nSimplifying this equation, we get:\n\n\\((\\text{Diagonal})^2 = a^2 + 4a^2\\)\n\n\\((\\text{Diagonal})^2 = 5a^2\\)\n\nTaking the square root of both sides, we have:\n\n\\(\\text{Diagonal} = \\sqrt{5a^2}\\)\n\nSimplifying further, we get:\n\n\\(\\text{Diagonal} = a\\sqrt{5}\\)\n\nTherefore, the length of the diagonal is \\(a\\sqrt{5}\\).\n\nThe answer is C) \\(a\\sqrt{5}\\)." } }, { "answer":"A", "choices":{ "A":"1", "B":"3", "C":"5", "D":"8", "E":"9" }, "id":11389, "question":"If \\(9\\sqrt{x} + 7 = 16\\), \\(x =\\)", "explanations":{ "correct":"To solve the equation \\(9\\sqrt{x} + 7 = 16\\), we need to isolate the variable \\(x\\).\n\nFirst, we subtract 7 from both sides of the equation:\n\\(9\\sqrt{x} = 16 - 7\\)\n\\(9\\sqrt{x} = 9\\)\n\nNext, we divide both sides of the equation by 9:\n\\(\\sqrt{x} = \\frac{9}{9}\\)\n\\(\\sqrt{x} = 1\\)\n\nTo solve for \\(x\\), we square both sides of the equation:\n\\(x = 1^2\\)\n\\(x = 1\\)\n\nTherefore, the answer is A) 1." } }, { "answer":"D", "choices":{ "A":"3(199)", "B":"3(200)", "C":"\\(3(3^200)\\)", "D":"\\(3^200\\)", "E":"\\(200^3\\)" }, "id":11391, "question":"\\begin { gather* } 3, 9, 27, 81 ....\\end { gather* } In the sequence above, the first term is 3 and each term after the first is 3 times the preceding term. Which of the following is an expression for the 200th term of the sequence?", "explanations":{ "correct":"To find the expression for the 200th term of the sequence, we need to determine the pattern and use it to calculate the term.\n\nGiven that the first term is 3 and each term after the first is 3 times the preceding term, we can see that the sequence is formed by repeatedly multiplying 3 by itself.\n\nTo find the 200th term, we need to multiply 3 by itself 199 times (since we already have the first term). This can be represented as \\(3^{199}\\).\n\nTherefore, the expression for the 200th term of the sequence is \\(3^{199}\\).\n\nThe answer is D) \\(3^{199}\\)." } }, { "answer":"B", "choices":{ "A":"213", "B":"222", "C":"231", "D":"233", "E":"333" }, "id":11392, "question":"Megan wrote down all the three-digit numbers that can be written using each of the numerals 1, 2 and 3 exactly once. What is the average (arithmetic mean) of the numbers Megan wrote?", "explanations":{ "correct":"To find the average of the three-digit numbers that can be written using each of the numerals 1, 2, and 3 exactly once, we need to find the sum of all these numbers and then divide it by the total number of numbers.\n\nFirst, let's list down all the possible three-digit numbers using the numerals 1, 2, and 3 exactly once:\n123, 132, 213, 231, 312, 321\n\nTo find the sum of these numbers, we add them up:\n123 + 132 + 213 + 231 + 312 + 321 = 1332\n\nSince there are 6 numbers in total, we divide the sum by 6 to find the average:\n1332 / 6 = 222\n\nTherefore, the average of the numbers Megan wrote is 222.\n\nThe answer is B) 222." } }, { "answer":"D", "choices":{ "A":"\\(a = b\\)", "B":"\\(a > 0\\)", "C":"\\(b > 0\\)", "D":"\\(a < b\\)", "E":"\\(a > b\\)" }, "id":11394, "question":"Which of the following conditions would make \\(2a - 2b < 0\\)?", "explanations":{ "correct":"To determine which condition would make \\(2a - 2b < 0\\), we need to analyze the inequality step-by-step.\n\nStep 1: Start with the given inequality \\(2a - 2b < 0\\).\n\nStep 2: Divide both sides of the inequality by 2 to simplify it: \\(\\frac{2a - 2b}{2} < \\frac{0}{2}\\).\n\nThis simplifies to \\(a - b < 0\\).\n\nStep 3: Rearrange the inequality to isolate the variable \\(a\\): \\(a < b\\).\n\nNow, let's analyze each option:\n\nA) \\(a = b\\): If \\(a\\) is equal to \\(b\\), then \\(a < b\\) is not true. Therefore, option A is not the correct answer.\n\nB) \\(a > 0\\): This condition does not provide any information about the relationship between \\(a\\) and \\(b\\). Therefore, option B is not the correct answer.\n\nC) \\(b > 0\\): This condition also does not provide any information about the relationship between \\(a\\) and \\(b\\). Therefore, option C is not the correct answer.\n\nD) \\(a < b\\): This condition directly matches the rearranged inequality \\(a < b\\). If \\(a\\) is less than \\(b\\), then \\(2a - 2b\\) will be negative, satisfying the inequality \\(2a - 2b < 0\\). Therefore, option D is the correct answer.\n\nE) \\(a > b\\): This condition does not match the rearranged inequality \\(a < b\\). Therefore, option E is not the correct answer.\n\nBased on the step-by-step analysis, the correct answer is D." } }, { "answer":"B", "choices":{ "A":"20", "B":"25", "C":"30", "D":"35", "E":"40" }, "id":11396, "question":"If \\(\\frac { a - b - 5 } { 2 } = 10\\), then what is the value of \\(a - b\\) ?", "explanations":{ "correct":"To find the value of \\(a - b\\), we need to solve the given equation \\(\\frac{a - b - 5}{2} = 10\\).\n\nFirst, we can start by multiplying both sides of the equation by 2 to eliminate the fraction:\n\n\\(\\frac{a - b - 5}{2} \\times 2 = 10 \\times 2\\)\n\nThis simplifies to:\n\n\\(a - b - 5 = 20\\)\n\nNext, we can add 5 to both sides of the equation to isolate the \\(a - b\\) term:\n\n\\(a - b - 5 + 5 = 20 + 5\\)\n\nThis simplifies to:\n\n\\(a - b = 25\\)\n\nTherefore, the value of \\(a - b\\) is 25.\n\nThe answer is B) 25." } }, { "answer":"D", "choices":{ "A":"\\(a = 0\\)", "B":"\\(\\sqrt{a + b} = -1\\)", "C":"\\(\\sqrt{a + b} = 0\\)", "D":"\\(a + b = 1\\)", "E":"\\((a + b)^2 = 0\\)" }, "id":11406, "question":"If \\(\\frac{1}{(a + b)^{-\\frac{1}{2}}} = (a + b)^{-\\frac{1}{2}}\\), which of the following must be true?", "explanations":{ "correct":"To solve this problem, let's start by simplifying the given equation:\n\n\\(\\frac{1}{(a + b)^{-\\frac{1}{2}}} = (a + b)^{-\\frac{1}{2}}\\)\n\nTo simplify the left side of the equation, we can apply the negative exponent rule, which states that \\(a^{-n} = \\frac{1}{a^n}\\). Applying this rule, we have:\n\n\\(\\frac{1}{(a + b)^{-\\frac{1}{2}}} = (a + b)^{\\frac{1}{2}}\\)\n\nNow, let's simplify the right side of the equation. The exponent \\(\\frac{1}{2}\\) represents the square root. So, we have:\n\n\\((a + b)^{\\frac{1}{2}} = \\sqrt{a + b}\\)\n\nTherefore, the given equation can be rewritten as:\n\n\\(\\frac{1}{\\sqrt{a + b}} = \\sqrt{a + b}\\)\n\nTo solve for \\(a\\) and \\(b\\), we can multiply both sides of the equation by \\(\\sqrt{a + b}\\):\n\n\\(\\frac{1}{\\sqrt{a + b}} \\cdot \\sqrt{a + b} = \\sqrt{a + b} \\cdot \\sqrt{a + b}\\)\n\nThis simplifies to:\n\n\\(1 = a + b\\)\n\nSo, the equation \\(a + b = 1\\) must be true.\n\nTherefore, the answer is D) \\(a + b = 1\\)." } }, { "answer":"D", "choices":{ "A":"6,300", "B":"10,000", "C":"10,300", "D":"20,000", "E":"31,400" }, "id":11407, "question":"A radio station emits a signal that can be received for 60 miles in all directions. If the intensity of the signal is strengthened so the reception increases by 40 miles in all directions, by approximately how many square miles is its region of reception increased?", "explanations":{ "correct":"To find the increase in the region of reception, we need to calculate the difference in the areas of the two regions.\n\nThe original region of reception can be represented by a circle with a radius of 60 miles. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.\n\nSo, the area of the original region of reception is A1 = π(60)^2.\n\nWhen the intensity of the signal is strengthened, the reception increases by 40 miles in all directions. This means the new radius is 60 + 40 = 100 miles.\n\nThe area of the new region of reception is A2 = π(100)^2.\n\nTo find the increase in the region of reception, we subtract the area of the original region from the area of the new region: A2 - A1 = π(100)^2 - π(60)^2.\n\nSimplifying this expression, we get A2 - A1 = π(100^2 - 60^2).\n\nCalculating the difference, we have A2 - A1 = π(10000 - 3600).\n\nFurther simplifying, we get A2 - A1 = π(6400).\n\nSince we are asked to find the increase in square miles, we need to approximate the value of π. Taking π as approximately 3.14, we can calculate the increase in square miles as 3.14 * 6400.\n\nMultiplying these values, we find the increase in square miles is approximately 20,096.\n\nTherefore, the answer is D) 20,000." } }, { "answer":"B", "choices":{ "A":"1", "B":"2", "C":"4", "D":"8", "E":"10" }, "id":11408, "question":"There are a total of 20 marbles in a bag containing only red marbles, blue marbles, and yellow marbles. If a marble is selected at random, the probability of getting a red marble is \\(\\frac{2}{5}\\) and the probability of getting a blue marble is \\(\\frac{1}{2}\\). How many yellow marbles are in the bag?", "explanations":{ "correct":"Let's assume that there are \\(x\\) yellow marbles in the bag.\n\nThe total number of marbles in the bag is given as 20. We know that the probability of getting a red marble is \\(\\frac{2}{5}\\), and the probability of getting a blue marble is \\(\\frac{1}{2}\\).\n\nThe probability of getting a yellow marble can be calculated by subtracting the sum of the probabilities of getting a red marble and a blue marble from 1, since these three events are mutually exclusive.\n\nSo, the probability of getting a yellow marble is \\(1 - \\left(\\frac{2}{5} + \\frac{1}{2}\\right)\\).\n\nSimplifying the expression inside the parentheses, we have \\(1 - \\left(\\frac{4}{10} + \\frac{5}{10}\\right)\\).\n\nFurther simplifying, we get \\(1 - \\frac{9}{10}\\).\n\nThis gives us the probability of getting a yellow marble as \\(\\frac{1}{10}\\).\n\nNow, we can set up an equation using the probability of getting a yellow marble and the total number of marbles:\n\n\\(\\frac{x}{20} = \\frac{1}{10}\\).\n\nCross-multiplying, we have \\(x = 2\\).\n\nTherefore, the answer is B) 2." } }, { "answer":"E", "choices":{ "A":"3, 6", "B":"4, 9", "C":"5, 15", "D":"6, 10", "E":"8, 12" }, "id":11412, "question":"Given the five pairs of numbers bellow, for which pair is the ratio of the smaller number to the larger number 2 to 3 ?", "explanations":{ "correct":"To find the pair of numbers where the ratio of the smaller number to the larger number is 2 to 3, we need to divide the smaller number by the larger number and check if the result is equal to 2/3.\n\nLet's go through each pair of numbers:\n\nA) 3 divided by 6 is 1/2, which is not equal to 2/3.\nB) 4 divided by 9 is not equal to 2/3.\nC) 5 divided by 15 is also not equal to 2/3.\nD) 6 divided by 10 is not equal to 2/3.\nE) 8 divided by 12 is equal to 2/3.\n\nTherefore, the pair of numbers where the ratio of the smaller number to the larger number is 2 to 3 is 8 and 12.\n\nThe answer is E." } }, { "answer":"D", "choices":{ "A":"-6", "B":"-2", "C":"2", "D":"3", "E":"4" }, "id":11417, "question":"Three times a number is the same as that number subtracted from 12. What is the number?", "explanations":{ "correct":"To solve this problem, let's assign a variable to the unknown number. Let's call the number \"x\".\n\nAccording to the problem, three times the number is the same as that number subtracted from 12. We can write this as the equation: 3x = 12 - x.\n\nTo solve for x, we need to isolate it on one side of the equation. Let's start by adding x to both sides of the equation: 3x + x = 12.\n\nCombining like terms, we get: 4x = 12.\n\nTo solve for x, we divide both sides of the equation by 4: x = 12/4.\n\nSimplifying the right side of the equation, we get: x = 3.\n\nTherefore, the number is 3.\n\nThe answer is D) 3." } }, { "answer":"C", "choices":{ "A":"\\(\\frac { \\pi } { 4 } \\)", "B":"\\(\\frac { \\pi - 4 } { 4 } \\)", "C":"\\(\\frac { 4 - \\pi } { 4 } \\)", "D":"\\(\\pi r\\)", "E":"\\(4r\\)" }, "id":11419, "question":"A circle with radius r is inscribed in a square. If a point within the square is randomly selected, what is the probability that the point will not be within the circle?", "explanations":{ "correct":"To find the probability that a randomly selected point within the square is not within the circle, we need to compare the areas of the square and the circle.\n\nThe area of the square is given by the formula \\(A_{\\text{square}} = s^2\\), where \\(s\\) is the length of a side of the square.\n\nThe area of the circle is given by the formula \\(A_{\\text{circle}} = \\pi r^2\\), where \\(r\\) is the radius of the circle.\n\nSince the circle is inscribed in the square, the diameter of the circle is equal to the length of a side of the square. Therefore, the radius of the circle is half the length of a side of the square, or \\(r = \\frac{s}{2}\\).\n\nSubstituting this value of \\(r\\) into the formula for the area of the circle, we get \\(A_{\\text{circle}} = \\pi \\left(\\frac{s}{2}\\right)^2 = \\frac{\\pi s^2}{4}\\).\n\nTo find the probability that a randomly selected point is not within the circle, we need to find the ratio of the area of the square that is not covered by the circle to the total area of the square.\n\nThe area of the square not covered by the circle is \\(A_{\\text{square}} - A_{\\text{circle}} = s^2 - \\frac{\\pi s^2}{4} = \\frac{4s^2 - \\pi s^2}{4} = \\frac{s^2(4 - \\pi)}{4}\\).\n\nThe total area of the square is \\(A_{\\text{square}} = s^2\\).\n\nTherefore, the probability that a randomly selected point is not within the circle is \\(\\frac{\\frac{s^2(4 - \\pi)}{4}}{s^2} = \\frac{4 - \\pi}{4}\\).\n\nHence, the answer is C) \\(\\frac{4 - \\pi}{4}\\)." } }, { "answer":"D", "choices":{ "A":"\\(4 \\pi\\)", "B":"\\(9 \\pi\\)", "C":"\\(16 \\pi\\)", "D":"\\(25 \\pi\\)", "E":"\\(36 \\pi\\)" }, "id":11420, "question":"In the xy-coordinate plane, the center of a circle is at ( 4, 0) and a point on the circle is at (7, 4). What is the area of the circle?", "explanations":{ "correct":"To find the area of a circle, we need to know the radius. The radius is the distance from the center of the circle to any point on the circle.\n\\(\\newline\\)In this case, the center of the circle is at (4, 0) and a point on the circle is at (7, 4). To find the radius, we can use the distance formula.\n\nThe distance formula is given by:\n\\(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\)\n\nUsing the coordinates (4, 0) and (7, 4), we can substitute the values into the distance formula:\n\n\\(d = \\sqrt{(7 - 4)^2 + (4 - 0)^2}\\)\n\\(d = \\sqrt{3^2 + 4^2}\\)\n\\(d = \\sqrt{9 + 16}\\)\n\\(d = \\sqrt{25}\\)\n\\(d = 5\\)\n\nSo, the radius of the circle is 5.\n\nThe formula for the area of a circle is given by:\n\\(A = \\pi r^2\\)\n\nSubstituting the value of the radius (5) into the formula, we get:\n\n\\(A = \\pi \\cdot 5^2\\)\n\\(A = \\pi \\cdot 25\\)\n\\(A = 25 \\pi\\)\n\nTherefore, the area of the circle is \\(25 \\pi\\).\n\nThe answer is D) \\(25 \\pi\\)." } }, { "answer":"A", "choices":{ "A":"2", "B":"3", "C":"4", "D":"6", "E":"8" }, "id":11421, "question":"Amanda, Ben, and Cindy-made a total of 34 greeting cards. Ben made 4 times as many as Amanda, and Cindy made 3 times as many as Ben. How many greeting cards did Amanda make?", "explanations":{ "correct":"Let's solve this problem step by step. \n\nLet's assume that Amanda made x greeting cards. \n\nAccording to the problem, Ben made 4 times as many as Amanda, so Ben made 4x greeting cards. \n\nCindy made 3 times as many as Ben, so Cindy made 3 * 4x = 12x greeting cards. \n\nThe total number of greeting cards made by all three is x + 4x + 12x = 17x. \n\nWe know that the total number of greeting cards made by all three is 34. \n\nSo, 17x = 34. \n\nDividing both sides of the equation by 17, we get x = 2. \n\nTherefore, Amanda made 2 greeting cards. \n\nThe answer is A) 2." } }, { "answer":"E", "choices":{ "A":"\\(\\sqrt { n } \\)", "B":"\\(\\sqrt { n^2 } \\)", "C":"\\(\\frac { 1 } { \\sqrt { n } } \\)", "D":"\\(\\frac { 1 } { \\sqrt { n^2 } } \\)", "E":"1" }, "id":11432, "question":"For all positive numbers \\(a\\) and \\(b\\), let \\(a \\square b = \\sqrt { ab } \\). If \\(n > 1\\), what does \\(n \\square \\frac { 1 } { n } \\) equal?", "explanations":{ "correct":"To find the value of \\(n \\square \\frac{1}{n}\\), we substitute \\(a = n\\) and \\(b = \\frac{1}{n}\\) into the given expression \\(a \\square b = \\sqrt{ab}\\).\n\nSo, \\(n \\square \\frac{1}{n} = \\sqrt{n \\cdot \\frac{1}{n}}\\).\n\nSimplifying the expression inside the square root, we have \\(\\sqrt{1}\\).\n\nSince the square root of 1 is 1, we can conclude that \\(n \\square \\frac{1}{n} = 1\\).\n\nTherefore, the answer is E) 1." } }, { "answer":"C", "choices":{ "A":"\\(-2\\)", "B":"1", "C":"\\(\\sqrt{2}\\)", "D":"2", "E":"\\(2\\sqrt{2}\\)" }, "id":11445, "question":"Let \\(\\_g\\_\\) be defined as \\(\\_g\\_ = 2g^2 - 2\\) for all values of g. If \\(\\_x\\_ = x^2\\), what is the value of x?", "explanations":{ "correct":"To find the value of x, we need to substitute the given expression for \\\\(\\_x\\\\_\\\\) into the equation \\\\(\\_g\\\\_ = 2g^2 - 2\\\\).\n\nGiven that \\\\(\\_x\\\\_ = x^2\\\\), we can substitute this expression into the equation:\n\n\\\\(x^2 = 2g^2 - 2\\\\)\n\nNow, let's solve for x by isolating it on one side of the equation:\n\n\\\\(x^2 + 2 = 2g^2\\\\)\n\nTo simplify further, we divide both sides of the equation by 2:\n\n\\\\(\\frac{x^2}{2} + 1 = g^2\\\\)\n\nTaking the square root of both sides, we get:\n\n\\\\(\\sqrt{\\frac{x^2}{2} + 1} = g\\\\)\n\nSince we are looking for the value of x, we need to find the value of g that satisfies this equation. Let's analyze the answer choices:\n\nA) \\(-2\\)\nB) 1\nC) \\(\\sqrt{2}\\)\nD) 2\nE) \\(2\\sqrt{2}\\)\n\nSubstituting each answer choice into the equation, we can determine which one satisfies the equation. Let's start with option B:\n\n\\\\(\\sqrt{\\frac{1^2}{2} + 1} = g\\\\)\n\nSimplifying the expression inside the square root:\n\n\\\\(\\sqrt{\\frac{1}{2} + 1} = g\\\\)\n\n\\\\(\\sqrt{\\frac{1}{2} + \\frac{2}{2}} = g\\\\)\n\n\\\\(\\sqrt{\\frac{3}{2}} = g\\\\)\n\nSince \\\\(\\sqrt{\\frac{3}{2}}\\\\) is not a rational number, option B is not a valid solution.\n\nLet's try option C:\n\n\\\\(\\sqrt{\\frac{(\\sqrt{2})^2}{2} + 1} = g\\\\)\n\nSimplifying the expression inside the square root:\n\n\\\\(\\sqrt{\\frac{2}{2} + 1} = g\\\\)\n\n\\\\(\\sqrt{\\frac{2}{2} + \\frac{2}{2}} = g\\\\)\n\n\\\\(\\sqrt{\\frac{4}{2}} = g\\\\)\n\n\\\\(\\sqrt{2} = g\\\\)\n\nOption C satisfies the equation, so \\\\(g = \\sqrt{2}\\\\). Since \\\\(\\_x\\\\_ = x^2\\\\), we can conclude that \\\\(x = \\sqrt{2}\\\\).\n\nTherefore, the answer is C." } }, { "answer":"C", "choices":{ "A":"\\(-\\frac { 7 } { 2 } \\)", "B":"\\(-\\frac { 2 } { 5 } \\)", "C":"\\(-\\frac { 2 } { 7 } \\)", "D":"\\(\\frac { 2 } { 7 } \\)", "E":"\\(\\frac { 2 } { 5 } \\)" }, "id":11447, "question":"If A is at (3, -1) and B is at (5, 6) what is the slope of the perpendicular bisector of segment \\(\\overline { AB } \\) ?", "explanations":{ "correct":"To find the slope of the perpendicular bisector of segment AB, we first need to find the slope of segment AB. \n\nThe slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:\n\n\\(m = \\frac { y2 - y1 } { x2 - x1 }\\)\n\nUsing the coordinates of points A (3, -1) and B (5, 6), we can calculate the slope of segment AB:\n\n\\(m_{AB} = \\frac { 6 - (-1) } { 5 - 3 } = \\frac { 7 } { 2 }\\)\n\nThe slope of the perpendicular bisector of segment AB is the negative reciprocal of the slope of segment AB. \n\nThe negative reciprocal of \\(\\frac { 7 } { 2 }\\) is \\(-\\frac { 2 } { 7 }\\).\n\nTherefore, the answer is C) \\(-\\frac { 2 } { 7 }\\)." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{1}{5}\\)", "B":"\\(\\frac{1}{4}\\)", "C":"\\(\\frac{3}{10}\\)", "D":"\\(\\frac{1}{2}\\)", "E":"\\(\\frac{3}{5}\\)" }, "id":11451, "question":"If Maurice has \\$ 80, and he spends \\$ 32.45 on clothes and gives \\$ 27.55 to his sister, what fraction of the original \\$ 80 does Maurice have left?", "explanations":{ "correct":"To find the fraction of the original \\$ 80 that Maurice has left, we need to subtract the amount he spent and gave away from the original amount and then divide it by the original amount.\n\nMaurice spent \\$ 32.45 on clothes and gave \\$ 27.55 to his sister. So, the total amount he spent and gave away is \\$ 32.45 + \\$ 27.55 = \\$ 60.\n\nTo find the amount Maurice has left, we subtract the total amount he spent and gave away from the original \\$ 80: \\$ 80 - \\$ 60 = \\$ 20.\n\nNow, we divide the amount Maurice has left ($ 20) by the original amount ($ 80) to find the fraction: \\(\\frac{20}{80}\\).\n\nSimplifying the fraction, we get \\(\\frac{1}{4}\\).\n\nTherefore, the answer is B) \\(\\frac{1}{4}\\)." } }, { "answer":"E", "choices":{ "A":"3", "B":"4", "C":"5", "D":"6", "E":"7" }, "id":11452, "question":"When 28 is divided by 8, the remainder is the same as that obtained when 53 is divided by which of the following numbers?", "explanations":{ "correct":"To find the remainder when 28 is divided by 8, we can perform the division: 28 ÷ 8 = 3 with a remainder of 4.\n\nNow, we need to find the number among the given options that will give us the same remainder when 53 is divided by that number.\n\nLet's check each option:\n\nA) When 53 is divided by 3, we get a quotient of 17 with a remainder of 2.\nB) When 53 is divided by 4, we get a quotient of 13 with a remainder of 1.\nC) When 53 is divided by 5, we get a quotient of 10 with a remainder of 3.\nD) When 53 is divided by 6, we get a quotient of 8 with a remainder of 5.\nE) When 53 is divided by 7, we get a quotient of 7 with a remainder of 4.\n\nFrom the options, we can see that the remainder obtained when 53 is divided by 7 is the same as the remainder obtained when 28 is divided by 8.\n\nTherefore, the answer is E) 7." } }, { "answer":"C", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III only", "E":"II and III only" }, "id":11453, "question":"If \\(x\\), \\(y\\), and \\(z\\) are positive and \\(x^4 y^3 z^3 > x^5 y^3 z^2\\). Which of the following must be true ? \\(\\newline\\)I. \\(x > y\\) \\(\\newline\\)II. \\(y > z\\) \\(\\newline\\)III. \\(z > x\\)", "explanations":{ "correct":"To determine which of the given statements must be true, let's simplify the inequality first.\n\nWe have \\(x^4 y^3 z^3 > x^5 y^3 z^2\\).\n\nDividing both sides of the inequality by \\(x^4 y^3 z^2\\) (since all variables are positive), we get:\n\n\\(\\frac{{x^4 y^3 z^3}}{{x^4 y^3 z^2}} > \\frac{{x^5 y^3 z^2}}{{x^4 y^3 z^2}}\\).\n\nSimplifying further, we have:\n\n\\(\\frac{z}{x} > x\\).\n\nNow, let's analyze each statement:\n\\(\\newline\\)I. \\(x > y\\): We cannot determine the relationship between \\(x\\) and \\(y\\) based on the given inequality. Therefore, statement I cannot be concluded.\n\\(\\newline\\)II. \\(y > z\\): From the simplified inequality, we have \\(\\frac{z}{x} > x\\). Since \\(z\\) and \\(x\\) are positive, we can conclude that \\(z < x\\). However, we cannot determine the relationship between \\(y\\) and \\(z\\) based on the given inequality. Therefore, statement II cannot be concluded.\n\\(\\newline\\)III. \\(z > x\\): From the simplified inequality, we have \\(\\frac{z}{x} > x\\). Since \\(z\\) and \\(x\\) are positive, we can conclude that \\(z > x\\). Therefore, statement III must be true.\n\nBased on our analysis, the only statement that must be true is \\(\\newline\\)III. Therefore, the answer is C) III only." } }, { "answer":"D", "choices":{ "A":"\\(170c\\)", "B":"\\(150 + c\\)", "C":"\\(150c + 20\\)", "D":"\\(150 + 20c\\)", "E":"\\(170 + 20c\\)" }, "id":11454, "question":"A chef prepares a multicourse meal at a client's home. For this service, she charges a base price of \\$ 150, and adds an additional \\$ 20 for each course ordered. If a client orders c courses, which of the following represents the total charge, in dollars, for the meal?", "explanations":{ "correct":"To find the total charge for the meal, we need to consider the base price and the additional charge for each course ordered.\n\nThe base price is \\$ 150, which is charged regardless of the number of courses ordered.\n\nFor each course ordered, there is an additional charge of \\$ 20. Since the client orders c courses, the additional charge for the courses would be \\$ 20 multiplied by c, which can be written as 20c.\n\nTo find the total charge, we need to add the base price and the additional charge for the courses. Therefore, the total charge can be represented as \\(150 + 20c\\).\n\nHence, the answer is D) \\(150 + 20c\\)." } }, { "answer":"A", "choices":{ "A":"28", "B":"30", "C":"32", "D":"34", "E":"36" }, "id":11456, "question":"If \\(5x + 3 = 21\\), then \\(5x + 10 =\\)", "explanations":{ "correct":"To find the value of \\\\(5x + 10\\\\), we need to first solve the equation \\\\(5x + 3 = 21\\\\) for \\\\(x\\\\).\n\nStep 1: Subtract 3 from both sides of the equation:\n\\\\(5x + 3 - 3 = 21 - 3\\\\)\n\\\\(5x = 18\\\\)\n\nStep 2: Divide both sides of the equation by 5 to isolate \\\\(x\\\\):\n\\\\(\\\\frac{{5x}}{5} = \\\\frac{{18}}{5}\\\\)\n\\\\(x = \\\\frac{18}{5}\\\\)\n\nNow that we have found the value of \\\\(x\\\\), we can substitute it into the expression \\\\(5x + 10\\\\) to find the final answer.\n\nStep 3: Substitute \\\\(x = \\\\frac{18}{5}\\\\) into \\\\(5x + 10\\\\):\n\\\\(5\\\\left(\\\\frac{18}{5}\\\\right) + 10\\\\)\n\\\\(\\\\frac{90}{5} + 10\\\\)\n\\\\(18 + 10\\\\)\n\\\\(28\\\\)\n\nTherefore, the answer is A) 28." } }, { "answer":"E", "choices":{ "A":"65", "B":"130", "C":"325", "D":"420", "E":"520" }, "id":11459, "question":"Of 650 vehicles in a parking lot, \\(\\frac { 1 } { 5 } \\) are trucks. How many of the vehicles are not trucks?", "explanations":{ "correct":"To find the number of vehicles that are not trucks, we need to subtract the number of trucks from the total number of vehicles.\n\nGiven that \\\\(\\frac{1}{5}\\\\) of the 650 vehicles are trucks, we can find the number of trucks by multiplying \\\\(\\frac{1}{5}\\\\) by 650:\n\n\\\\(\\frac{1}{5} \\times 650 = \\frac{650}{5} = 130\\\\)\n\nSo, there are 130 trucks in the parking lot.\n\nTo find the number of vehicles that are not trucks, we subtract the number of trucks from the total number of vehicles:\n\n650 - 130 = 520\n\nTherefore, the answer is E) 520." } }, { "answer":"E", "choices":{ "A":"-4", "B":"-2", "C":"0", "D":"2", "E":"4" }, "id":11460, "question":"What is the value of \\(m\\) if \\(\\frac { 4 } { 3m - 2 } = \\frac { 4 } { m + 6 } \\) ?", "explanations":{ "correct":"To find the value of \\\\(m\\\\), we can start by cross-multiplying the fractions. \n\nCross-multiplying means multiplying the numerator of the first fraction with the denominator of the second fraction, and vice versa. \n\nSo, we have \\\\(4 \\cdot (m + 6) = 4 \\cdot (3m - 2)\\\\).\n\nExpanding both sides of the equation, we get \\\\(4m + 24 = 12m - 8\\\\).\n\nNext, we can simplify the equation by combining like terms. \n\nSubtracting \\\\(4m\\\\) from both sides, we have \\\\(24 = 8m - 8\\\\).\n\nAdding 8 to both sides, we get \\\\(32 = 8m\\\\).\n\nFinally, we can solve for \\\\(m\\\\) by dividing both sides by 8. \n\nThis gives us \\\\(m = 4\\\\).\n\nTherefore, the answer is E) 4." } }, { "answer":"D", "choices":{ "A":"\\(y(n) = 1,000 + 0.12n\\)", "B":"\\(y(n) = 1,000 + 1,000(1 + 0.12n)\\)", "C":"\\(y(n) = 1,000(0.88)^n\\)", "D":"\\(y(n) = 1,000(1.12)^n\\)", "E":"\\(y(n) = 1,000(1.12n)\\)" }, "id":11467, "question":"Josephine invests \\$ 1,000 in a savings account paying 12 percent annual interest. If she invests \\(n\\) years, which of the following functions represents the amount in her account, \\(y\\) in dollars, after \\(n\\) years?", "explanations":{ "correct":"To determine the correct function that represents the amount in Josephine's account after \\(n\\) years, we need to consider the compound interest formula.\n\nThe compound interest formula is given by:\n\\[A = P(1 + r)^n\\]\nwhere:\n- \\(A\\) is the final amount\n- \\(P\\) is the principal amount (initial investment)\n- \\(r\\) is the interest rate per period\n- \\(n\\) is the number of periods\n\\(\\newline\\)In this case, Josephine invests \\$ 1,000 (principal) at an annual interest rate of 12 percent (0.12) for \\(n\\) years.\n\nTherefore, the correct function that represents the amount in her account after \\(n\\) years is:\n\\[y(n) = 1,000(1 + 0.12)^n\\]\n\nSimplifying the expression:\n\\[y(n) = 1,000(1.12)^n\\]\n\nHence, the answer is D) \\(y(n) = 1,000(1.12)^n\\)." } }, { "answer":"D", "choices":{ "A":"I only", "B":"III only", "C":"I and II only", "D":"I and III only", "E":"I, II, and III" }, "id":11468, "question":"Two sides of a right triangle are 5 and 6. Which of the following could be the length of the third side? \\(\\newline\\)I. \\(\\sqrt { 11 } \\) \\(\\newline\\)II. \\(\\sqrt { 31 } \\) \\(\\newline\\)III. \\(\\sqrt { 61 } \\)", "explanations":{ "correct":"To determine the length of the third side of a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.\n\nLet's calculate the possible lengths of the third side using the given information:\n\nFor the first option, \\\\(\\sqrt{11}\\\\), we can square it to get 11. Since 5^2 + 6^2 = 25 + 36 = 61, which is not equal to 11, \\\\(\\sqrt{11}\\\\) cannot be the length of the third side.\n\nFor the second option, \\\\(\\sqrt{31}\\\\), we can square it to get 31. Since 5^2 + 6^2 = 25 + 36 = 61, which is equal to 31, \\\\(\\sqrt{31}\\\\) could be the length of the third side.\n\nFor the third option, \\\\(\\sqrt{61}\\\\), we can square it to get 61. Since 5^2 + 6^2 = 25 + 36 = 61, which is equal to 61, \\\\(\\sqrt{61}\\\\) could be the length of the third side.\n\nTherefore, the lengths of the third side that are possible are \\\\(\\sqrt{31}\\\\) and \\\\(\\sqrt{61}\\\\). \n\nThe answer is D) I and III only." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{1}{48}\\)", "B":"\\(\\frac{1}{8}\\)", "C":"\\(\\frac{1}{6}\\)", "D":"6", "E":"8" }, "id":11469, "question":"If \\(6a = 48\\) and \\(ab = 1\\), what is the value of b?", "explanations":{ "correct":"To find the value of \\(b\\), we can use the given equations \\(6a = 48\\) and \\(ab = 1\\).\n\nFirst, let's solve the equation \\(6a = 48\\) to find the value of \\(a\\). We divide both sides of the equation by 6:\n\n\\(\\frac{6a}{6} = \\frac{48}{6}\\)\n\nThis simplifies to:\n\n\\(a = 8\\)\n\nNow that we know \\(a = 8\\), we can substitute this value into the equation \\(ab = 1\\) to solve for \\(b\\):\n\n\\(8b = 1\\)\n\nTo isolate \\(b\\), we divide both sides of the equation by 8:\n\n\\(\\frac{8b}{8} = \\frac{1}{8}\\)\n\nThis simplifies to:\n\n\\(b = \\frac{1}{8}\\)\n\nTherefore, the value of \\(b\\) is \\(\\frac{1}{8}\\).\n\nThe answer is B) \\(\\frac{1}{8}\\)." } }, { "answer":"E", "choices":{ "A":"3,000", "B":"2,020", "C":"1,800", "D":"1,470", "E":"1,200" }, "id":11471, "question":"If 70 percent of x is 2,100, then what is 40 percent of x?", "explanations":{ "correct":"To find the answer, we need to set up an equation using the given information. We are told that 70 percent of x is equal to 2,100. \n\nLet's represent x as a variable. We can set up the equation as follows:\n\n0.70x = 2,100\n\nTo find 40 percent of x, we need to multiply x by 0.40. \n\n0.40x = (0.40)(2,100)\n\nSimplifying the right side of the equation:\n\n0.40x = 840\n\nTherefore, 40 percent of x is equal to 840.\n\nThe answer is E) 1,200." } }, { "answer":"E", "choices":{ "A":"19", "B":"95", "C":"105", "D":"190", "E":"200" }, "id":11473, "question":"Stephanie has ten shirts, four pairs of pants, and five pairs of shoes. If an outfit consists of exactly one shirt, one pair of pants and one pair of shoes, how many different outfits could she wear?", "explanations":{ "correct":"To find the number of different outfits Stephanie can wear, we need to multiply the number of choices for each item of clothing together. \n\nStephanie has 10 choices for a shirt, 4 choices for a pair of pants, and 5 choices for a pair of shoes. \n\nTo find the total number of outfits, we multiply these choices together: 10 * 4 * 5 = 200. \n\nTherefore, Stephanie can wear 200 different outfits. \n\nThe answer is E) 200." } }, { "answer":"D", "choices":{ "A":"\\(C(p) = 6p - 15\\)", "B":"\\(C(p) = 6p - 9\\)", "C":"\\(C(p) = 6p\\)", "D":"\\(C(p) = 6p + 9\\)", "E":"\\(C(p) = 6p + 15\\)" }, "id":11476, "question":"The cost for coal from a certain company is \\$ 15 for the first pound plus \\$ 6 for each additional pound of coal. Which of the following functions gives the total cost, in dollars, for p pounds of coal?", "explanations":{ "correct":"To find the total cost for p pounds of coal, we need to consider the cost for the first pound and the cost for each additional pound.\n\nAccording to the given information, the cost for the first pound of coal is \\$ 15. This means that regardless of the number of pounds, we will always have \\$ 15 as part of the total cost.\n\nFor each additional pound of coal, the cost is \\$ 6. So, if we have p pounds of coal, we will have (p-1) additional pounds, since we already accounted for the first pound.\n\nTherefore, the total cost for p pounds of coal can be calculated as:\nTotal cost = Cost for the first pound + Cost for additional pounds\nTotal cost = \\$ 15 + ($ 6 * (p-1))\nTotal cost = \\$ 15 + \\$ 6p - \\$ 6\nTotal cost = \\$ 6p + \\$ 9\n\nHence, the correct function that gives the total cost, in dollars, for p pounds of coal is:\nC(p) = 6p + 9\n\nTherefore, the answer is D." } }, { "answer":"D", "choices":{ "A":"50", "B":"80", "C":"100", "D":"120", "E":"150" }, "id":11478, "question":"In a class, 20 children were sharing equally the cost of a present for their teacher. When 4 of the children decided not to contribute, each of the other children had to pay \\$ 1.50 more. How much, in dollars, did the present cost?", "explanations":{ "correct":"Let's assume that the initial cost of the present was x dollars. \n\\(\\newline\\)Initially, there were 20 children sharing the cost equally, so each child had to pay x/20 dollars.\n\nWhen 4 children decided not to contribute, the remaining children had to cover their share. So now, each child had to pay x/16 dollars.\n\nAccording to the problem, each child had to pay \\$ 1.50 more than before. So we can set up the equation:\n\nx/16 = x/20 + 1.50\n\nTo solve this equation, we can start by multiplying both sides by 16 * 20 to eliminate the denominators:\n\n20x = 16x + 1.50 * 16 * 20\n\nSimplifying the equation:\n\n20x = 16x + 480\n\nSubtracting 16x from both sides:\n\n4x = 480\n\nDividing both sides by 4:\n\nx = 120\n\nTherefore, the present cost 120 dollars.\n\nThe answer is D) 120." } }, { "answer":"C", "choices":{ "A":"\\(f(5) = -3\\)", "B":"\\(f(0) = -5\\)", "C":"\\(f(2) = 4\\)", "D":"\\(f(1) = k\\)", "E":"\\(f(-2) = b\\)" }, "id":11484, "question":"The function f is defined by \\(f(x) = a (x - b)2 + k\\) for all real \\(x\\), where \\(a\\), \\(b\\), and \\(k\\) are constants. If \\(a\\) and \\(k\\) are negative, which of the following CANNOT be true?", "explanations":{ "correct":"To determine which statement cannot be true, we need to analyze the given function \\(f(x) = a(x - b)^2 + k\\) and the given conditions that \\(a\\) and \\(k\\) are negative.\n\n1. Statement A: \\(f(5) = -3\\)\nSubstituting \\(x = 5\\) into the function, we get \\(f(5) = a(5 - b)^2 + k\\). Since \\(a\\) and \\(k\\) are negative, the term \\(a(5 - b)^2\\) will always be positive. Therefore, it is not possible for \\(f(5)\\) to equal -3. \n\n2. Statement B: \\(f(0) = -5\\)\nSubstituting \\(x = 0\\) into the function, we get \\(f(0) = a(0 - b)^2 + k\\). Again, since \\(a\\) and \\(k\\) are negative, the term \\(a(0 - b)^2\\) will always be positive. Therefore, it is not possible for \\(f(0)\\) to equal -5.\n\n3. Statement C: \\(f(2) = 4\\)\nSubstituting \\(x = 2\\) into the function, we get \\(f(2) = a(2 - b)^2 + k\\). Since \\(a\\) and \\(k\\) are negative, the term \\(a(2 - b)^2\\) will always be positive. Therefore, it is not possible for \\(f(2)\\) to equal 4.\n\n4. Statement D: \\(f(1) = k\\)\nSubstituting \\(x = 1\\) into the function, we get \\(f(1) = a(1 - b)^2 + k\\). Since \\(a\\) and \\(k\\) are negative, the term \\(a(1 - b)^2\\) will always be positive. However, since \\(f(1)\\) is equal to \\(k\\), which is negative, it is possible for \\(f(1)\\) to equal \\(k\\).\n\n5. Statement E: \\(f(-2) = b\\)\nSubstituting \\(x = -2\\) into the function, we get \\(f(-2) = a(-2 - b)^2 + k\\). Since \\(a\\) and \\(k\\) are negative, the term \\(a(-2 - b)^2\\) will always be positive. Therefore, it is not possible for \\(f(-2)\\) to equal \\(b\\).\n\nBased on the analysis above, the statement that CANNOT be true is Statement C: \\(f(2) = 4\\).\n\nTherefore, the answer is C." } }, { "answer":"C", "choices":{ "A":"8", "B":"10", "C":"12", "D":"14", "E":"20" }, "id":11487, "question":"Some boys and girls were having a car wash to raise money for a class trip. Initially, \\(\\frac { 2 } { 5 } \\) of the group were girls. Shortly thereafter two girls left and two boys entered, then \\(\\frac { 1 } { 3 } \\) of the group are girls. How many girls were initially in the group?", "explanations":{ "correct":"Let's assume the total number of boys and girls in the group initially is represented by the variable \\(x\\).\n\nAccording to the problem, initially, \\(\\frac{2}{5}\\) of the group were girls. This means there were \\(\\frac{2}{5}x\\) girls in the group.\n\nAfter two girls left and two boys entered, the total number of boys and girls in the group remains the same, which is \\(x\\). At this point, \\(\\frac{1}{3}\\) of the group are girls. This means there were \\(\\frac{1}{3}x\\) girls in the group.\n\nWe can set up an equation based on the given information:\n\n\\(\\frac{2}{5}x - 2 = \\frac{1}{3}x\\)\n\nTo solve for \\(x\\), we can start by getting rid of the fractions by multiplying both sides of the equation by the least common multiple (LCM) of 5 and 3, which is 15:\n\n\\(15 \\cdot \\left(\\frac{2}{5}x - 2\\right) = 15 \\cdot \\left(\\frac{1}{3}x\\right)\\)\n\n\\(6x - 30 = 5x\\)\n\nNext, we can isolate the variable \\(x\\) by subtracting \\(5x\\) from both sides of the equation:\n\n\\(6x - 5x - 30 = 5x - 5x\\)\n\n\\(x - 30 = 0\\)\n\nFinally, we can solve for \\(x\\) by adding 30 to both sides of the equation:\n\n\\(x - 30 + 30 = 0 + 30\\)\n\n\\(x = 30\\)\n\nTherefore, the initial total number of boys and girls in the group is 30. To find the number of girls initially, we can substitute \\(x = 30\\) into the expression \\(\\frac{2}{5}x\\):\n\n\\(\\frac{2}{5} \\cdot 30 = 12\\)\n\nSo, there were initially 12 girls in the group.\n\nThe answer is C) 12." } }, { "answer":"A", "choices":{ "A":"-4", "B":"-1", "C":"6", "D":"8", "E":"36" }, "id":11488, "question":"If \\(3\\sqrt { k^2 } + 8 = 20\\), which of the following could be the value of \\(k\\) ?", "explanations":{ "correct":"To solve the equation \\(3\\sqrt{k^2} + 8 = 20\\), we need to isolate the variable \\(k\\).\n\nFirst, let's subtract 8 from both sides of the equation:\n\\(3\\sqrt{k^2} = 20 - 8\\)\n\\(3\\sqrt{k^2} = 12\\)\n\nNext, divide both sides of the equation by 3:\n\\(\\sqrt{k^2} = \\frac{12}{3}\\)\n\\(\\sqrt{k^2} = 4\\)\n\nSince we are looking for the possible values of \\(k\\), we need to consider both the positive and negative square roots.\n\nTaking the positive square root:\n\\(k = \\sqrt{4}\\)\n\\(k = 2\\)\n\nTaking the negative square root:\n\\(k = -\\sqrt{4}\\)\n\\(k = -2\\)\n\nTherefore, the possible values of \\(k\\) are 2 and -2.\n\nChecking the answer choices, we see that the only option that matches one of the possible values of \\(k\\) is:\n\nThe answer is A) -4." } }, { "answer":"C", "choices":{ "A":"82", "B":"82.5", "C":"83", "D":"84", "E":"84.5" }, "id":11496, "question":"On a certain test, a class of 12 students has an average (arithmetic mean) score of 80. A second class of 18 students has an average score of 85. What is the average score of the combined classes?", "explanations":{ "correct":"To find the average score of the combined classes, we need to calculate the total sum of scores for both classes and divide it by the total number of students.\n\nFor the first class with 12 students and an average score of 80, the total sum of scores is 12 * 80 = 960.\n\nFor the second class with 18 students and an average score of 85, the total sum of scores is 18 * 85 = 1530.\n\nTo find the average score of the combined classes, we add the total sum of scores for both classes (960 + 1530 = 2490) and divide it by the total number of students (12 + 18 = 30).\n\nTherefore, the average score of the combined classes is 2490 / 30 = 83.\n\nThe answer is C) 83." } }, { "answer":"D", "choices":{ "A":"\\(|p| \\leq 2\\)", "B":"\\(|p - 2| \\leq 2\\)", "C":"\\(|p + 2| \\leq 0.25\\)", "D":"\\(|p - 2| \\leq 0.25\\)", "E":"\\(|p - 17.5 \\leq 0.25\\)" }, "id":11503, "question":"The weight of a box of apples ranges from 1.75 pounds to 2.25 pounds. If \\(p\\) is the weight, in pounds, of the box, which of the following must be true?", "explanations":{ "correct":"To determine which of the given options must be true, we need to analyze the given information about the weight of the box of apples.\n\nThe weight of the box of apples ranges from 1.75 pounds to 2.25 pounds. Let's break this down into two separate inequalities:\n\n1.75 ≤ p ≤ 2.25\n\nNow, let's analyze each option:\n\nA) |p| ≤ 2\nTo check if this option is true, we need to consider the maximum and minimum values of p within the given range. The maximum value of p is 2.25, which is greater than 2. Therefore, |p| ≤ 2 is not true.\n\nB) |p - 2| ≤ 2\nTo check if this option is true, we need to consider the maximum and minimum values of p within the given range. The maximum value of p is 2.25, so |2.25 - 2| = |0.25| = 0.25, which is less than or equal to 2. Therefore, |p - 2| ≤ 2 is true.\n\nC) |p + 2| ≤ 0.25\nTo check if this option is true, we need to consider the maximum and minimum values of p within the given range. The maximum value of p is 2.25, so |2.25 + 2| = |4.25| = 4.25, which is greater than 0.25. Therefore, |p + 2| ≤ 0.25 is not true.\n\nD) |p - 2| ≤ 0.25\nTo check if this option is true, we need to consider the maximum and minimum values of p within the given range. The maximum value of p is 2.25, so |2.25 - 2| = |0.25| = 0.25, which is less than or equal to 0.25. Therefore, |p - 2| ≤ 0.25 is true.\n\nE) |p - 17.5| ≤ 0.25\nThis option is not relevant to the given information because it compares p to a value of 17.5, which is not mentioned in the problem. Therefore, |p - 17.5| ≤ 0.25 is not true.\n\nBased on our analysis, the only option that must be true is D) |p - 2| ≤ 0.25.\n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"5w", "B":"\\(w^5\\)", "C":"8w", "D":"\\(8w^2\\)", "E":"\\(9w^2\\)" }, "id":11504, "question":"If \\(w = 3^3\\), which of the following expressions is equal to \\(3^8\\)?", "explanations":{ "correct":"To find the expression that is equal to \\(3^8\\), we need to simplify the given expression \\(w = 3^3\\) first.\n\n\\(w = 3^3\\) means that we need to multiply 3 by itself three times: \\(w = 3 \\times 3 \\times 3 = 27\\).\n\nNow, let's find the expression that is equal to \\(3^8\\).\n\n\\(3^8\\) means that we need to multiply 3 by itself eight times: \\(3^8 = 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3\\).\n\nWe can rewrite this expression using the value of \\(w\\) that we found earlier:\n\n\\(3^8 = 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 \\times 3 = 3^3 \\times 3^3 \\times 3^2\\).\n\nUsing the exponent rule that states \\(a^m \\times a^n = a^{m+n}\\), we can simplify this expression further:\n\n\\(3^8 = 3^3 \\times 3^3 \\times 3^2 = 3^{3+3+2} = 3^8\\).\n\nTherefore, the expression that is equal to \\(3^8\\) is \\(3^8\\) itself.\n\nThe answer is E." } }, { "answer":"D", "choices":{ "A":"20%", "B":"40%", "C":"50%", "D":"60%", "E":"70%" }, "id":11506, "question":"A 20-gallon water tank is filled to capacity. If 8 gallons of water are pumped out of the tank, what percent of the tank is now filled?", "explanations":{ "correct":"To find the percent of the tank that is now filled after 8 gallons of water are pumped out, we need to determine the remaining amount of water in the tank.\n\nGiven that the tank is initially filled to capacity with 20 gallons of water, and 8 gallons are pumped out, we can subtract 8 from 20 to find the remaining amount of water in the tank: 20 - 8 = 12 gallons.\n\nTo find the percent of the tank that is now filled, we need to calculate the ratio of the remaining amount of water to the tank's capacity, and then multiply by 100.\n\nThe remaining amount of water is 12 gallons, and the tank's capacity is 20 gallons. So, the ratio of the remaining amount of water to the tank's capacity is 12/20.\n\nTo convert this ratio to a percentage, we multiply by 100: (12/20) * 100 = 60%.\n\nTherefore, the answer is D) 60%." } }, { "answer":"D", "choices":{ "A":"10", "B":"12", "C":"30", "D":"40", "E":"80" }, "id":11509, "question":"\\(r = 4c - d\\) \\(t = (c +1)r\\) If r and t are defined by the equations above, what is the value of \\(t\\) when \\(c = 3\\) and \\(d = 2\\) ?", "explanations":{ "correct":"To find the value of \\\\(t\\\\) when \\\\(c = 3\\\\) and \\\\(d = 2\\\\), we need to substitute these values into the given equations and solve for \\\\(t\\\\).\n\nGiven: \\\\(r = 4c - d\\\\) and \\\\(t = (c + 1)r\\\\)\n\nSubstituting \\\\(c = 3\\\\) and \\\\(d = 2\\\\) into the first equation:\n\\\\(r = 4(3) - 2\\\\)\n\\\\(r = 12 - 2\\\\)\n\\\\(r = 10\\\\)\n\nNow, substituting \\\\(c = 3\\\\) and \\\\(r = 10\\\\) into the second equation:\n\\\\(t = (3 + 1)(10)\\\\)\n\\\\(t = 4 \\times 10\\\\)\n\\\\(t = 40\\\\)\n\nTherefore, the value of \\\\(t\\\\) when \\\\(c = 3\\\\) and \\\\(d = 2\\\\) is 40.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"9", "B":"12", "C":"18", "D":"27", "E":"36" }, "id":11512, "question":"If \\(a\\) and \\(b\\) are positive integers, and \\(3^a \\times 3^b = 27\\), what is the value of \\(3^a + 3^b\\)?", "explanations":{ "correct":"To find the value of \\(3^a + 3^b\\), we need to determine the values of \\(a\\) and \\(b\\) that satisfy the equation \\(3^a \\times 3^b = 27\\).\n\nSince \\(3^a \\times 3^b\\) is equal to \\(27\\), we can rewrite this equation using the properties of exponents: \\(3^{a+b} = 27\\).\n\nTo solve for \\(a+b\\), we need to find the exponent that gives us \\(27\\) when the base is \\(3\\). We know that \\(3^3 = 27\\), so \\(a+b = 3\\).\n\nNow, we need to find the values of \\(a\\) and \\(b\\) that satisfy \\(a+b = 3\\). Since \\(a\\) and \\(b\\) are positive integers, the possible pairs of values are:\n\n\\(a = 1\\) and \\(b = 2\\)\n\\(a = 2\\) and \\(b = 1\\)\n\\(a = 3\\) and \\(b = 0\\)\n\nNow, let's substitute these values into \\(3^a + 3^b\\) to find the sum:\n\nFor \\(a = 1\\) and \\(b = 2\\), \\(3^a + 3^b = 3^1 + 3^2 = 3 + 9 = 12\\).\nFor \\(a = 2\\) and \\(b = 1\\), \\(3^a + 3^b = 3^2 + 3^1 = 9 + 3 = 12\\).\nFor \\(a = 3\\) and \\(b = 0\\), \\(3^a + 3^b = 3^3 + 3^0 = 27 + 1 = 28\\).\n\nTherefore, the only possible value for \\(3^a + 3^b\\) is \\(12\\).\n\nThe answer is B) 12." } }, { "answer":"D", "choices":{ "A":"16", "B":"24", "C":"32", "D":"\\(37 \\frac { 1 } { 2 } \\)", "E":"50" }, "id":11516, "question":"In the figure above, line \\(\\ell\\) is given by the equation \\(y = -x + 8\\), and line \\(m\\) is given by \\(y = 2x - 7\\). If line \\(\\ell\\) intersects line \\(m\\) at point \\(B\\), what is the area of \\(\\triangle { ABC } \\) ?", "explanations":{ "correct":"To find the area of triangle ABC, we first need to find the coordinates of points A and C. \n\nGiven that line \\( \\ell \\) is given by the equation \\( y = -x + 8 \\), we can substitute this equation into the equation of line \\( m \\) to find the x-coordinate of point B. \n\nSubstituting \\( -x + 8 \\) for y in the equation \\( y = 2x - 7 \\), we get:\n\n\\( -x + 8 = 2x - 7 \\)\n\nSimplifying the equation, we have:\n\n\\( 3x = 15 \\)\n\nDividing both sides by 3, we find:\n\n\\( x = 5 \\)\n\nNow, we can substitute this value of x back into the equation of line \\( \\ell \\) to find the y-coordinate of point B. \n\nSubstituting \\( x = 5 \\) into the equation \\( y = -x + 8 \\), we get:\n\n\\( y = -5 + 8 \\)\n\nSimplifying, we find:\n\n\\( y = 3 \\)\n\nSo, the coordinates of point B are (5, 3).\n\nTo find the coordinates of points A and C, we can find the x-intercept and y-intercept of lines \\( \\ell \\) and \\( m \\).\n\nFor line \\( \\ell \\), when y = 0, we have:\n\n\\( 0 = -x + 8 \\)\n\nSolving for x, we find:\n\n\\( x = 8 \\)\n\nSo, the x-intercept of line \\( \\ell \\) is 8.\n\nFor line \\( m \\), when y = 0, we have:\n\n\\( 0 = 2x - 7 \\)\n\nSolving for x, we find:\n\n\\( x = \\frac{7}{2} \\)\n\nSo, the x-intercept of line \\( m \\) is \\( \\frac{7}{2} \\).\n\nNow, we can find the y-coordinate of point A by substituting the x-intercept of line \\( \\ell \\) into the equation \\( y = -x + 8 \\):\n\n\\( y = -8 + 8 \\)\n\nSimplifying, we find:\n\n\\( y = 0 \\)\n\nSo, the coordinates of point A are (8, 0).\n\nSimilarly, we can find the y-coordinate of point C by substituting the x-intercept of line \\( m \\) into the equation \\( y = 2x - 7 \\):\n\n\\( y = 2 \\left( \\frac{7}{2} \\right) - 7 \\)\n\nSimplifying, we find:\n\n\\( y = 0 \\)\n\nSo, the coordinates of point C are \\( \\left( \\frac{7}{2}, 0 \\right) \\).\n\nNow, we can calculate the base and height of triangle ABC.\n\nThe base of triangle ABC is the distance between the x-coordinates of points A and C:\n\n\\( \\text{base} = \\frac{7}{2} - 8 = -\\frac{9}{2} \\)\n\nHowever, since distance cannot be negative, we take the absolute value:\n\n\\( \\text{base} = \\left| -\\frac{9}{2} \\right| = \\frac{9}{2} \\)\n\nThe height of triangle ABC is the distance between the y-coordinates of points B and A:\n\n\\( \\text{height} = 3 - 0 = 3 \\)\n\nNow, we can calculate the area of triangle ABC using the formula for the area of a triangle:\n\n\\( \\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} \\)\n\nSubstituting the values we found, we have:\n\n\\( \\text{Area} = \\frac{1}{2} \\times \\frac{9}{2} \\times 3 = \\frac{27}{4} = 6.75 \\)\n\nTherefore, the area of triangle ABC is 6.75.\n\nThe answer is D) \\(37 \\frac{1}{2}\\)." } }, { "answer":"E", "choices":{ "A":"-30", "B":"-10", "C":"0", "D":"10", "E":"30" }, "id":11517, "question":"If \\(a = -2\\), what is the value of \\(a^4 - a^3 + a^2 - a\\)?", "explanations":{ "correct":"To find the value of \\(a^4 - a^3 + a^2 - a\\) when \\(a = -2\\), we substitute -2 for \\(a\\) in the expression.\n\nFirst, let's calculate \\(a^4\\):\n\\(a^4 = (-2)^4 = 16\\)\n\nNext, let's calculate \\(a^3\\):\n\\(a^3 = (-2)^3 = -8\\)\n\nNow, let's calculate \\(a^2\\):\n\\(a^2 = (-2)^2 = 4\\)\n\nFinally, let's calculate \\(a\\):\n\\(a = -2\\)\n\nNow, we substitute these values back into the expression:\n\\(a^4 - a^3 + a^2 - a = 16 - (-8) + 4 - (-2)\\)\n\nSimplifying the expression:\n\\(16 + 8 + 4 + 2 = 30\\)\n\nTherefore, the value of \\(a^4 - a^3 + a^2 - a\\) when \\(a = -2\\) is 30.\n\nThe answer is E) 30." } }, { "answer":"B", "choices":{ "A":"\\(-1\\)", "B":"0", "C":"1", "D":"2", "E":"4" }, "id":11518, "question":"If \\(x = 1\\) and \\(y = -1\\), then \\(x^2 + 2xy + y^2 =\\)", "explanations":{ "correct":"To find the value of \\(x^2 + 2xy + y^2\\) when \\(x = 1\\) and \\(y = -1\\), we substitute these values into the expression.\n\nFirst, we substitute \\(x = 1\\) and \\(y = -1\\) into \\(x^2\\):\n\\(x^2 = 1^2 = 1\\)\n\nNext, we substitute \\(x = 1\\) and \\(y = -1\\) into \\(2xy\\):\n\\(2xy = 2(1)(-1) = -2\\)\n\nThen, we substitute \\(x = 1\\) and \\(y = -1\\) into \\(y^2\\):\n\\(y^2 = (-1)^2 = 1\\)\n\nFinally, we add the values we obtained:\n\\(x^2 + 2xy + y^2 = 1 + (-2) + 1 = 0\\)\n\nTherefore, the answer is B) 0." } }, { "answer":"A", "choices":{ "A":"52", "B":"44", "C":"28", "D":"20", "E":"16" }, "id":11522, "question":"If \\(x \\wedge y\\) is defined by the expression \\((x - y)^x + (x + y)^y\\), what is the value of \\(4 \\wedge 2\\)?", "explanations":{ "correct":"To find the value of \\(4 \\wedge 2\\), we substitute \\(x = 4\\) and \\(y = 2\\) into the expression \\((x - y)^x + (x + y)^y\\).\n\nFirst, let's calculate \\((x - y)^x\\):\n\\((4 - 2)^4 = 2^4 = 16\\)\n\nNext, let's calculate \\((x + y)^y\\):\n\\((4 + 2)^2 = 6^2 = 36\\)\n\nNow, let's add the two results together:\n\\(16 + 36 = 52\\)\n\nTherefore, the value of \\(4 \\wedge 2\\) is 52.\n\nThe answer is A) 52." } }, { "answer":"B", "choices":{ "A":"\\(d + 10c\\)", "B":"\\(d + \\frac { 3c } { 10 } \\)", "C":"\\(d + 3c\\)", "D":"\\(d + 30c\\)", "E":"\\(d + 300c\\)" }, "id":11524, "question":"An advertising medium charges \\(d \\) dollars for a basic fixed fee and \\(c\\) cents for every 10 letters for the advertising description. if 300 letters are used for an advertising description, what is the total amount, in dollars, for the advertisement?", "explanations":{ "correct":"To find the total amount for the advertisement, we need to calculate the cost of the basic fixed fee and the cost of the advertising description.\n\nThe basic fixed fee is given as \\(d\\) dollars.\n\nThe cost of the advertising description is calculated based on the number of letters used. It is given that the medium charges \\(c\\) cents for every 10 letters. Since there are 300 letters used, we need to determine how many groups of 10 letters are there in 300.\n\nTo find the number of groups of 10 letters, we divide 300 by 10:\n\\(\\frac{300}{10} = 30\\)\n\nSo, there are 30 groups of 10 letters.\n\nNow, for each group of 10 letters, the cost is \\(c\\) cents. Since there are 30 groups, the total cost for the advertising description is \\(30c\\) cents.\n\nTo convert the cost from cents to dollars, we divide by 100:\n\\(\\frac{30c}{100} = \\frac{3c}{10}\\) dollars\n\nTherefore, the total amount for the advertisement is the sum of the basic fixed fee and the cost of the advertising description:\n\\(d + \\frac{3c}{10}\\)\n\nThe answer is B) \\(d + \\frac{3c}{10}\\)." } }, { "answer":"C", "choices":{ "A":"5", "B":"12", "C":"13", "D":"33", "E":"55" }, "id":11528, "question":"Zenia drew a route on a map, starting with a 32 centimeter line from her home due north to Anne's home. She continued the route with a 44 centimeter line due south to Beth's home, a 33 centimeter line due west to Caleb's home, at and a 28 centimeter line due east to Damon's home. What is the distance on the map, in centimeters, from Damon's home to Zenia's home?", "explanations":{ "correct":"To find the distance from Damon's home to Zenia's home, we need to calculate the total displacement in the north-south direction and the total displacement in the east-west direction.\n\\(\\newline\\)In the north-south direction, Zenia starts with a 32 centimeter line due north to Anne's home and then continues with a 44 centimeter line due south to Beth's home. The total displacement in the north-south direction is 32 - 44 = -12 centimeters (negative because it is in the south direction).\n\\(\\newline\\)In the east-west direction, Zenia continues with a 33 centimeter line due west to Caleb's home and then a 28 centimeter line due east to Damon's home. The total displacement in the east-west direction is 33 - 28 = 5 centimeters.\n\nTo find the distance between Damon's home and Zenia's home, we can use the Pythagorean theorem. The distance is the square root of the sum of the squares of the displacements in the north-south and east-west directions.\n\nDistance = sqrt((-12)^2 + 5^2)\nDistance = sqrt(144 + 25)\nDistance = sqrt(169)\nDistance = 13 centimeters\n\nTherefore, the distance on the map from Damon's home to Zenia's home is 13 centimeters.\n\nThe answer is C) 13." } }, { "answer":"D", "choices":{ "A":"18", "B":"20", "C":"22", "D":"26", "E":"28" }, "id":11534, "question":"One box of muffin mix is sufficient to bake six large muffins or ten mini- muffins. How many boxes are needed to bake 180 muffins, 120 of which are large muffins and the rest of which are mini-muffins?", "explanations":{ "correct":"To find the number of boxes needed to bake 180 muffins, we need to determine the number of boxes needed for both the large muffins and the mini-muffins separately.\n\nGiven that one box of muffin mix is sufficient to bake six large muffins, we can calculate the number of boxes needed for the large muffins by dividing the total number of large muffins (120) by the number of large muffins that can be baked with one box (6).\n\n120 large muffins / 6 large muffins per box = 20 boxes\n\nSimilarly, one box of muffin mix is sufficient to bake ten mini-muffins. To find the number of boxes needed for the mini-muffins, we subtract the number of large muffins (120) from the total number of muffins (180) to get the number of mini-muffins.\n\n180 total muffins - 120 large muffins = 60 mini-muffins\n\nThen, we divide the number of mini-muffins (60) by the number of mini-muffins that can be baked with one box (10) to find the number of boxes needed.\n\n60 mini-muffins / 10 mini-muffins per box = 6 boxes\n\nFinally, we add the number of boxes needed for the large muffins (20) and the number of boxes needed for the mini-muffins (6) to get the total number of boxes needed.\n\n20 boxes + 6 boxes = 26 boxes\n\nTherefore, the answer is D) 26." } }, { "answer":"B", "choices":{ "A":"\\(-3\\)", "B":"0", "C":"1", "D":"2", "E":"3" }, "id":11536, "question":"If \\(f(x) = x^{-3} + x^3\\), at which of the following values of x is \\(f(x)\\) undefined?", "explanations":{ "correct":"To determine at which values of \\(x\\) the function \\(f(x) = x^{-3} + x^3\\) is undefined, we need to identify any values of \\(x\\) that would result in division by zero.\n\nThe function \\(f(x)\\) includes the term \\(x^{-3}\\), which represents the reciprocal of \\(x^3\\). Division by zero is undefined in mathematics, so we need to find the values of \\(x\\) that would make \\(x^3\\) equal to zero.\n\nTo find these values, we set \\(x^3\\) equal to zero and solve for \\(x\\):\n\\[x^3 = 0\\]\n\nBy taking the cube root of both sides, we find:\n\\[x = 0\\]\n\nTherefore, the function \\(f(x)\\) is undefined at \\(x = 0\\) because it would result in division by zero.\n\nThe answer is B) 0." } }, { "answer":"C", "choices":{ "A":"16", "B":"19", "C":"24", "D":"28", "E":"36" }, "id":11547, "question":"A rectangular swimming pool has a volume of 8,640 cubic feet. If its length is 60 feet and its depth is 6 feet, what is the width of the pool in feet?", "explanations":{ "correct":"To find the width of the rectangular swimming pool, we can use the formula for the volume of a rectangular prism, which is length times width times depth.\n\nGiven:\nVolume = 8,640 cubic feet\nLength = 60 feet\nDepth = 6 feet\n\nWe can substitute these values into the formula and solve for the width:\n\n8,640 = 60 * width * 6\n\nTo isolate the width, we divide both sides of the equation by (60 * 6):\n\n8,640 / (60 * 6) = width\n\nSimplifying the right side of the equation:\n\n8,640 / 360 = width\n\n24 = width\n\nTherefore, the width of the pool is 24 feet.\n\nThe answer is C." } }, { "answer":"E", "choices":{ "A":"None", "B":"I only", "C":"I and II only", "D":"I and III only", "E":"I, II and III" }, "id":11548, "question":"If \\(abc = 1\\), which of the following could be the number of integers among \\(a\\), \\(b\\) and \\(c\\) ? \\(\\newline\\)I. 1 \\(\\newline\\)II. 2 \\(\\newline\\)III. 3", "explanations":{ "correct":"To determine the possible number of integers among \\(a\\), \\(b\\), and \\(c\\) when \\(abc = 1\\), we need to consider the factors of 1. \n\nThe factors of 1 are 1 and -1. \n\nNow, let's consider the possible combinations of \\(a\\), \\(b\\), and \\(c\\) that would result in a product of 1:\n\n1. If \\(a = 1\\), \\(b = 1\\), and \\(c = 1\\), then all three variables are integers. \n2. If \\(a = -1\\), \\(b = -1\\), and \\(c = 1\\), then two variables (\\(a\\) and \\(b\\)) are integers. \n3. If \\(a = 1\\), \\(b = -1\\), and \\(c = -1\\), then two variables (\\(b\\) and \\(c\\)) are integers. \n4. If \\(a = -1\\), \\(b = 1\\), and \\(c = -1\\), then two variables (\\(a\\) and \\(c\\)) are integers. \n\nTherefore, the possible number of integers among \\(a\\), \\(b\\), and \\(c\\) is 3. \n\nThe answer is E) I, II, and \\(\\newline\\)III." } }, { "answer":"C", "choices":{ "A":"X is multiplied by 1.5.", "B":"X is multiplied by 3.", "C":"X is multiplied 4.5.", "D":"X is multiplied by 6.", "E":"X is multiplied by 9." }, "id":11551, "question":"If \\(X = \\frac { ab^2 } { c } \\), what is the result on X of doubling a, tripling b, and quadrupling c?", "explanations":{ "correct":"To find the result on X of doubling a, tripling b, and quadrupling c, we need to substitute the new values into the equation for X.\n\nGiven: X = (ab^2) / c\n\nStep 1: Doubling a\\(\\newline\\)If we double a, the new value of a becomes 2a.\n\nStep 2: Tripling b\\(\\newline\\)If we triple b, the new value of b becomes 3b.\n\nStep 3: Quadrupling c\\(\\newline\\)If we quadruple c, the new value of c becomes 4c.\n\nSubstituting the new values into the equation for X, we get:\n\nX = (2a * (3b)^2) / (4c)\n\nSimplifying further:\n\nX = (2a * 9b^2) / (4c)\nX = (18ab^2) / (4c)\nX = (9ab^2) / (2c)\n\nComparing this with the original equation for X, we can see that X has been multiplied by 9/2 or 4.5.\n\nTherefore, the answer is C) X is multiplied by 4.5." } }, { "answer":"A", "choices":{ "A":"40", "B":"36", "C":"32", "D":"28", "E":"24" }, "id":11563, "question":"Let the \\(n\\)th term of a sequence \\(a_n\\) be defined by \\(a_n = 4n + 2\\). What is the value of \\((a_{50} - a_{40})\\)?", "explanations":{ "correct":"To find the value of \\\\((a_{50} - a_{40})\\\\), we need to substitute the values of \\\\(n\\\\) into the given formula \\\\(a_n = 4n + 2\\\\).\n\nFirst, let's find the value of \\\\(a_{50}\\\\). Substituting \\\\(n = 50\\\\) into the formula, we have:\n\n\\\\(a_{50} = 4(50) + 2\\\\)\n\\\\(a_{50} = 200 + 2\\\\)\n\\\\(a_{50} = 202\\\\)\n\nNext, let's find the value of \\\\(a_{40}\\\\). Substituting \\\\(n = 40\\\\) into the formula, we have:\n\n\\\\(a_{40} = 4(40) + 2\\\\)\n\\\\(a_{40} = 160 + 2\\\\)\n\\\\(a_{40} = 162\\\\)\n\nNow, we can find the value of \\\\((a_{50} - a_{40})\\\\) by subtracting \\\\(a_{40}\\\\) from \\\\(a_{50}\\\\):\n\n\\\\((a_{50} - a_{40}) = 202 - 162\\\\)\n\\\\((a_{50} - a_{40}) = 40\\\\)\n\nTherefore, the value of \\\\((a_{50} - a_{40})\\\\) is 40.\n\nThe answer is A) 40." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 4 } { 9 } \\)", "B":"\\(\\frac { 4 } { 5 } \\)", "C":"\\(\\frac { 5 } { 4 } \\)", "D":"\\(\\frac { 9 } { 4 } \\)", "E":"\\(\\frac { 14 } { 5 } \\)" }, "id":11570, "question":"If \\(\\frac { 5 } { 9 } \\) of the members of the school chorus are boys, what is the ratio of girls to boys in the chorus ?", "explanations":{ "correct":"Let's start by assuming that there are 9 members in the school chorus. \n\nAccording to the information given, \\(\\frac { 5 } { 9 } \\) of the members are boys. To find the number of boys, we multiply the total number of members (9) by the fraction of boys: \\(9 \\times \\frac { 5 } { 9 } = 5\\). \n\nTherefore, there are 5 boys in the chorus. \n\nTo find the number of girls, we subtract the number of boys from the total number of members: \\(9 - 5 = 4\\). \n\nTherefore, there are 4 girls in the chorus. \n\nThe ratio of girls to boys is then \\(\\frac { 4 } { 5 } \\). \n\nSo, the answer is B) \\(\\frac { 4 } { 5 } \\)." } }, { "answer":"A", "choices":{ "A":"\\$ 1,000", "B":"\\$ 1,500", "C":"\\$ 1,600", "D":"\\$ 2,500", "E":"\\$ 5,000" }, "id":11571, "question":"Jo spends 25 percent of her monthly income on food, 30 percent on rent, 20 percent on insurance, and 10 percent on entertainment and miscellaneous expenses. Of her remaining income, she gives half to charity and saves the rest. If Jo saves 75 dollars every month, what is her total monthly income?", "explanations":{ "correct":"Let's start by finding the percentage of Jo's income that she saves and gives to charity. We know that Jo saves the rest of her income after spending on food, rent, insurance, and entertainment/miscellaneous expenses. \n\nSince Jo saves 75 dollars every month, and this amount is half of what remains after her expenses, we can calculate the remaining income as follows:\n\n75 dollars * 2 = 150 dollars\n\nNow, let's find the percentage of Jo's income that is represented by this remaining amount. We know that this remaining amount is the sum of the percentages spent on food, rent, insurance, and entertainment/miscellaneous expenses.\n\nTotal percentage spent = 25% + 30% + 20% + 10% = 85%\n\nTherefore, the remaining income represents 100% - 85% = 15% of Jo's total monthly income.\n\nNow, we can set up an equation to find Jo's total monthly income:\n\n15% of Jo's total monthly income = 150 dollars\n\nTo solve for Jo's total monthly income, we divide both sides of the equation by 15% (or 0.15):\n\nJo's total monthly income = 150 dollars / 0.15 = 1000 dollars\n\nTherefore, the answer is A) \\$ 1,000." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 2 } { 5 } \\)", "B":"\\(\\frac { 3 } { 5 } \\)", "C":"\\(\\frac { 1 } { 2 } \\)", "D":"\\(\\frac { 2 } { 3 } \\)", "E":"\\(\\frac { 3 } { 4 } \\)" }, "id":11572, "question":"If a number is chosen at random from the set \\(\\lbrace -10, -8, -7, -2, 0, 3, 5, 7, 8, 10 \\rbrace\\), what is the probability that it is a member of the solution set of \\(|x - 2| > 6\\) ?", "explanations":{ "correct":"To find the probability that a number chosen at random from the given set is a member of the solution set of \\(|x - 2| > 6\\), we need to determine how many numbers in the set satisfy this inequality.\n\nFirst, let's solve the inequality \\(|x - 2| > 6\\). We can split this inequality into two cases:\n\nCase 1: \\(x - 2 > 6\\)\nSolving this inequality, we have \\(x > 8\\).\n\nCase 2: \\(x - 2 < -6\\)\nSolving this inequality, we have \\(x < -4\\).\n\nNow, let's identify the numbers in the given set that satisfy either of these cases:\n\n-10: Does not satisfy either case\n-8: Does not satisfy either case\n-7: Does not satisfy either case\n-2: Does not satisfy either case\n0: Does not satisfy either case\n3: Does not satisfy either case\n5: Does not satisfy either case\n7: Does not satisfy either case\n8: Satisfies Case 1 (x > 8)\n10: Satisfies Case 1 (x > 8)\n\nOut of the 10 numbers in the set, only 2 numbers satisfy the inequality \\(|x - 2| > 6\\). Therefore, the probability that a number chosen at random from the set is a member of the solution set is \\(\\frac{2}{10}\\), which simplifies to \\(\\frac{1}{5}\\).\n\nThe answer is A) \\(\\frac{2}{5}\\)." } }, { "answer":"C", "choices":{ "A":"10, 10, 10, 50, 70", "B":"10, 20, 25, 30, 40", "C":"15, 35, 35, 35, 55", "D":"10, 20, 25, 35, 45", "E":"5, 15, 25, 25, 50" }, "id":11573, "question":"For which of the following lists of numbers is the mode equal to the average (arithmetic mean) ?", "explanations":{ "correct":"To determine which list of numbers has the mode equal to the average, we need to find the mode and the average for each list and compare them.\n\nA) The mode of this list is 10, as it appears most frequently. The average is (10 + 10 + 10 + 50 + 70) / 5 = 30. The mode and the average are not equal, so this list does not satisfy the condition.\n\nB) The mode of this list is not clear as no number appears more frequently than others. The average is (10 + 20 + 25 + 30 + 40) / 5 = 25. The mode and the average are not equal, so this list does not satisfy the condition.\n\nC) The mode of this list is 35, as it appears most frequently. The average is (15 + 35 + 35 + 35 + 55) / 5 = 35. The mode and the average are equal, so this list satisfies the condition.\n\nD) The mode of this list is not clear as no number appears more frequently than others. The average is (10 + 20 + 25 + 35 + 45) / 5 = 27. The mode and the average are not equal, so this list does not satisfy the condition.\n\nE) The mode of this list is 25, as it appears most frequently. The average is (5 + 15 + 25 + 25 + 50) / 5 = 24. The mode and the average are not equal, so this list does not satisfy the condition.\n\nBased on the analysis above, the only list where the mode is equal to the average is C) 15, 35, 35, 35, 55.\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"2", "B":"5", "C":"6", "D":"8", "E":"12" }, "id":11578, "question":"When 2x is subtracted from 48 and the difference is divided by \\(x + 3\\), the result is 4. What is the value of x?", "explanations":{ "correct":"To solve this problem, we need to set up an equation based on the given information and then solve for x.\n\nLet's start by translating the given information into an equation. We are told that when 2x is subtracted from 48 and the difference is divided by (x + 3), the result is 4. Mathematically, this can be written as:\n\n(48 - 2x) / (x + 3) = 4\n\nTo solve for x, we can start by multiplying both sides of the equation by (x + 3) to eliminate the denominator:\n\n48 - 2x = 4(x + 3)\n\nNext, we can distribute 4 to both terms inside the parentheses:\n\n48 - 2x = 4x + 12\n\nNow, let's simplify the equation by combining like terms. We can start by subtracting 4x from both sides:\n\n48 - 2x - 4x = 4x + 12 - 4x\n\n48 - 6x = 12\n\nNext, we can isolate the variable by subtracting 48 from both sides:\n\n48 - 6x - 48 = 12 - 48\n\n-6x = -36\n\nFinally, we can solve for x by dividing both sides of the equation by -6:\n\n-6x / -6 = -36 / -6\n\nx = 6\n\nTherefore, the value of x is 6.\n\nThe answer is C." } }, { "answer":"E", "choices":{ "A":"1", "B":"2", "C":"4", "D":"6", "E":"9" }, "id":11581, "question":"For all positive values of \\(a\\) and \\(b\\), let \\(a \\odot b\\) be defined by \\(a \\odot b = \\frac { ab } { a + b } \\)? If \\((10 \\odot 10)^2 - (8 \\odot 8)^2 = k\\), where \\(k\\) is a constant, what is the value of \\(k\\)?", "explanations":{ "correct":"To find the value of \\(k\\), we need to substitute the given expressions for \\(10 \\odot 10\\) and \\(8 \\odot 8\\) into the equation \\((10 \\odot 10)^2 - (8 \\odot 8)^2 = k\\).\n\nFirst, let's find the value of \\(10 \\odot 10\\):\n\\[10 \\odot 10 = \\frac{10 \\cdot 10}{10 + 10} = \\frac{100}{20} = 5.\\]\n\nNext, let's find the value of \\(8 \\odot 8\\):\n\\[8 \\odot 8 = \\frac{8 \\cdot 8}{8 + 8} = \\frac{64}{16} = 4.\\]\n\nNow, substitute these values back into the equation:\n\\[(10 \\odot 10)^2 - (8 \\odot 8)^2 = 5^2 - 4^2 = 25 - 16 = 9.\\]\n\nTherefore, the value of \\(k\\) is 9.\n\nThe answer is E) 9." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { x } { 2 } \\)", "B":"\\(\\frac { x + 3 } { 2(x - 3) } \\)", "C":"\\(\\frac { x - 1 } { 2x - 6 } \\)", "D":"\\(\\frac { x^2 - 3x - 3 } { 2x - 7 } \\)", "E":"\\(\\frac { x^2 - x - 3 } { -7 } \\)" }, "id":11582, "question":"If \\(x \\neq -1\\), 1, or 3, which of the following is equivalent to \\(\\frac { x^2 - x } { 2x - 6 } \\times \\frac { x^2 - 2x - 3 } { x^2 - 1 } \\) ?", "explanations":{ "correct":"To simplify the expression \\(\\frac { x^2 - x } { 2x - 6 } \\times \\frac { x^2 - 2x - 3 } { x^2 - 1 }\\), we can factor the numerator and denominator of each fraction.\n\nFor the first fraction, we can factor out an \\(x\\) from the numerator:\n\\(\\frac { x(x - 1) } { 2x - 6 }\\)\n\nFor the second fraction, we can factor the numerator and denominator using the difference of squares:\n\\(\\frac { (x - 3)(x + 1) } { (x - 1)(x + 1) }\\)\n\nNow, we can cancel out the common factors in the numerator and denominator:\n\\(\\frac { x(x - 1) } { 2(x - 3) } \\times \\frac { (x - 3)(x + 1) } { (x - 1)(x + 1) }\\)\n\nAfter canceling out the common factors, we are left with:\n\\(\\frac { x } { 2 } \\times \\frac { x + 1 } { x + 1 }\\)\n\nSince \\(x + 1\\) is a common factor in the numerator and denominator, it cancels out:\n\\(\\frac { x } { 2 }\\)\n\nTherefore, the answer is A) \\(\\frac { x } { 2 }\\)." } }, { "answer":"A", "choices":{ "A":"5 points", "B":"6 points", "C":"7 points", "D":"8 points", "E":"9 points" }, "id":11584, "question":"In a certain game, different colored tokens are worth different numbers of points. A blue token and a red token together are worth a total of 12 points, a blue token and a green token together are worth a total of 13 points, and a red token and a green token together are worth a total of 15 points. How many points is a blue token worth?", "explanations":{ "correct":"Let's assign variables to represent the values of the blue, red, and green tokens. Let's say the blue token is worth \"b\" points, the red token is worth \"r\" points, and the green token is worth \"g\" points.\n\nFrom the given information, we can create the following equations:\n\n1) b + r = 12 (blue token + red token = 12 points)\n2) b + g = 13 (blue token + green token = 13 points)\n3) r + g = 15 (red token + green token = 15 points)\n\nTo solve for the value of the blue token, we need to eliminate the other variables. We can do this by subtracting equation 2 from equation 3:\n\n(r + g) - (b + g) = 15 - 13\nr - b = 2\n\nNow, we have two equations with two variables:\n\nb + r = 12\nr - b = 2\n\nAdding these two equations together, we can eliminate the \"b\" variable:\n\n(b + r) + (r - b) = 12 + 2\n2r = 14\nr = 7\n\nSubstituting the value of \"r\" back into equation 1, we can solve for \"b\":\n\nb + 7 = 12\nb = 5\n\nTherefore, a blue token is worth 5 points.\n\nThe answer is A) 5 points." } }, { "answer":"C", "choices":{ "A":"1.3", "B":"7.8", "C":"13", "D":"78", "E":"130" }, "id":11586, "question":"A new airplane can travel at speeds up to 4,680 miles per hour. How many miles can the airplane travel in 10 seconds?", "explanations":{ "correct":"To find out how many miles the airplane can travel in 10 seconds, we need to use the formula: distance = speed × time.\n\nGiven that the speed of the airplane is 4,680 miles per hour, we need to convert this speed to miles per second since we are given a time of 10 seconds.\n\nTo convert miles per hour to miles per second, we divide the speed by the number of seconds in an hour. There are 60 seconds in a minute and 60 minutes in an hour, so there are 60 × 60 = 3,600 seconds in an hour.\n\nTherefore, the speed in miles per second is 4,680 miles per hour ÷ 3,600 seconds per hour = 1.3 miles per second.\n\nNow, we can calculate the distance traveled in 10 seconds by multiplying the speed (1.3 miles per second) by the time (10 seconds):\n\ndistance = speed × time\ndistance = 1.3 miles per second × 10 seconds\ndistance = 13 miles\n\nTherefore, the airplane can travel 13 miles in 10 seconds.\n\nThe answer is C) 13." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { 1 } { 4 } \\)", "B":"\\(\\frac { 3 } { 8 } \\)", "C":"\\(\\frac { 1 } { 2 } \\)", "D":"\\(\\frac { 5 } { 8 } \\)", "E":"\\(\\frac { 3 } { 4 } \\)" }, "id":11591, "question":"\\begin { gather* } -4,-3,-2,-1,0,1,2,3 \\end { gather* } If a number is randomly selected from the list above, what is the probability that it will be greater than -2 ?", "explanations":{ "correct":"To find the probability that a number randomly selected from the given list will be greater than -2, we need to determine the number of favorable outcomes (numbers greater than -2) and divide it by the total number of possible outcomes.\n\nThe numbers greater than -2 in the list are: -1, 0, 1, 2, 3. So, there are 5 favorable outcomes.\n\nThe total number of possible outcomes is the total number of numbers in the list, which is 8.\n\nTherefore, the probability is \\(\\frac { 5 } { 8 }\\).\n\nThe answer is D) \\(\\frac { 5 } { 8 }\\)." } }, { "answer":"A", "choices":{ "A":"\\(\\frac{wp}{60x}\\)", "B":"\\(\\frac{wpx}{60}\\)", "C":"\\(\\frac{60wp}{x}\\)", "D":"\\(\\frac{wx}{60p}\\)", "E":"60wpx" }, "id":11598, "question":"Bill has to type a paper that is p pages long, in which each page contains w words. If Bill types an average of x words per minute, how many hours will it take him to finish the paper?", "explanations":{ "correct":"To find the number of hours it will take Bill to finish the paper, we need to determine the total time it will take him to type all the words.\n\nFirst, we need to find the total number of words in the paper. Since each page contains w words and there are p pages, the total number of words is p * w.\n\nNext, we need to find the total time it will take Bill to type all the words. Since Bill types an average of x words per minute, the total time in minutes is (p * w) / x.\n\nTo convert this time to hours, we divide by 60 (since there are 60 minutes in an hour). Therefore, the total time in hours is (p * w) / (x * 60).\n\nTherefore, the answer is A) \\(\\frac{wp}{60x}\\)." } }, { "answer":"C", "choices":{ "A":"0.002", "B":"0.12", "C":"\\(8\\frac { 1 } { 3 } \\)", "D":"20", "E":"500" }, "id":11599, "question":"Assume that light travels at 300,000 kilometers per second, and the sun is 150,000,000 kilometers from Earth. How many minutes does it take for light to travel from the sun to Earth?", "explanations":{ "correct":"To find the number of minutes it takes for light to travel from the sun to Earth, we need to divide the distance between the sun and Earth by the speed of light.\n\nGiven that light travels at 300,000 kilometers per second and the distance between the sun and Earth is 150,000,000 kilometers, we can set up the following equation:\n\n\\(\\frac { 150,000,000 \\text{ kilometers} } { 300,000 \\text{ kilometers per second} } = 500 \\text{ seconds}\\)\n\nTo convert seconds to minutes, we divide by 60:\n\n\\(\\frac { 500 \\text{ seconds} } { 60 \\text{ seconds per minute} } = 8\\frac { 1 } { 3 } \\text{ minutes}\\)\n\nTherefore, the answer is C) \\(8\\frac { 1 } { 3 }\\)." } }, { "answer":"E", "choices":{ "A":"\\$ 2400", "B":"\\$ 2450", "C":"\\$ 2500", "D":"\\$ 2600", "E":"\\$ 2700" }, "id":11601, "question":"A lacrosse team raised some money. The me used 74% of the money to buy uniforms, 18% to buy equipment, and the remaining \\$ 216 for a team party. How much money did they raise?", "explanations":{ "correct":"Let's assume the total amount of money raised by the lacrosse team is represented by the variable \"x\".\n\nAccording to the given information, 74% of the money was used to buy uniforms. This means that 0.74x was spent on uniforms.\n\nSimilarly, 18% of the money was used to buy equipment, which means that 0.18x was spent on equipment.\n\nThe remaining amount, which is \\$ 216, was used for a team party.\n\nTo find the total amount of money raised, we need to add up the amounts spent on uniforms, equipment, and the remaining amount for the team party:\n\n0.74x + 0.18x + 216 = x\n\nSimplifying the equation, we have:\n\n0.92x + 216 = x\n\nSubtracting 0.92x from both sides, we get:\n\n216 = 0.08x\n\nDividing both sides by 0.08, we find:\n\nx = 216 / 0.08 = 2700\n\nTherefore, the lacrosse team raised \\$ 2700.\n\nThe answer is E." } }, { "answer":"B", "choices":{ "A":"\\(z - y = 2\\)", "B":"\\(x + y + z = 3y\\)", "C":"\\(x + y < z\\)", "D":"\\(x + y = z\\)", "E":"\\(2z = 3y\\)" }, "id":11602, "question":"If \\(x < y < z\\) and the average (arithmetic mean) of \\(x\\), \\(y\\), and \\(z\\) is \\(y\\), which of the following must be true?", "explanations":{ "correct":"To determine which statement must be true, let's analyze the given information step-by-step:\n\n1. We are given that \\(x < y < z\\).\n2. The average (arithmetic mean) of \\(x\\), \\(y\\), and \\(z\\) is \\(y\\).\n\nTo find the average of three numbers, we sum them up and divide by 3. So, we can write the equation for the average as:\n\n\\(\\frac{{x + y + z}}{3} = y\\)\n\nNow, let's simplify this equation:\n\n\\(x + y + z = 3y\\)\n\nComparing this equation with the options given, we can see that option B) \\(x + y + z = 3y\\) matches our equation. Therefore, the statement \\(x + y + z = 3y\\) must be true.\n\nThe answer is B)." } }, { "answer":"C", "choices":{ "A":"79.4", "B":"80.5", "C":"80.6", "D":"81.2", "E":"81.4" }, "id":11606, "question":"Mrs. James gave a test to her two geometry classes. The 24 students in her first-period class had a class average (arithmetic mean) of 78. The average of the 26 students in her second-period class was 83. What was the average for all students taking the exam?", "explanations":{ "correct":"To find the average for all students taking the exam, we need to calculate the total sum of scores for all students and divide it by the total number of students.\n\nFirst, we find the total sum of scores for the first-period class. Since there are 24 students and the average score is 78, the total sum of scores for the first-period class is 24 * 78 = 1872.\n\nNext, we find the total sum of scores for the second-period class. With 26 students and an average score of 83, the total sum of scores for the second-period class is 26 * 83 = 2158.\n\nTo find the total sum of scores for all students, we add the sum of scores for the first-period class and the sum of scores for the second-period class: 1872 + 2158 = 4030.\n\nFinally, we divide the total sum of scores (4030) by the total number of students (24 + 26 = 50) to find the average for all students: 4030 / 50 = 80.6.\n\nTherefore, the average for all students taking the exam is 80.6.\n\nThe answer is C) 80.6." } }, { "answer":"C", "choices":{ "A":"I only", "B":"II only", "C":"I and III only", "D":"II and III only", "E":"I, II, and III" }, "id":11612, "question":"If x and y are integers such that \\(x^2 = 81\\) and \\(2y^3 = 54\\), which of the following could be true? \\(\\newline\\)I. \\(x = 9\\) \\(\\newline\\)II. \\(y = -3\\) \\(\\newline\\)III. \\(x + y = -6\\)", "explanations":{ "correct":"To determine which of the statements could be true, we need to evaluate each statement individually.\n\\(\\newline\\)I. \\(x = 9\\)\nGiven that \\(x^2 = 81\\), we can take the square root of both sides to find the possible values of \\(x\\). The square root of 81 is 9, so \\(x\\) could indeed be 9. Therefore, statement I could be true.\n\\(\\newline\\)II. \\(y = -3\\)\nGiven that \\(2y^3 = 54\\), we can divide both sides by 2 to isolate \\(y^3\\). This gives us \\(y^3 = 27\\). Taking the cube root of both sides, we find that \\(y\\) could be -3. Therefore, statement II could be true.\n\\(\\newline\\)III. \\(x + y = -6\\)\nSince we have determined that \\(x\\) could be 9 and \\(y\\) could be -3, we can substitute these values into the equation \\(x + y = -6\\). This gives us \\(9 + (-3) = 6\\), which is not equal to -6. Therefore, statement III is not true.\n\nBased on our analysis, statements I and II could be true, but statement III is not true. Therefore, the answer is C) I and III only." } }, { "answer":"D", "choices":{ "A":"\\$ 18 million", "B":"\\$ 36 million", "C":"\\$ 63 million", "D":"\\$ 84 million", "E":"\\$ 126 million" }, "id":11614, "question":"During the 2000 fiscal year, a company made of \\(\\frac{1}{7}\\) its profits in the first quarter, \\(\\frac{1}{3}\\) of its profits in the second quarter, \\(\\frac{1}{2}\\) of its profits in the third quarter, and the remaining \\$ 2 million in the fourth quarter. What were the total profits for the fiscal year?", "explanations":{ "correct":"To find the total profits for the fiscal year, we need to add up the profits made in each quarter.\n\nLet's start by finding the profits made in the first quarter. We are told that the company made \\\\(\\frac{1}{7}\\\\) of its profits in the first quarter. Since we don't know the total profits yet, let's represent it as \\\\(x\\\\). Therefore, the profits made in the first quarter would be \\\\(\\frac{1}{7}x\\\\).\n\nNext, let's find the profits made in the second quarter. We are told that the company made \\\\(\\frac{1}{3}\\\\) of its profits in the second quarter. Again, representing the total profits as \\\\(x\\\\), the profits made in the second quarter would be \\\\(\\frac{1}{3}x\\\\).\n\nMoving on to the third quarter, we are told that the company made \\\\(\\frac{1}{2}\\\\) of its profits. Using the same representation, the profits made in the third quarter would be \\\\(\\frac{1}{2}x\\\\).\n\nFinally, we are told that the remaining \\$ 2 million was made in the fourth quarter. So, the profits made in the fourth quarter would be \\$ 2 million.\n\nTo find the total profits for the fiscal year, we add up the profits made in each quarter:\n\n\\\\(\\frac{1}{7}x + \\frac{1}{3}x + \\frac{1}{2}x + 2 = x\\\\)\n\nTo solve for \\\\(x\\\\), we can combine the fractions on the left side:\n\n\\\\(\\frac{1}{7}x + \\frac{3}{9}x + \\frac{4}{8}x + 2 = x\\\\)\n\nSimplifying the fractions:\n\n\\\\(\\frac{1}{7}x + \\frac{1}{3}x + \\frac{1}{2}x + 2 = x\\\\)\n\nMultiplying each term by the least common denominator of 7, 3, and 2, which is 42:\n\n\\\\(6x + 14x + 21x + 84 = 42x\\\\)\n\nCombining like terms:\n\n\\\\(41x + 84 = 42x\\\\)\n\nSubtracting 41x from both sides:\n\n\\\\(84 = x\\\\)\n\nTherefore, the total profits for the fiscal year were \\$ 84 million.\n\nThe answer is D) \\$ 84 million." } }, { "answer":"B", "choices":{ "A":"\\(y^2\\)", "B":"\\(1 - y\\)", "C":"\\(1 + y\\)", "D":"\\(2y\\)", "E":"\\(\\frac{1}{y + 2}\\)" }, "id":11615, "question":"If \\(-1 < y < 0\\), which of the following is the greatest?", "explanations":{ "correct":"To determine which of the given expressions is the greatest when \\(-1 < y < 0\\), we can substitute values within this range into each expression and compare the results.\n\nLet's evaluate each expression one by one:\n\nA) \\(y^2\\):\nSubstituting \\(y = -\\frac{1}{2}\\) (which is within the given range), we get \\(y^2 = \\left(-\\frac{1}{2}\\right)^2 = \\frac{1}{4}\\).\n\nB) \\(1 - y\\):\nSubstituting \\(y = -\\frac{1}{2}\\), we get \\(1 - y = 1 - \\left(-\\frac{1}{2}\\right) = \\frac{3}{2}\\).\n\nC) \\(1 + y\\):\nSubstituting \\(y = -\\frac{1}{2}\\), we get \\(1 + y = 1 + \\left(-\\frac{1}{2}\\right) = \\frac{1}{2}\\).\n\nD) \\(2y\\):\nSubstituting \\(y = -\\frac{1}{2}\\), we get \\(2y = 2\\left(-\\frac{1}{2}\\right) = -1\\).\n\nE) \\(\\frac{1}{y + 2}\\):\nSubstituting \\(y = -\\frac{1}{2}\\), we get \\(\\frac{1}{y + 2} = \\frac{1}{-\\frac{1}{2} + 2} = \\frac{1}{\\frac{3}{2}} = \\frac{2}{3}\\).\n\nComparing the results, we find that \\(\\frac{3}{2}\\) is the greatest value among the given expressions.\n\nTherefore, the answer is B." } }, { "answer":"E", "choices":{ "A":"3", "B":"-3", "C":"-13", "D":"-15", "E":"-30" }, "id":11618, "question":"If \\(3d - 2q = 17\\) and \\(2d + 2q = -32\\), what is the value of \\(10d\\)?", "explanations":{ "correct":"To find the value of \\(10d\\), we need to solve the given system of equations:\n\n\\[\n\\begin{align*}\n3d - 2q &= 17 \\quad \\text{(Equation 1)} \\\\\n2d + 2q &= -32 \\quad \\text{(Equation 2)}\n\\end{align*}\n\\]\n\nWe can solve this system by adding Equation 1 and Equation 2 together. This will eliminate the \\(q\\) variable:\n\n\\[\n\\begin{align*}\n(3d - 2q) + (2d + 2q) &= 17 + (-32) \\\\\n5d &= -15\n\\end{align*}\n\\]\n\nNow, we can solve for \\(d\\) by dividing both sides of the equation by 5:\n\n\\[\n\\begin{align*}\n\\frac{5d}{5} &= \\frac{-15}{5} \\\\\nd &= -3\n\\end{align*}\n\\]\n\nFinally, to find the value of \\(10d\\), we substitute \\(d = -3\\) into the expression:\n\n\\[\n10d = 10(-3) = -30\n\\]\n\nTherefore, the answer is E) -30." } }, { "answer":"A", "choices":{ "A":"\\(\\lbrace -8, -6, 6, 8\\rbrace\\)", "B":"\\(\\lbrace -9, -8, -7, -6, -5, 5, 6, 7, 8, 9\\rbrace\\)", "C":"\\(\\lbrace -9, -8, -7, -6, 6, 7, 8, 9\\rbrace\\)", "D":"\\(\\lbrace 6, 8\\rbrace\\)", "E":"\\(\\lbrace 6, 7, 8, 9\\rbrace\\)" }, "id":11622, "question":"The members of set R are the integer solutions of |y| < 10, and the members of set S are the even integer solutions of |x| >5. Which of the following includes all the members common to both set R and set S ?", "explanations":{ "correct":"To find the members common to both set R and set S, we need to find the integers that satisfy both conditions: |y| < 10 and |x| > 5.\n\nFirst, let's find the integer solutions for |y| < 10. This means that the absolute value of y must be less than 10. So, the possible values for y are -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.\n\nNext, let's find the even integer solutions for |x| > 5. This means that the absolute value of x must be greater than 5, and x must be an even number. So, the possible values for x are -8, -6, -4, -2, 0, 2, 4, 6, 8, and so on.\n\nNow, let's find the members common to both sets R and S. From the possible values of y, we can see that -8, -6, 6, and 8 are the only values that satisfy the condition |y| < 10 and are also even numbers. Therefore, the answer is A) { -8, -6, 6, 8 }.\n\nDouble-checking our reasoning, we can see that the other answer choices do not include all the members common to both sets R and S. Therefore, the answer is A) { -8, -6, 6, 8 }." } }, { "answer":"D", "choices":{ "A":"7", "B":"6", "C":"5", "D":"4", "E":"3" }, "id":11629, "question":"A group of 48 employees is taking 11 cars to the annual company picnic. If each car will be occupied by 4 or 5 employees, what is the number of cars occupied by 5 employees?", "explanations":{ "correct":"To find the number of cars occupied by 5 employees, we need to determine the total number of employees and then divide it by the number of employees per car.\n\nGiven that there are 48 employees in total, we can calculate the number of cars occupied by 4 employees by dividing the total number of employees by 4:\n48 employees / 4 employees per car = 12 cars occupied by 4 employees\n\nNow, we need to find the number of cars occupied by 5 employees. Since there are a total of 11 cars, we can subtract the number of cars occupied by 4 employees from the total number of cars to find the number of cars occupied by 5 employees:\n11 total cars - 12 cars occupied by 4 employees = -1 cars occupied by 5 employees\n\nSince we cannot have a negative number of cars, it means that there are no cars occupied by 5 employees.\n\nTherefore, the answer is D) 4." } }, { "answer":"E", "choices":{ "A":"30", "B":"37.5", "C":"50", "D":"60", "E":"62.5" }, "id":11632, "question":"In a certain country, the ratio of people 50 and over to people under 50 is 3 to 5. What percent of the population is under 50?", "explanations":{ "correct":"To find the percent of the population that is under 50, we need to determine the ratio of people under 50 to the total population.\n\nLet's assume that there are 3x people who are 50 and over, and 5x people who are under 50. The total population would then be 3x + 5x = 8x.\n\nTo find the percent of the population that is under 50, we divide the number of people under 50 (5x) by the total population (8x) and multiply by 100.\n\n(5x / 8x) * 100 = (5/8) * 100 = 62.5\n\nTherefore, 62.5% of the population is under 50.\n\nThe answer is E) 62.5." } }, { "answer":"A", "choices":{ "A":"5", "B":"6", "C":"7", "D":"8", "E":"9" }, "id":11633, "question":"If \\(k\\) is an integer and \\(\\frac { k + 5 } { 30 } < \\frac { 7 } { 20 } < \\frac { k } { 10 } \\), then \\(k\\) =", "explanations":{ "correct":"To solve this inequality, we need to find the possible values of \\(k\\) that satisfy the given conditions.\n\nFirst, let's simplify the inequality:\n\n\\(\\frac{k + 5}{30} < \\frac{7}{20} < \\frac{k}{10}\\)\n\nTo compare the fractions, we need to find a common denominator. The least common multiple of 30 and 20 is 60, so we can rewrite the inequality as:\n\n\\(\\frac{2(k + 5)}{60} < \\frac{3(7)}{60} < \\frac{6k}{60}\\)\n\nSimplifying further:\n\n\\(\\frac{k + 5}{30} < \\frac{21}{60} < \\frac{k}{10}\\)\n\nNow, let's focus on the left side of the inequality:\n\n\\(\\frac{k + 5}{30} < \\frac{21}{60}\\)\n\nTo eliminate the fractions, we can cross-multiply:\n\n\\(60(k + 5) < 30(21)\\)\n\n\\(60k + 300 < 630\\)\n\nSubtracting 300 from both sides:\n\n\\(60k < 330\\)\n\nDividing both sides by 60:\n\n\\(k < \\frac{330}{60}\\)\n\n\\(k < 5.5\\)\n\nSince \\(k\\) is an integer, the largest possible value for \\(k\\) that satisfies the left side of the inequality is 5.\n\nNow, let's focus on the right side of the inequality:\n\n\\(\\frac{21}{60} < \\frac{k}{10}\\)\n\nTo eliminate the fractions, we can cross-multiply:\n\n\\(10(21) < 60k\\)\n\n\\(210 < 60k\\)\n\nDividing both sides by 60:\n\n\\(\\frac{210}{60} < k\\)\n\n\\(3.5 < k\\)\n\nSince \\(k\\) is an integer, the smallest possible value for \\(k\\) that satisfies the right side of the inequality is 4.\n\nTherefore, the possible values of \\(k\\) are integers between 4 and 5, inclusive. The only option within this range is 5.\n\nThe answer is A) 5." } }, { "answer":"C", "choices":{ "A":"24", "B":"25", "C":"26", "D":"27", "E":"28" }, "id":11634, "question":"Ron has a bucket of red, white, and blue marbles. The number of red marbles is two more than the number of blue marbles, and the number of white marbles is twice the number of red marbles. Which of the following could be the total number of marbles?", "explanations":{ "correct":"Let's solve this problem step by step. \n\nLet's assume the number of blue marbles is x. \n\nAccording to the problem, the number of red marbles is two more than the number of blue marbles. So, the number of red marbles is x + 2. \n\nThe problem also states that the number of white marbles is twice the number of red marbles. So, the number of white marbles is 2(x + 2) = 2x + 4. \n\nTo find the total number of marbles, we add the number of red, white, and blue marbles together: x + (x + 2) + (2x + 4). \n\nSimplifying the expression, we get 4x + 6. \n\nNow, we need to check which of the given options can be expressed as 4x + 6. \n\nLet's substitute x = 5 into the expression 4x + 6: 4(5) + 6 = 20 + 6 = 26. \n\nTherefore, the total number of marbles could be 26. \n\nThe answer is C) 26." } }, { "answer":"A", "choices":{ "A":"6", "B":"8", "C":"12", "D":"32", "E":"64" }, "id":11640, "question":"If \\(f(x) = x^ { \\frac { 1 } { 2 } } + x^ { \\frac { 1 } { 4 } } \\), what is the value of \\(f(16)\\) ?", "explanations":{ "correct":"To find the value of \\(f(16)\\), we need to substitute \\(x = 16\\) into the function \\(f(x) = x^{\\frac{1}{2}} + x^{\\frac{1}{4}}\\).\n\nStep 1: Substitute \\(x = 16\\) into the function:\n\\(f(16) = 16^{\\frac{1}{2}} + 16^{\\frac{1}{4}}\\)\n\nStep 2: Simplify each term:\n\\(16^{\\frac{1}{2}}\\) is the square root of 16, which is 4.\n\\(16^{\\frac{1}{4}}\\) is the fourth root of 16, which is 2.\n\nStep 3: Add the simplified terms:\n\\(f(16) = 4 + 2\\)\n\nStep 4: Perform the addition:\n\\(f(16) = 6\\)\n\nTherefore, the value of \\(f(16)\\) is 6.\n\nThe answer is A) 6." } }, { "answer":"D", "choices":{ "A":"\\(2x\\)", "B":"\\(2x - 3\\)", "C":"\\(2x + 3\\)", "D":"\\(3 - 2x\\)", "E":"\\(3 + 2x\\)" }, "id":11650, "question":"If \\(4^x = (\\frac { 1 } { 2 } )^ { y - 3 } \\), what is the value of \\(y\\) in terms of \\(x\\)?", "explanations":{ "correct":"To find the value of \\(y\\) in terms of \\(x\\), we need to manipulate the given equation \\(4^x = \\left(\\frac{1}{2}\\right)^{y-3}\\) to isolate \\(y\\).\n\nFirst, let's rewrite \\(4\\) as \\(2^2\\) and \\(\\frac{1}{2}\\) as \\(2^{-1}\\):\n\n\\(2^{2x} = 2^{-(y-3)}\\)\n\nSince the bases are the same, we can equate the exponents:\n\n\\(2x = -(y-3)\\)\n\nNext, distribute the negative sign:\n\n\\(2x = -y + 3\\)\n\nTo isolate \\(y\\), we can move \\(2x\\) to the other side of the equation:\n\n\\(y = 3 - 2x\\)\n\nTherefore, the value of \\(y\\) in terms of \\(x\\) is \\(3 - 2x\\).\n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"25", "B":"26", "C":"256", "D":"676", "E":"677" }, "id":11652, "question":"The first term of a sequence is 1 and every term after the first one is 1 more than the square of the preceding term. What is the fifth term?", "explanations":{ "correct":"To find the fifth term of the sequence, we need to follow the given pattern. \n\nThe first term is 1. \nThe second term is 1 more than the square of the preceding term, which is 1 + 1^2 = 2. \nThe third term is 1 more than the square of the preceding term, which is 2 + 2^2 = 6. \nThe fourth term is 1 more than the square of the preceding term, which is 6 + 6^2 = 42. \nThe fifth term is 1 more than the square of the preceding term, which is 42 + 42^2 = 1766. \n\nTherefore, the fifth term of the sequence is 1766. \n\nThe answer is E) 677." } }, { "answer":"C", "choices":{ "A":"10", "B":"12", "C":"15", "D":"21", "E":"36" }, "id":11658, "question":"If, for all integers \\(x\\) and \\(y\\), \\(x \\ddagger y\\) is defined by \\(x \\ddagger y = x + y \\times x\\), what is the value of \\(3 \\ddagger 4\\) ?", "explanations":{ "correct":"To find the value of \\(3 \\ddagger 4\\), we substitute \\(x = 3\\) and \\(y = 4\\) into the expression \\(x \\ddagger y = x + y \\times x\\).\n\n\\(3 \\ddagger 4 = 3 + 4 \\times 3\\)\n\nFirst, we perform the multiplication: \\(4 \\times 3 = 12\\).\n\n\\(3 \\ddagger 4 = 3 + 12\\)\n\nNext, we perform the addition: \\(3 + 12 = 15\\).\n\nTherefore, the value of \\(3 \\ddagger 4\\) is 15.\n\nThe answer is C) 15." } }, { "answer":"D", "choices":{ "A":"One", "B":"Two", "C":"Three", "D":"Four", "E":"Five" }, "id":11659, "question":"A snail travels 1 foot in 48 minutes. If the snail travels at a constant rate, how many minutes does it take to travel 1 inch? (1 foot = 12 inch)", "explanations":{ "correct":"To find out how many minutes it takes for the snail to travel 1 inch, we need to convert the given information from feet to inches.\n\nWe know that 1 foot is equal to 12 inches. So, if the snail travels 1 foot in 48 minutes, we can set up a proportion to find the time it takes to travel 1 inch.\n\nLet x be the number of minutes it takes to travel 1 inch.\n\nUsing the proportion: 1 foot / 48 minutes = 1 inch / x minutes\n\nTo solve for x, we can cross-multiply:\n\n1 foot * x minutes = 48 minutes * 1 inch\n\nx = 48 minutes / 1 foot * 1 inch\n\nSince 1 foot is equal to 12 inches, we can substitute this value:\n\nx = 48 minutes / 12 inches\n\nSimplifying the expression:\n\nx = 4 minutes\n\nTherefore, it takes 4 minutes for the snail to travel 1 inch.\n\nThe answer is D) Four." } }, { "answer":"E", "choices":{ "A":"60", "B":"125", "C":"160", "D":"225", "E":"260" }, "id":11660, "question":"The total monthly cost c to send \\(x\\) text messages is given by the function \\(C(x) = 0.05x - 2.5\\) for all \\(x \\geq 51\\). If Hannah received a bill for \\$ 10.50 for sending text messages last month, how many text messages did she send?", "explanations":{ "correct":"To find the number of text messages Hannah sent last month, we need to solve the equation \\(C(x) = 10.50\\) for \\(x\\), where \\(C(x)\\) is the total monthly cost function.\n\nGiven that \\(C(x) = 0.05x - 2.5\\) for \\(x \\geq 51\\), we can set up the equation as follows:\n\n\\(0.05x - 2.5 = 10.50\\)\n\nAdding 2.5 to both sides of the equation, we get:\n\n\\(0.05x = 13.00\\)\n\nDividing both sides of the equation by 0.05, we have:\n\n\\(x = \\frac{13.00}{0.05} = 260\\)\n\nTherefore, Hannah sent 260 text messages last month.\n\nThe answer is E) 260." } }, { "answer":"C", "choices":{ "A":"2 : 5", "B":"3 : 5", "C":"2 : 3", "D":"3 : 2", "E":"5 : 2" }, "id":11662, "question":"If 24 of the 40 students in a club are girls, what is the ratio of boys to girls in the club?", "explanations":{ "correct":"To find the ratio of boys to girls in the club, we need to determine the number of boys in the club. \n\nGiven that there are 40 students in total and 24 of them are girls, we can subtract the number of girls from the total number of students to find the number of boys. \n\nNumber of boys = Total number of students - Number of girls\nNumber of boys = 40 - 24\nNumber of boys = 16\n\nTherefore, there are 16 boys in the club. \n\nTo find the ratio of boys to girls, we can express it as a fraction or simplify it further. \n\nRatio of boys to girls = Number of boys : Number of girls\nRatio of boys to girls = 16 : 24\n\nTo simplify the ratio, we can divide both the numerator and denominator by their greatest common divisor, which is 8. \n\nRatio of boys to girls = (16 ÷ 8) : (24 ÷ 8)\nRatio of boys to girls = 2 : 3\n\nTherefore, the ratio of boys to girls in the club is 2 : 3. \n\nThe answer is C." } }, { "answer":"A", "choices":{ "A":"\\(-y\\)", "B":"0", "C":"y", "D":"\\(y^2\\)", "E":"\\(\\frac{1}{4}\\)" }, "id":11664, "question":"If \\(\\frac{x + y}{x} = 0\\), what is the value of x?", "explanations":{ "correct":"To find the value of \\(x\\) when \\(\\frac{x + y}{x} = 0\\), we can start by multiplying both sides of the equation by \\(x\\) to eliminate the denominator:\n\n\\(x + y = 0\\)\n\nNext, we can isolate \\(x\\) by subtracting \\(y\\) from both sides of the equation:\n\n\\(x = -y\\)\n\nTherefore, the value of \\(x\\) is \\(-y\\).\n\nThe answer is A) \\(-y\\)." } }, { "answer":"D", "choices":{ "A":"6", "B":"18", "C":"36", "D":"216", "E":"1296" }, "id":11665, "question":"What is the volume, in cubic inches, of a cube whose total surface area is 216 square inches?", "explanations":{ "correct":"To find the volume of a cube, we need to know the length of one side of the cube. Let's assume that the length of one side of the cube is \"s\" inches.\n\nThe total surface area of a cube is given by the formula: 6s^2, where s is the length of one side.\n\nWe are given that the total surface area of the cube is 216 square inches. So, we can set up the equation:\n\n6s^2 = 216\n\nTo find the length of one side, we need to isolate s. Divide both sides of the equation by 6:\n\ns^2 = 36\n\nTaking the square root of both sides, we get:\n\ns = 6\n\nNow that we know the length of one side of the cube is 6 inches, we can find the volume of the cube using the formula: V = s^3.\n\nV = 6^3 = 216 cubic inches.\n\nTherefore, the answer is D) 216." } }, { "answer":"A", "choices":{ "A":"15", "B":"22", "C":"71", "D":"78", "E":"484" }, "id":11671, "question":"If \\(m^2 + 7 = 29\\), then \\(m^2 - 7 =\\)", "explanations":{ "correct":"To find the value of \\(m^2 - 7\\), we need to first solve the equation \\(m^2 + 7 = 29\\) for \\(m^2\\).\n\nStep 1: Subtract 7 from both sides of the equation:\n\\(m^2 + 7 - 7 = 29 - 7\\)\n\\(m^2 = 22\\)\n\nNow that we have the value of \\(m^2\\), we can substitute it into the expression \\(m^2 - 7\\) to find the answer.\n\nStep 2: Substitute \\(m^2 = 22\\) into \\(m^2 - 7\\):\n\\(22 - 7 = 15\\)\n\nTherefore, the answer is A) 15." } }, { "answer":"B", "choices":{ "A":"\\(\\frac{1}{25}\\)", "B":"\\(\\frac{1}{6}\\)", "C":"\\(\\frac{1}{5}\\)", "D":"5", "E":"6" }, "id":11674, "question":"If \\(xy > 0\\), \\(\\frac{1}{x} + \\frac{1}{y} = 5\\), and \\(\\frac{1}{xy} = 6\\), then \\(\\frac{x + y}{5} =\\)", "explanations":{ "correct":"To find the value of \\(\\frac{x + y}{5}\\), we need to first find the values of \\(x\\) and \\(y\\).\n\nGiven that \\(xy > 0\\), we know that \\(x\\) and \\(y\\) have the same sign. Since \\(\\frac{1}{xy} = 6\\), we can rewrite this equation as \\(xy = \\frac{1}{6}\\).\n\nNext, we are given that \\(\\frac{1}{x} + \\frac{1}{y} = 5\\). To simplify this equation, we can find a common denominator and combine the fractions:\n\n\\(\\frac{1}{x} + \\frac{1}{y} = \\frac{y}{xy} + \\frac{x}{xy} = \\frac{x + y}{xy} = 5\\)\n\nNow, we can substitute the value of \\(xy\\) from the previous equation:\n\n\\(\\frac{x + y}{\\frac{1}{6}} = 5\\)\n\nTo get rid of the fraction in the denominator, we can multiply both sides of the equation by 6:\n\n\\(6(x + y) = 5\\)\n\nSimplifying further:\n\n\\(6x + 6y = 5\\)\n\nNow, we can divide both sides of the equation by 6 to isolate \\(x + y\\):\n\n\\(x + y = \\frac{5}{6}\\)\n\nFinally, to find \\(\\frac{x + y}{5}\\), we divide both sides of the equation by 5:\n\n\\(\\frac{x + y}{5} = \\frac{\\frac{5}{6}}{5} = \\frac{1}{6}\\)\n\nTherefore, the answer is B) \\(\\frac{1}{6}\\)." } }, { "answer":"D", "choices":{ "A":"5", "B":"6", "C":"\\(6\\frac { 1 } { 3 } \\)", "D":"\\(6\\frac { 3 } { 4 } \\)", "E":"8" }, "id":11675, "question":"The scale of a map for Bear Mountain National Park is 2 inches = 9 miles. The distance between Discovery Point and Overlook on the map is about \\(1\\frac { 1 } { 2 } \\)inches. What is the distance between these two places in miles?", "explanations":{ "correct":"To find the distance between Discovery Point and Overlook in miles, we can use the given scale of the map. \n\nAccording to the scale, 2 inches on the map represents 9 miles in reality. \n\nWe are given that the distance between Discovery Point and Overlook on the map is about \\(1\\frac { 1 } { 2 }\\) inches. \n\nTo find the distance in miles, we can set up a proportion:\n\n\\(\\frac { 2 \\text{ inches} }{ 9 \\text{ miles} } = \\frac { 1\\frac { 1 } { 2 } \\text{ inches} }{ x \\text{ miles} }\\)\n\nTo solve for x, we can cross-multiply:\n\n\\(2 \\times x = 9 \\times 1\\frac { 1 } { 2 }\\)\n\nSimplifying the right side:\n\n\\(2x = 9 \\times \\frac { 3 } { 2 }\\)\n\n\\(2x = \\frac { 27 } { 2 }\\)\n\nDividing both sides by 2:\n\n\\(x = \\frac { 27 } { 4 }\\)\n\nTherefore, the distance between Discovery Point and Overlook is \\(6\\frac { 3 } { 4 }\\) miles.\n\nThe answer is D." } }, { "answer":"A", "choices":{ "A":"\\(-8\\)", "B":"\\(-4\\)", "C":"0", "D":"4", "E":"8" }, "id":11679, "question":"\\begin{gather*} 3x - 4y = 10 \\\\ 6x + wy = 16 \\end{gather*} For which of the following values of w will the system of equations above have no solutions?", "explanations":{ "correct":"To determine the value of \\(w\\) for which the system of equations has no solutions, we can use the concept of determinants. \n\nFirst, let's write the system of equations in matrix form:\n\n\\[\n\\begin{bmatrix}\n3 & -4 \\\\\n6 & w\n\\end{bmatrix}\n\\begin{bmatrix}\nx \\\\\ny\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n10 \\\\\n16\n\\end{bmatrix}\n\\]\n\nThe system of equations has no solutions if and only if the determinant of the coefficient matrix is zero. Therefore, we need to find the determinant of the matrix:\n\n\\[\n\\begin{vmatrix}\n3 & -4 \\\\\n6 & w\n\\end{vmatrix}\n\\]\n\nTo find the determinant, we use the formula \\(ad - bc\\), where \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are the entries of the matrix. In this case, \\(a = 3\\), \\(b = -4\\), \\(c = 6\\), and \\(d = w\\). Substituting these values into the formula, we have:\n\n\\[\n(3)(w) - (-4)(6) = 3w + 24 = 0\n\\]\n\nTo solve for \\(w\\), we subtract 24 from both sides of the equation:\n\n\\[\n3w = -24\n\\]\n\nDividing both sides by 3, we find:\n\n\\[\nw = -8\n\\]\n\nTherefore, the system of equations will have no solutions when \\(w = -8\\).\n\nThe answer is A) \\(-8\\)." } }, { "answer":"B", "choices":{ "A":"115", "B":"120", "C":"125", "D":"130", "E":"145" }, "id":11680, "question":"Three lines, \\(\\ell_1\\), \\(\\ell_2\\) and \\(\\ell_3\\), intersect as shown. What is the value of \\(m\\) ?", "explanations":{ "correct":"To find the value of \\(m\\), we need to analyze the angles formed by the intersecting lines. \n\nFirst, let's look at the angle formed by \\(m\\) and the line \\(\\ell_1\\). This angle is an exterior angle to the triangle formed by \\(\\ell_1\\), \\(\\ell_2\\), and \\(\\ell_3\\). \n\nAccording to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In this case, the two non-adjacent interior angles are the angles formed by \\(\\ell_2\\) and \\(\\ell_3\\). \n\nSince \\(\\ell_2\\) and \\(\\ell_3\\) are parallel lines, the angles formed by them are corresponding angles. Corresponding angles are congruent, so the measure of each angle is equal. Let's call this measure \\(x\\). \n\nTherefore, the measure of the exterior angle \\(m\\) is \\(2x\\). \n\nNow, let's look at the angle formed by \\(m\\) and the line \\(\\ell_2\\). This angle is an interior angle to the triangle formed by \\(\\ell_1\\), \\(\\ell_2\\), and \\(\\ell_3\\). \n\nSince \\(\\ell_2\\) and \\(\\ell_3\\) are parallel lines, the angle formed by \\(m\\) and \\(\\ell_2\\) is an alternate interior angle with the angle formed by \\(\\ell_1\\) and \\(\\ell_3\\). Alternate interior angles are congruent, so the measure of this angle is also \\(x\\). \n\nTherefore, the measure of the interior angle \\(m\\) is \\(x\\). \n\nSince the sum of the measures of the interior angles of a triangle is \\(180^\\circ\\), we can set up the following equation: \n\n\\(x + x + m = 180^\\circ\\) \n\nSimplifying the equation, we have: \n\n\\(2x + m = 180^\\circ\\) \n\nSubstituting \\(2x\\) for \\(m\\), we get: \n\n\\(2x + 2x = 180^\\circ\\) \n\nCombining like terms, we have: \n\n\\(4x = 180^\\circ\\) \n\nDividing both sides by 4, we find: \n\n\\(x = 45^\\circ\\) \n\nSince \\(m\\) is equal to \\(2x\\), we have: \n\n\\(m = 2 \\times 45^\\circ = 90^\\circ\\) \n\nTherefore, the value of \\(m\\) is 90. \n\nThe answer is B) 120." } }, { "answer":"D", "choices":{ "A":"\\(-3a^5b^8\\)", "B":"\\(-3a^6b^15\\)", "C":"\\(-27a^5b^8\\)", "D":"\\(-27a^6b^15\\)", "E":"\\(27a^6b^15\\)" }, "id":11685, "question":"\\((-3a^2b^5)^3\\) =", "explanations":{ "correct":"To simplify the expression \\((-3a^2b^5)^3\\), we need to apply the exponent to each term inside the parentheses.\n\nFirst, let's focus on the coefficient \\(-3\\). Raising \\(-3\\) to the power of \\(3\\) gives us \\((-3)^3 = -27\\).\n\nNext, let's consider the variables \\(a\\) and \\(b\\). When we raise \\(a^2\\) to the power of \\(3\\), we multiply the exponents, giving us \\(a^{2 \\cdot 3} = a^6\\). Similarly, when we raise \\(b^5\\) to the power of \\(3\\), we multiply the exponents, resulting in \\(b^{5 \\cdot 3} = b^{15}\\).\n\nPutting it all together, we have \\((-3a^2b^5)^3 = -27a^6b^{15}\\).\n\nTherefore, the answer is D) \\(-27a^6b^{15}\\)." } }, { "answer":"B", "choices":{ "A":"20", "B":"18", "C":"16", "D":"14", "E":"12" }, "id":11686, "question":"If \\(n = 4p^2q\\), where \\(p\\) and \\(q\\) are distinct prime numbers, and not equal to 2, how many factors does the number \\(n\\) have?", "explanations":{ "correct":"To find the number of factors of a given number, we need to consider its prime factorization. \n\\(\\newline\\)In this case, the number \\(n\\) is given as \\(n = 4p^2q\\), where \\(p\\) and \\(q\\) are distinct prime numbers, and not equal to 2. \n\nLet's break down the prime factorization of \\(n\\):\n\n1. The number 4 can be written as \\(2^2\\).\n2. The prime factorization of \\(p^2\\) is \\(p \\cdot p\\).\n3. The prime factorization of \\(q\\) is simply \\(q\\).\n\nCombining these factors, we have \\(n = 2^2 \\cdot p \\cdot p \\cdot q\\).\n\nTo find the number of factors, we need to consider the exponents of each prime factor and add 1 to each exponent, then multiply the results.\n\\(\\newline\\)In this case, we have:\n- The exponent of 2 is 2, so we have \\(2+1 = 3\\) possible choices for the exponent of 2.\n- The exponent of \\(p\\) is 2, so we have \\(2+1 = 3\\) possible choices for the exponent of \\(p\\).\n- The exponent of \\(q\\) is 1, so we have \\(1+1 = 2\\) possible choices for the exponent of \\(q\\).\n\nMultiplying these results together, we get \\(3 \\cdot 3 \\cdot 2 = 18\\).\n\nTherefore, the number \\(n\\) has 18 factors.\n\nThe answer is B) 18." } }, { "answer":"A", "choices":{ "A":"4", "B":"6", "C":"9", "D":"11", "E":"32" }, "id":11689, "question":"What is the value of x if \\(2x + 3y = 11\\) and \\(2x - y = 7\\)?", "explanations":{ "correct":"To find the value of x, we can solve the given system of equations using the method of elimination. \n\nFirst, let's eliminate the variable y by adding the two equations together. \n\n(2x + 3y) + (2x - y) = 11 + 7\n\nSimplifying the left side of the equation, we get:\n\n4x + 2y = 18\n\nNow, let's isolate the variable x by subtracting 2y from both sides of the equation:\n\n4x = 18 - 2y\n\nNext, divide both sides of the equation by 4 to solve for x:\n\nx = (18 - 2y) / 4\n\nNow, let's substitute the value of y from the second equation into the expression for x:\n\nx = (18 - 2(2x - 7)) / 4\n\nSimplifying further:\n\nx = (18 - 4x + 14) / 4\n\nCombining like terms:\n\nx = (32 - 4x) / 4\n\nMultiplying both sides of the equation by 4 to eliminate the fraction:\n\n4x = 32 - 4x\n\nAdding 4x to both sides of the equation:\n\n8x = 32\n\nDividing both sides of the equation by 8 to solve for x:\n\nx = 32 / 8\n\nSimplifying:\n\nx = 4\n\nTherefore, the value of x is 4.\n\nThe answer is A) 4." } }, { "answer":"A", "choices":{ "A":"130", "B":"225", "C":"331", "D":"463", "E":"563" }, "id":11691, "question":"A bookseller's net profit, in dollars, from the sale of b books is given by \\(P(b) = 2.5b - 100\\). How many books must she sell in order to earn a net profit of \\$ 225?", "explanations":{ "correct":"To find the number of books the bookseller must sell in order to earn a net profit of \\$ 225, we need to set up an equation using the given profit function.\n\nThe profit function is given by \\(P(b) = 2.5b - 100\\), where \\(P(b)\\) represents the net profit in dollars and \\(b\\) represents the number of books sold.\n\nWe want to find the value of \\(b\\) when the net profit is \\$ 225. So we set up the equation:\n\n\\(2.5b - 100 = 225\\)\n\nTo solve for \\(b\\), we first add 100 to both sides of the equation:\n\n\\(2.5b = 225 + 100\\)\n\n\\(2.5b = 325\\)\n\nNext, we divide both sides of the equation by 2.5 to isolate \\(b\\):\n\n\\(b = \\frac{325}{2.5}\\)\n\nSimplifying the right side of the equation:\n\n\\(b = 130\\)\n\nTherefore, the bookseller must sell 130 books in order to earn a net profit of \\$ 225.\n\nThe answer is A) 130." } }, { "answer":"E", "choices":{ "A":"2", "B":"4", "C":"20", "D":"25", "E":"100" }, "id":11695, "question":"If \\(3x = 10 - 2x\\), then \\((5x)(5x)\\) =", "explanations":{ "correct":"To find the value of \\\\((5x)(5x)\\\\), we first need to solve the equation \\\\(3x = 10 - 2x\\\\).\n\nStep 1: Combine like terms by adding \\\\(2x\\\\) to both sides of the equation:\n\\\\(3x + 2x = 10 - 2x + 2x\\\\)\nSimplifying, we get:\n\\\\(5x = 10\\\\)\n\nStep 2: Divide both sides of the equation by 5 to solve for \\\\(x\\\\):\n\\\\(\\\\frac{{5x}}{5} = \\\\frac{{10}}{5}\\\\)\nSimplifying, we get:\n\\\\(x = 2\\\\)\n\nNow that we have found the value of \\\\(x\\\\) as 2, we can substitute it into \\\\((5x)(5x)\\\\) to find the final answer.\n\nStep 3: Substitute \\\\(x = 2\\\\) into \\\\((5x)(5x)\\\\):\n\\\\((5(2))(5(2))\\\\)\nSimplifying, we get:\n\\\\(10 \\\\cdot 10\\\\)\n\\\\(100\\\\)\n\nTherefore, the answer is E) 100." } }, { "answer":"E", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III only", "E":"I, II, and III" }, "id":11699, "question":"If \\(k(a^2 - b^2) = 0\\), which of the following could be true? \\(\\newline\\)I. \\(k = 0\\) \\(\\newline\\)II. \\(a = 5\\) and \\(b = 5\\) \\(\\newline\\)III. \\(a = -b\\)", "explanations":{ "correct":"To determine which of the given statements could be true when \\(k(a^2 - b^2) = 0\\), we need to analyze each statement individually.\n\\(\\newline\\)I. \\(k = 0\\):\\(\\newline\\)If \\(k = 0\\), then the entire equation becomes \\(0(a^2 - b^2) = 0\\). This means that no matter the values of \\(a\\) and \\(b\\), the equation will always be true. Therefore, statement I could be true.\n\\(\\newline\\)II. \\(a = 5\\) and \\(b = 5\\):\\(\\newline\\)If \\(a = 5\\) and \\(b = 5\\), the equation becomes \\(k(5^2 - 5^2) = 0\\), which simplifies to \\(k(0) = 0\\). This means that no matter the value of \\(k\\), the equation will always be true. Therefore, statement II could be true.\n\\(\\newline\\)III. \\(a = -b\\):\\(\\newline\\)If \\(a = -b\\), the equation becomes \\(k((-b)^2 - b^2) = 0\\), which simplifies to \\(k(b^2 - b^2) = 0\\), and further simplifies to \\(k(0) = 0\\). This means that no matter the value of \\(k\\), the equation will always be true. Therefore, statement III could be true.\n\nBased on the analysis above, all three statements could be true. Therefore, the answer is E) I, II, and \\(\\newline\\)III." } }, { "answer":"A", "choices":{ "A":"-1", "B":"\\(-\\frac { 1 } { 2 } \\)", "C":"0", "D":"\\(\\frac { 1 } { 2 } \\)", "E":"1" }, "id":11700, "question":"If \\(2x + 4x + 6x = -12\\) ,then \\(x\\) =", "explanations":{ "correct":"To solve the equation \\(2x + 4x + 6x = -12\\), we need to combine like terms on the left side of the equation. \n\nFirst, let's add the coefficients of the \\(x\\) terms: \\(2 + 4 + 6 = 12\\). \n\nSo, the equation becomes \\(12x = -12\\). \n\nTo isolate \\(x\\), we need to divide both sides of the equation by 12: \n\n\\(\\frac{12x}{12} = \\frac{-12}{12}\\). \n\nThis simplifies to \\(x = -1\\). \n\nTherefore, the answer is A) -1." } }, { "answer":"D", "choices":{ "A":"\\(y = \\frac { 3 } { 5 } x + 4\\)", "B":"\\(y = \\frac { 5 } { 3 } x + 4\\)", "C":"\\(y = -\\frac { 3 } { 5 } x - 4\\)", "D":"\\(y = \\frac { 3 } { 5 } x - 4\\)", "E":"\\(y = \\frac { 5 } { 3 } x - 4\\)" }, "id":11702, "question":"In the xy- plane, the equation of line \\(\\ell\\) is \\(y = -\\frac { 3 } { 5 } x + 4\\). If line \\(m\\) is the reflection of line \\(\\ell\\) over the x- axis, what is the equation of line \\(m\\)?", "explanations":{ "correct":"To find the equation of line \\(m\\), which is the reflection of line \\(\\ell\\) over the x-axis, we need to change the sign of the y-coordinate in the equation of line \\(\\ell\\).\n\nThe equation of line \\(\\ell\\) is \\(y = -\\frac{3}{5}x + 4\\). To reflect this line over the x-axis, we change the sign of the y-coordinate, resulting in \\(y = -\\left(-\\frac{3}{5}x + 4\\right)\\).\n\nSimplifying the equation, we have \\(y = \\frac{3}{5}x - 4\\).\n\nTherefore, the equation of line \\(m\\) is \\(y = \\frac{3}{5}x - 4\\).\n\nThe answer is D) \\(y = \\frac{3}{5}x - 4\\)." } }, { "answer":"A", "choices":{ "A":"-5", "B":"0", "C":"5", "D":"7", "E":"9" }, "id":11706, "question":"If \\(|t| = 2\\), what is the value of \\((t + 3)(t - 3)\\)?", "explanations":{ "correct":"To find the value of \\((t + 3)(t - 3)\\) when \\(|t| = 2\\), we can substitute \\(|t|\\) with its value of 2.\n\nFirst, let's simplify the expression \\((t + 3)(t - 3)\\) by using the distributive property. \n\n\\((t + 3)(t - 3) = t(t) + t(-3) + 3(t) + 3(-3)\\)\n\nSimplifying further, we have:\n\n\\(t^2 - 3t + 3t - 9\\)\n\nCombining like terms, we get:\n\n\\(t^2 - 9\\)\n\nNow, let's substitute \\(|t|\\) with 2:\n\n\\(2^2 - 9 = 4 - 9 = -5\\)\n\nTherefore, the value of \\((t + 3)(t - 3)\\) when \\(|t| = 2\\) is -5.\n\nThe answer is A) -5." } }, { "answer":"B", "choices":{ "A":"5", "B":"\\(5\\sqrt { 2 } \\)", "C":"10", "D":"\\(10\\sqrt { 2 } \\)", "E":"20" }, "id":11709, "question":"A certain flag has a length that is 2 times its width. What is the width of a flag that has an area of 100?", "explanations":{ "correct":"To find the width of the flag, we need to set up an equation using the given information. Let's assume the width of the flag is \\(x\\). \n\nAccording to the problem, the length of the flag is 2 times its width. So, the length would be \\(2x\\). \n\nThe formula for the area of a rectangle is \\(A = \\text{length} \\times \\text{width}\\). In this case, the area is given as 100. So, we can write the equation as:\n\n\\(100 = 2x \\times x\\)\n\nSimplifying the equation, we have:\n\n\\(100 = 2x^2\\)\n\nDividing both sides of the equation by 2, we get:\n\n\\(50 = x^2\\)\n\nTaking the square root of both sides, we find:\n\n\\(x = \\sqrt{50}\\)\n\nSimplifying the square root, we have:\n\n\\(x = \\sqrt{25 \\times 2}\\)\n\nSince the width cannot be negative, we take the positive square root:\n\n\\(x = 5\\sqrt{2}\\)\n\nTherefore, the width of the flag that has an area of 100 is \\(5\\sqrt{2}\\).\n\nThe answer is B." } }, { "answer":"B", "choices":{ "A":"\\(-10\\)", "B":"\\(-5\\)", "C":"\\(-1\\)", "D":"5", "E":"10" }, "id":11713, "question":"If \\(5x - x = 2x + x - 5\\), then \\(x =\\)", "explanations":{ "correct":"To solve the equation \\(5x - x = 2x + x - 5\\), we need to simplify both sides of the equation and then solve for \\(x\\).\n\nOn the left side of the equation, we have \\(5x - x\\). This simplifies to \\(4x\\).\n\nOn the right side of the equation, we have \\(2x + x - 5\\). This simplifies to \\(3x - 5\\).\n\nSo, our equation becomes \\(4x = 3x - 5\\).\n\nTo isolate \\(x\\), we need to get rid of the \\(3x\\) term on the right side. We can do this by subtracting \\(3x\\) from both sides of the equation:\n\n\\(4x - 3x = 3x - 5 - 3x\\)\n\nThis simplifies to \\(x = -5\\).\n\nTherefore, the answer is B) \\(-5\\)." } }, { "answer":"A", "choices":{ "A":"\\(\\frac{1}{12}\\)", "B":"\\(\\frac{1}{7}\\)", "C":"\\(\\frac{2}{7}\\)", "D":"\\(\\frac{7}{12}\\)", "E":"\\(\\frac{12}{7}\\)" }, "id":11719, "question":"In a certain pet store, every third goldfish has long fins, and every fourth goldfish has a long tail. If a fish is selected at random from a tank of 120 fish, what is the probability that the fish will have long fins and a long tail?", "explanations":{ "correct":"To find the probability that a fish will have long fins and a long tail, we need to determine the number of fish that satisfy both conditions and divide it by the total number of fish in the tank.\n\nLet's first find the number of fish with long fins. Every third goldfish has long fins, so we can calculate the number of fish with long fins by dividing the total number of fish by 3: 120 ÷ 3 = 40.\n\nNext, let's find the number of fish with a long tail. Every fourth goldfish has a long tail, so we can calculate the number of fish with a long tail by dividing the total number of fish by 4: 120 ÷ 4 = 30.\n\nNow, we need to find the number of fish that satisfy both conditions. To do this, we need to find the least common multiple (LCM) of 3 and 4, which is 12. This means that every 12th fish will have both long fins and a long tail.\n\nTo find the number of fish that satisfy both conditions, we divide the total number of fish by 12: 120 ÷ 12 = 10.\n\nFinally, we can calculate the probability by dividing the number of fish that satisfy both conditions by the total number of fish: 10 ÷ 120 = \\(\\frac{1}{12}\\).\n\nTherefore, the answer is A) \\(\\frac{1}{12}\\)." } }, { "answer":"C", "choices":{ "A":"80", "B":"100", "C":"120", "D":"140", "E":"160" }, "id":11723, "question":"The measures of the angles of a triangle, in degrees, can be expressed by the ratio 4:5:6. What is the sum of the measures of the largest and smallest angles?", "explanations":{ "correct":"To find the sum of the measures of the largest and smallest angles of a triangle, we need to determine the measures of each angle. \n\nLet's assume the measures of the angles are 4x, 5x, and 6x, where x is a constant. \n\nAccording to the given ratio, the measures of the angles can be expressed as 4x:5x:6x. \n\nThe sum of the measures of all angles in a triangle is always 180 degrees. \n\nSo, we can set up the equation: \n\n4x + 5x + 6x = 180 \n\nCombining like terms, we get: \n\n15x = 180 \n\nDividing both sides of the equation by 15, we find: \n\nx = 12 \n\nNow, we can substitute the value of x back into the expressions for the angles: \n\n4x = 4 * 12 = 48 degrees \n\n5x = 5 * 12 = 60 degrees \n\n6x = 6 * 12 = 72 degrees \n\nThe largest angle is 72 degrees, and the smallest angle is 48 degrees. \n\nTo find the sum of the largest and smallest angles, we add these two angles together: \n\n72 + 48 = 120 \n\nTherefore, the sum of the measures of the largest and smallest angles is 120 degrees. \n\nThe answer is C) 120." } }, { "answer":"E", "choices":{ "A":"\\(-1\\)", "B":"0", "C":"1", "D":"2", "E":"3" }, "id":11725, "question":"If \\(a^2 - 16 = b^2\\), and \\(2a = 10\\), which of the following could be a value for b?", "explanations":{ "correct":"To find the possible value for \\(b\\), we need to substitute the given value of \\(a\\) into the equation \\(a^2 - 16 = b^2\\).\n\nGiven that \\(2a = 10\\), we can solve for \\(a\\) by dividing both sides of the equation by 2:\n\\[2a = 10\\]\n\\[\\frac{2a}{2} = \\frac{10}{2}\\]\n\\[a = 5\\]\n\nNow, substitute \\(a = 5\\) into the equation \\(a^2 - 16 = b^2\\):\n\\[5^2 - 16 = b^2\\]\n\\[25 - 16 = b^2\\]\n\\[9 = b^2\\]\n\nTo find the possible values for \\(b\\), we need to find the square root of both sides of the equation:\n\\[\\sqrt{9} = \\sqrt{b^2}\\]\n\\[3 = b\\] or \\[-3 = b\\]\n\nTherefore, the possible values for \\(b\\) are 3 and -3.\n\nLooking at the answer choices, the only possible value for \\(b\\) is 3.\n\nThe answer is E." } }, { "answer":"D", "choices":{ "A":"\\(8x^3\\)", "B":"\\(27x^3\\)", "C":"\\(64x^3\\)", "D":"\\(125x^3\\)", "E":"\\(150x^3\\)" }, "id":11726, "question":"If the surface area of a cube is \\(150x^2\\), what is the volume of the cube?", "explanations":{ "correct":"To find the volume of a cube, we need to know the length of one side of the cube. Let's assume that the length of one side of the cube is \\(s\\).\n\nThe formula for the surface area of a cube is \\(6s^2\\). We are given that the surface area of the cube is \\(150x^2\\), so we can set up the equation:\n\n\\(6s^2 = 150x^2\\)\n\nTo find the volume, we need to find the length of one side of the cube. We can solve for \\(s\\) by dividing both sides of the equation by 6:\n\n\\(s^2 = \\frac{150x^2}{6}\\)\n\n\\(s^2 = 25x^2\\)\n\nTaking the square root of both sides, we get:\n\n\\(s = 5x\\)\n\nNow that we know the length of one side of the cube is \\(5x\\), we can find the volume by cubing the length of one side:\n\n\\(V = (5x)^3\\)\n\n\\(V = 125x^3\\)\n\nTherefore, the volume of the cube is \\(125x^3\\).\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"-1", "B":"0", "C":"1", "D":"3", "E":"5" }, "id":11727, "question":"If \\(5c + 3 = 3c + 5\\), what is the value of \\(c\\)?", "explanations":{ "correct":"To find the value of \\(c\\) in the equation \\(5c + 3 = 3c + 5\\), we need to isolate the variable \\(c\\) on one side of the equation.\n\nStep 1: Start by subtracting \\(3c\\) from both sides of the equation to eliminate the \\(3c\\) term on the right side:\n\\[5c + 3 - 3c = 3c + 5 - 3c\\]\nSimplifying, we get:\n\\[2c + 3 = 5\\]\n\nStep 2: Next, subtract \\(3\\) from both sides of the equation to isolate the \\(2c\\) term on the left side:\n\\[2c + 3 - 3 = 5 - 3\\]\nSimplifying, we get:\n\\[2c = 2\\]\n\nStep 3: Finally, divide both sides of the equation by \\(2\\) to solve for \\(c\\):\n\\[\\frac{2c}{2} = \\frac{2}{2}\\]\nSimplifying, we get:\n\\[c = 1\\]\n\nTherefore, the value of \\(c\\) that satisfies the equation \\(5c + 3 = 3c + 5\\) is \\(c = 1\\).\n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"\\$ 26", "B":"\\$ 28", "C":"\\$ 16", "D":"\\$ 8", "E":"\\$ 2" }, "id":11732, "question":"If, at a certain clothing store, three pairs of dress socks and four pairs of athletic socks cost a total of \\$ 27 and four pairs of dress socks and three pairs of athletic socks cost a total of \\$ 29, what is the combined cost of one pair of dress socks and one pair of athletic socks?", "explanations":{ "correct":"Let's solve this problem step-by-step.\n\nLet's assume the cost of one pair of dress socks is \"x\" dollars and the cost of one pair of athletic socks is \"y\" dollars.\n\nAccording to the given information, three pairs of dress socks and four pairs of athletic socks cost a total of \\$ 27. So, we can write the equation:\n\n3x + 4y = 27 ---(Equation 1)\n\nSimilarly, four pairs of dress socks and three pairs of athletic socks cost a total of \\$ 29. So, we can write the equation:\n\n4x + 3y = 29 ---(Equation 2)\n\nTo find the combined cost of one pair of dress socks and one pair of athletic socks, we need to find the value of x + y.\n\nTo solve this system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.\n\nFrom Equation 1, we can solve for x in terms of y:\n\n3x = 27 - 4y\nx = (27 - 4y)/3 ---(Equation 3)\n\nNow, substitute the value of x from Equation 3 into Equation 2:\n\n4((27 - 4y)/3) + 3y = 29\n\nMultiply both sides of the equation by 3 to eliminate the fraction:\n\n4(27 - 4y) + 9y = 87\n\nDistribute 4 to the terms inside the parentheses:\n\n108 - 16y + 9y = 87\n\nCombine like terms:\n\n-7y = -21\n\nDivide both sides of the equation by -7:\n\ny = 3\n\nNow, substitute the value of y = 3 into Equation 3 to find the value of x:\n\nx = (27 - 4(3))/3\nx = (27 - 12)/3\nx = 15/3\nx = 5\n\nTherefore, the combined cost of one pair of dress socks and one pair of athletic socks is x + y = 5 + 3 = \\$ 8.\n\nThe answer is D) \\$ 8." } }, { "answer":"C", "choices":{ "A":"\\(\\frac{n + 16}{4}\\)", "B":"\\(\\frac{n + 4}{4}\\)", "C":"\\(\\frac{n + 16}{4n}\\)", "D":"\\(\\frac{n + 4}{4n}\\)", "E":"\\(\\frac{16 - n}{4}\\)" }, "id":11733, "question":"After the first term, each term in a sequence is 4 greater than \\(\\frac{1}{4}\\) of the preceding term. If n is the first term of the sequence and \\(n \\neq 0\\), what is the ratio of the second term to the first term?", "explanations":{ "correct":"To find the ratio of the second term to the first term, we need to understand the pattern of the sequence.\n\nWe are given that after the first term, each term in the sequence is 4 greater than \\(\\frac{1}{4}\\) of the preceding term. Let's break this down step-by-step:\n\n1. The first term is \\(n\\).\n2. The second term is \\(\\frac{1}{4}\\) of the first term plus 4. So, the second term is \\(\\frac{1}{4}n + 4\\).\n\nTo find the ratio of the second term to the first term, we divide the second term by the first term:\n\n\\(\\frac{\\frac{1}{4}n + 4}{n}\\)\n\nTo simplify this expression, we can multiply the numerator and denominator by 4 to get rid of the fraction:\n\n\\(\\frac{4(\\frac{1}{4}n + 4)}{4n}\\)\n\nSimplifying further:\n\n\\(\\frac{n + 16}{4n}\\)\n\nTherefore, the correct answer is C) \\(\\frac{n + 16}{4n}\\).\n\nThe answer is C)." } }, { "answer":"B", "choices":{ "A":"25%", "B":"30%", "C":"35%", "D":"40%", "E":"45%" }, "id":11734, "question":"If Jorge earns \\$ 2,000 a month and spends \\$ 600 a month on rent, what percent of Jorge's monthly earnings does he spend on rent?", "explanations":{ "correct":"To find the percent of Jorge's monthly earnings that he spends on rent, we need to divide the amount he spends on rent by his monthly earnings and then multiply by 100.\n\nJorge spends \\$ 600 on rent per month and earns \\$ 2,000 per month. \n\nDividing \\$ 600 by \\$ 2,000 gives us 0.3. \n\nMultiplying 0.3 by 100 gives us 30.\n\nTherefore, Jorge spends 30% of his monthly earnings on rent.\n\nThe answer is B) 30%." } }, { "answer":"D", "choices":{ "A":"4", "B":"16", "C":"\\(16 \\sqrt { 2 } \\)", "D":"32", "E":"64" }, "id":11736, "question":"If \\(|frac { \\sqrt { x } } { 2 } | = 2\\sqrt { 2 } \\), what is the value of \\(x\\) ?", "explanations":{ "correct":"To find the value of \\(x\\), we need to isolate \\(x\\) in the given equation: \\(\\left|\\frac{\\sqrt{x}}{2}\\right| = 2\\sqrt{2}\\).\n\nFirst, we can remove the absolute value by considering both positive and negative cases.\n\nCase 1: \\(\\frac{\\sqrt{x}}{2} = 2\\sqrt{2}\\)\nMultiplying both sides by 2, we get \\(\\sqrt{x} = 4\\sqrt{2}\\).\nSquaring both sides, we have \\(x = (4\\sqrt{2})^2 = 16 \\cdot 2 = 32\\).\n\nCase 2: \\(\\frac{\\sqrt{x}}{2} = -2\\sqrt{2}\\)\nMultiplying both sides by 2, we get \\(\\sqrt{x} = -4\\sqrt{2}\\).\nSince the square root of a number cannot be negative, this case is not valid.\n\nTherefore, the value of \\(x\\) is 32.\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"2", "B":"8", "C":"80", "D":"81", "E":"82" }, "id":11740, "question":"If \\(\\sqrt{x} = 3\\), then \\(x^2 - 1 =\\)", "explanations":{ "correct":"To find the value of \\(x^2 - 1\\) when \\(\\sqrt{x} = 3\\), we need to substitute the value of \\(\\sqrt{x}\\) into the expression.\n\nGiven that \\(\\sqrt{x} = 3\\), we can square both sides of the equation to find the value of \\(x\\). Squaring both sides gives us \\(x = 3^2 = 9\\).\n\nNow, we can substitute the value of \\(x\\) into the expression \\(x^2 - 1\\). Plugging in \\(x = 9\\), we get \\(9^2 - 1 = 81 - 1 = 80\\).\n\nTherefore, the answer is C) 80." } }, { "answer":"D", "choices":{ "A":"0.01", "B":"0.1", "C":"1", "D":"10", "E":"100" }, "id":11748, "question":"If \\((0.0013)x = 0.013\\), then \\(x =\\)", "explanations":{ "correct":"To solve the equation \\\\((0.0013)x = 0.013\\\\) for \\\\(x\\\\), we need to isolate \\\\(x\\\\) on one side of the equation. \n\nFirst, we can divide both sides of the equation by \\\\(0.0013\\\\) to eliminate the coefficient of \\\\(x\\\\). This gives us:\n\n\\\\(x = \\\\frac{0.013}{0.0013}\\\\)\n\nTo simplify the right side of the equation, we divide \\\\(0.013\\\\) by \\\\(0.0013\\\\):\n\n\\\\(x = 10\\\\)\n\nTherefore, the value of \\\\(x\\\\) is 10.\n\nThe answer is D) 10." } }, { "answer":"D", "choices":{ "A":"\\(a + 2\\)", "B":"\\(a^2\\)", "C":"\\(a^3\\)", "D":"\\(a^2 - a\\)", "E":"\\((a + 2)^2\\)" }, "id":11757, "question":"If \\(a\\) is an odd integer, which of the following is an even integer?", "explanations":{ "correct":"To determine which of the given expressions is an even integer when \\(a\\) is an odd integer, we need to understand the properties of odd and even numbers.\n\nAn odd integer can be represented as \\(2n + 1\\), where \\(n\\) is an integer. This means that any odd integer can be obtained by adding an even integer (\\(2n\\)) with 1.\n\nNow let's analyze each expression:\n\nA) \\(a + 2\\):\nSince \\(a\\) is an odd integer, we can represent it as \\(2n + 1\\). Adding 2 to an odd integer will result in \\(2n + 1 + 2 = 2n + 3\\), which is an odd integer. Therefore, option A is not an even integer.\n\nB) \\(a^2\\):\nSquaring an odd integer will result in \\((2n + 1)^2 = 4n^2 + 4n + 1\\). Notice that the first two terms, \\(4n^2\\) and \\(4n\\), are both divisible by 2 since they contain a factor of 2. Adding 1 to this expression will not change its evenness. Therefore, option B is an odd integer.\n\nC) \\(a^3\\):\nCubing an odd integer will result in \\((2n + 1)^3 = 8n^3 + 12n^2 + 6n + 1\\). Similar to the previous explanation, the first three terms, \\(8n^3\\), \\(12n^2\\), and \\(6n\\), are all divisible by 2. Adding 1 to this expression will not change its evenness. Therefore, option C is an odd integer.\n\nD) \\(a^2 - a\\):\nSubstituting \\(a = 2n + 1\\) into the expression, we get \\((2n + 1)^2 - (2n + 1) = 4n^2 + 4n + 1 - 2n - 1 = 4n^2 + 2n\\). Both terms, \\(4n^2\\) and \\(2n\\), are divisible by 2. Therefore, option D is an even integer.\n\nE) \\((a + 2)^2\\):\nExpanding the expression, we have \\((2n + 1 + 2)^2 = (2n + 3)^2 = 4n^2 + 12n + 9\\). The first two terms, \\(4n^2\\) and \\(12n\\), are divisible by 2. Adding 9 to this expression will not change its evenness. Therefore, option E is an odd integer.\n\nBased on the analysis above, the only expression that represents an even integer when \\(a\\) is an odd integer is option D.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"0", "B":"5", "C":"11", "D":"62", "E":"67" }, "id":11758, "question":"If \\(A\\) is the set of positive multiples of 5 less than 200 and \\(B\\) is the set of positive multiples of 7 less than 200, how many numbers are in both set \\(A\\) and set \\(B\\) ?", "explanations":{ "correct":"To find the numbers that are in both set A and set B, we need to determine the common multiples of 5 and 7 that are less than 200.\n\nFirst, let's find the multiples of 5 less than 200. We start with 5 and keep adding 5 to get the next multiple: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195.\n\nNext, let's find the multiples of 7 less than 200. We start with 7 and keep adding 7 to get the next multiple: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196.\n\nNow, let's identify the common multiples of 5 and 7 from the two lists above: 35, 70, 105, 140, 175.\n\nThere are 5 numbers that are in both set A and set B.\n\nTherefore, the answer is B) 5." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 19 } { 50 } \\)", "B":"\\(\\frac { 39 } { 100 } \\)", "C":"\\(\\frac { 2 } { 5 } \\)", "D":"\\(\\frac { 31 } { 50 } \\)", "E":"\\(\\frac { 9 } { 10 } \\)" }, "id":11761, "question":"Set A consists of all even integers from 1 to 100, inclusive. If a number is selected at random from set A, what is the probability that the number is less than 40?", "explanations":{ "correct":"To find the probability that a number selected at random from set A is less than 40, we need to determine the number of even integers in set A that are less than 40 and divide it by the total number of even integers in set A.\n\nFirst, let's find the number of even integers in set A that are less than 40. We know that set A consists of all even integers from 1 to 100, inclusive. To find the number of even integers less than 40, we divide 40 by 2 (since every even integer is divisible by 2) and subtract 1 (since we want to exclude 40 itself). So, the number of even integers less than 40 is (40/2) - 1 = 19 - 1 = 18.\n\nNext, let's find the total number of even integers in set A. We know that set A consists of all even integers from 1 to 100, inclusive. To find the total number of even integers, we divide 100 by 2 and add 1 (since we want to include 100 itself). So, the total number of even integers in set A is (100/2) + 1 = 50 + 1 = 51.\n\nFinally, we can calculate the probability by dividing the number of favorable outcomes (even integers less than 40) by the total number of possible outcomes (total number of even integers in set A). The probability is 18/51, which simplifies to 6/17.\n\nTherefore, the answer is A) \\(\\frac { 19 } { 50 } \\)." } }, { "answer":"E", "choices":{ "A":"-32", "B":"-10", "C":"-6", "D":"-4", "E":"-2" }, "id":11762, "question":"If \\(a < 0\\) and \\((a - 4)^2 = 36\\), what is \\(a\\) ?", "explanations":{ "correct":"To solve the equation \\\\((a - 4)^2 = 36\\\\), we need to find the value of \\\\(a\\\\) when \\\\(a < 0\\\\).\n\nStep 1: Take the square root of both sides of the equation to eliminate the square. \n\\\\(\\sqrt{(a - 4)^2} = \\sqrt{36}\\\\)\n\nStep 2: Simplify the square root of the left side of the equation.\n\\\\(|a - 4| = 6\\\\)\n\nStep 3: Since \\\\(a < 0\\\\), the expression \\\\(|a - 4|\\) can be rewritten as \\\\(-(a - 4)\\).\n\\\\(-(a - 4) = 6\\\\)\n\nStep 4: Distribute the negative sign to the terms inside the parentheses.\n\\\\-a + 4 = 6\\\\\n\nStep 5: Subtract 4 from both sides of the equation to isolate \\\\(a\\\\).\n\\\\-a = 6 - 4\\\\\n\\\\-a = 2\\\\)\n\nStep 6: Multiply both sides of the equation by -1 to solve for \\\\(a\\\\).\n\\\\a = -2\\\\)\n\nTherefore, the value of \\\\(a\\\\) is -2. \n\nThe answer is E." } }, { "answer":"A", "choices":{ "A":"1.014", "B":"10.14", "C":"1,014", "D":"10,140", "E":"101,400" }, "id":11771, "question":"For all positive integers f and g, let f \\# g be defined as \\(\\frac{f + 2g}{f - 2g}\\). What is the value of 1,007 \\# 3.5?", "explanations":{ "correct":"To find the value of 1,007 \\\\# 3.5, we substitute the values of f and g into the given expression \\\\(\\\\frac{f + 2g}{f - 2g}\\\\).\n\nSubstituting f = 1,007 and g = 3.5, we have:\n\n1,007 \\\\# 3.5 = \\\\(\\\\frac{1,007 + 2(3.5)}{1,007 - 2(3.5)}\\\\)\n\nSimplifying the numerator and denominator, we get:\n\n1,007 \\\\# 3.5 = \\\\(\\\\frac{1,007 + 7}{1,007 - 7}\\\\)\n\n1,007 \\\\# 3.5 = \\\\(\\\\frac{1,014}{1,000}\\\\)\n\nDividing 1,014 by 1,000, we find:\n\n1,007 \\\\# 3.5 = 1.014\n\nTherefore, the answer is A) 1.014." } }, { "answer":"C", "choices":{ "A":"60", "B":"80", "C":"100", "D":"120", "E":"160" }, "id":11773, "question":"In a certain triangle, the measure of the largest angle is 40 degrees more than the measure of the middle-sized angle. If the measure of the smallest angle is 20 degrees, what is the degree measure of the largest angle?", "explanations":{ "correct":"Let's denote the measure of the middle-sized angle as x degrees. According to the problem, the largest angle is 40 degrees more than the middle-sized angle. Therefore, the measure of the largest angle is x + 40 degrees.\n\nWe know that the sum of the angles in a triangle is always 180 degrees. So, we can set up the equation:\n\nx + (x + 40) + 20 = 180\n\nCombining like terms, we have:\n\n2x + 60 = 180\n\nSubtracting 60 from both sides, we get:\n\n2x = 120\n\nDividing both sides by 2, we find:\n\nx = 60\n\nTherefore, the measure of the middle-sized angle is 60 degrees. \n\nTo find the measure of the largest angle, we substitute x = 60 into the expression x + 40:\n\n60 + 40 = 100\n\nSo, the degree measure of the largest angle is 100 degrees.\n\nThe answer is C) 100." } }, { "answer":"C", "choices":{ "A":"1,680", "B":"2,520", "C":"2,800", "D":"4,075", "E":"6,300" }, "id":11775, "question":"A survey of all the students who attend Central Valley High School showed that there was an average (arithmetic mean) of 33.6 students per class room and and average of 22.4 electronic devices brought to class by students per classroom. If 4,200 students attend Central Valley High School, which of the following is the best estimate of the total number of electronic devices in the classrooms of Central Valley?", "explanations":{ "correct":"To find the best estimate of the total number of electronic devices in the classrooms of Central Valley High School, we need to multiply the average number of electronic devices brought to class per classroom by the total number of classrooms.\n\nGiven that the average number of electronic devices brought to class per classroom is 22.4 and the average number of students per classroom is 33.6, we can assume that each student brings one electronic device to class.\n\nTo find the total number of classrooms, we divide the total number of students by the average number of students per classroom:\n4,200 students / 33.6 students per classroom = 125 classrooms\n\nNow, we can find the best estimate of the total number of electronic devices by multiplying the average number of electronic devices per classroom by the total number of classrooms:\n22.4 electronic devices per classroom * 125 classrooms = 2,800 electronic devices\n\nTherefore, the best estimate of the total number of electronic devices in the classrooms of Central Valley High School is 2,800.\n\nThe answer is C) 2,800." } }, { "answer":"D", "choices":{ "A":"\\(1\\frac{1}{2}\\)", "B":"2", "C":"3", "D":"4", "E":"\\(4\\frac{1}{2}\\)" }, "id":11786, "question":"The average (arithmetic mean) of six numbers is 6. If 3 is subtracted from each of four of the numbers, what is the new average?", "explanations":{ "correct":"To find the new average after subtracting 3 from each of four numbers, we need to consider the effect of this subtraction on the sum of the numbers and the total count of numbers.\n\nGiven that the average of the six numbers is 6, we can determine the sum of the numbers by multiplying the average by the count of numbers. In this case, the sum of the six numbers is \\(6 \\times 6 = 36\\).\n\nTo find the new sum after subtracting 3 from each of four numbers, we need to subtract 3 four times from the original sum. This gives us \\(36 - (3 \\times 4) = 36 - 12 = 24\\).\n\nNow, we need to determine the new count of numbers. Since we subtracted 3 from each of four numbers, the count of numbers remains the same at 6.\n\nTo find the new average, we divide the new sum by the new count of numbers. Therefore, the new average is \\(24 \\div 6 = 4\\).\n\nHence, the new average after subtracting 3 from each of four numbers is 4.\n\nThe answer is D." } }, { "answer":"B", "choices":{ "A":"\\(y < 5\\)", "B":"\\(y \\geq -5\\)", "C":"\\(y < -5\\)", "D":"\\(y > 5\\)", "E":"All real numbers" }, "id":11789, "question":"\\(f(x) = x^2 - 5\\) If \\(y = f(x)\\), which of the following represents all possible values of y?", "explanations":{ "correct":"To find the possible values of \\(y\\) when \\(y = f(x)\\), we need to substitute \\(f(x) = x^2 - 5\\) into the equation.\n\nSo, we have \\(y = x^2 - 5\\).\n\nTo determine the possible values of \\(y\\), we need to consider the range of the function \\(f(x)\\), which is the set of all possible output values.\n\nThe graph of \\(f(x) = x^2 - 5\\) is a parabola that opens upward. Since the coefficient of \\(x^2\\) is positive, the parabola opens upward and does not have a maximum value. Therefore, the range of \\(f(x)\\) is all real numbers greater than or equal to the vertex of the parabola.\n\nThe vertex of the parabola is located at the point \\((0, -5)\\). This means that the minimum value of \\(f(x)\\) is -5.\n\nTherefore, the possible values of \\(y\\) are all real numbers greater than or equal to -5.\n\nThe answer is B) \\(y \\geq -5\\)." } }, { "answer":"A", "choices":{ "A":"40", "B":"160", "C":"320", "D":"1,280", "E":"2,560" }, "id":11798, "question":"A scientist found that the time, in seconds, a mouse required to complete a maze test successfully varied inversely with the number of hours of sleep the mouse had the previous night, such that time = \\(\\frac { 320 } { sleep } \\). If the mouse had 8 hours of sleep the night before a maze test, how long, in seconds, would the mouse take to complete the maze test?", "explanations":{ "correct":"To find the time it takes for the mouse to complete the maze test, we can substitute the given value of sleep into the equation: time = \\(\\frac{320}{sleep}\\).\n\nGiven that the mouse had 8 hours of sleep the night before the maze test, we can substitute sleep = 8 into the equation: time = \\(\\frac{320}{8}\\).\n\nSimplifying the equation, we have: time = 40.\n\nTherefore, the mouse would take 40 seconds to complete the maze test.\n\nThe answer is A) 40." } }, { "answer":"A", "choices":{ "A":"\\(\\frac { 1 } { 4 } \\)", "B":"\\(\\frac { 1 } { 2 } \\)", "C":"\\(\\frac { 2 } { 3 } \\)", "D":"\\(\\frac { 3 } { 4 } \\)", "E":"\\(\\frac { 4 } { 5 } \\)" }, "id":11799, "question":"If \\(\\frac { m + m } { m \\times m } = 8\\) and \\(m \\neq 0\\), what is the value of \\(m\\)?", "explanations":{ "correct":"To find the value of \\(m\\), we can start by simplifying the given equation: \n\n\\(\\frac{m + m}{m \\times m} = 8\\)\n\nSimplifying the numerator, we have:\n\n\\(\\frac{2m}{m \\times m} = 8\\)\n\nNext, we can simplify the denominator:\n\n\\(\\frac{2m}{m^2} = 8\\)\n\nTo simplify further, we can multiply both sides of the equation by \\(m^2\\) to eliminate the denominator:\n\n\\(2m = 8m^2\\)\n\nRearranging the equation, we have:\n\n\\(8m^2 - 2m = 0\\)\n\nFactoring out \\(2m\\), we get:\n\n\\(2m(4m - 1) = 0\\)\n\nSetting each factor equal to zero, we have two possible solutions:\n\n\\(2m = 0\\) or \\(4m - 1 = 0\\)\n\nFor the first equation, if \\(2m = 0\\), then \\(m = 0\\). However, the given condition states that \\(m \\neq 0\\), so this solution is not valid.\n\nFor the second equation, if \\(4m - 1 = 0\\), then \\(4m = 1\\) and \\(m = \\frac{1}{4}\\).\n\nTherefore, the value of \\(m\\) is \\(\\frac{1}{4}\\).\n\nThe answer is A) \\(\\frac{1}{4}\\)." } }, { "answer":"C", "choices":{ "A":"16", "B":"24", "C":"32", "D":"48", "E":"96" }, "id":11803, "question":"\\begin { gather* } a, ar, ar^2, ar^3,...\\end { gather* } In the sequence above, the first term is \\(a\\) and each term after the first is obtained by multiplying the preceding term by \\(r\\). If the third and fifth terms in sequence are 2 and 8, what is the value of the 7th term?", "explanations":{ "correct":"To find the value of the 7th term in the sequence, we need to determine the common ratio (r) and the first term (a).\n\nGiven that the third term is 2 and the fifth term is 8, we can set up two equations using the formula for the nth term of a geometric sequence:\n\n\\(ar^2 = 2\\) (equation 1)\n\\(ar^4 = 8\\) (equation 2)\n\nTo eliminate the variable 'a', we can divide equation 2 by equation 1:\n\n\\(\\frac{{ar^4}}{{ar^2}} = \\frac{8}{2}\\)\n\\(r^2 = 4\\)\n\nTaking the square root of both sides, we find:\n\n\\(r = \\pm 2\\)\n\nSince the common ratio cannot be negative (as it would result in negative terms), we can conclude that \\(r = 2\\).\n\nNow, substituting the value of 'r' into equation 1, we can solve for 'a':\n\n\\(a(2)^2 = 2\\)\n\\(4a = 2\\)\n\\(a = \\frac{1}{2}\\)\n\nSo, the first term (a) is \\(\\frac{1}{2}\\) and the common ratio (r) is 2.\n\nTo find the 7th term, we can use the formula for the nth term of a geometric sequence:\n\n\\(a_n = ar^{n-1}\\)\n\nSubstituting the values we found:\n\n\\(a_7 = \\frac{1}{2} \\cdot 2^{7-1}\\)\n\\(a_7 = \\frac{1}{2} \\cdot 2^6\\)\n\\(a_7 = \\frac{1}{2} \\cdot 64\\)\n\\(a_7 = 32\\)\n\nTherefore, the value of the 7th term in the sequence is 32.\n\nThe answer is C) 32." } }, { "answer":"E", "choices":{ "A":"6", "B":"12", "C":"54", "D":"108", "E":"162" }, "id":11805, "question":"For all numbers a and b, let a \\(\\Phi b = a^2b + ab^2\\). What is the value of \\((1 \\Phi 2) \\Phi 3\\)?", "explanations":{ "correct":"To find the value of \\((1 \\Phi 2) \\Phi 3\\), we need to substitute the values of \\(a\\) and \\(b\\) into the expression \\(a^2b + ab^2\\) step by step.\n\nFirst, let's find the value of \\(1 \\Phi 2\\):\n\\(1 \\Phi 2 = 1^2 \\cdot 2 + 1 \\cdot 2^2\\)\nSimplifying this expression, we get:\n\\(1 \\Phi 2 = 2 + 4 = 6\\)\n\nNow, we substitute the value of \\(1 \\Phi 2\\) into the expression \\((1 \\Phi 2) \\Phi 3\\):\n\\((1 \\Phi 2) \\Phi 3 = 6 \\Phi 3\\)\n\nNext, let's find the value of \\(6 \\Phi 3\\):\n\\(6 \\Phi 3 = 6^2 \\cdot 3 + 6 \\cdot 3^2\\)\nSimplifying this expression, we get:\n\\(6 \\Phi 3 = 36 \\cdot 3 + 6 \\cdot 9\\)\n\\(6 \\Phi 3 = 108 + 54 = 162\\)\n\nTherefore, the value of \\((1 \\Phi 2) \\Phi 3\\) is 162.\n\nThe answer is E) 162." } }, { "answer":"C", "choices":{ "A":"\\(\\frac{1 + 3\\sqrt{3}}{27}\\)", "B":"\\(\\frac{1 + 3\\sqrt{3}}{81}\\)", "C":"\\(\\frac{1 + 81\\sqrt{3}}{27}\\)", "D":"\\(1 + 27\\sqrt{3}\\)", "E":"\\(27 + 81\\sqrt{3}\\)" }, "id":11809, "question":"If \\(f(x) = x^{-3} + x^{\\frac{3}{2}}\\), what is the value of \\(f(3)\\)?", "explanations":{ "correct":"To find the value of \\(f(3)\\), we need to substitute \\(x = 3\\) into the function \\(f(x) = x^{-3} + x^{\\frac{3}{2}}\\).\n\nStep 1: Substitute \\(x = 3\\) into the function:\n\\(f(3) = 3^{-3} + 3^{\\frac{3}{2}}\\)\n\nStep 2: Simplify the exponents:\n\\(f(3) = \\frac{1}{3^3} + 3^{\\frac{3}{2}}\\)\n\nStep 3: Evaluate the exponents:\n\\(f(3) = \\frac{1}{27} + 3^{\\frac{3}{2}}\\)\n\nStep 4: Simplify the square root:\n\\(f(3) = \\frac{1}{27} + \\sqrt{3^3}\\)\n\nStep 5: Evaluate the cube:\n\\(f(3) = \\frac{1}{27} + \\sqrt{27}\\)\n\nStep 6: Simplify the square root:\n\\(f(3) = \\frac{1}{27} + 3\\sqrt{3}\\)\n\nTherefore, the value of \\(f(3)\\) is \\(\\frac{1}{27} + 3\\sqrt{3}\\).\n\nThe answer is C) \\(\\frac{1 + 81\\sqrt{3}}{27}\\)." } }, { "answer":"B", "choices":{ "A":"\\(16\\pi\\)", "B":"\\(17\\pi\\)", "C":"\\(18\\pi\\)", "D":"\\(144\\pi\\)", "E":"\\(1156\\pi\\)" }, "id":11813, "question":"If A (2, -1) and B (4, 7) are the endpoints of a diameter of a circle, what is the area of the circle?", "explanations":{ "correct":"To find the area of a circle, we need to know the radius of the circle. Since we are given the endpoints of a diameter, we can find the length of the diameter and then divide it by 2 to get the radius.\n\nUsing the distance formula, we can find the length of the diameter AB:\n\n\\(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\)\n\nSubstituting the coordinates of A (2, -1) and B (4, 7):\n\n\\(d = \\sqrt{(4 - 2)^2 + (7 - (-1))^2}\\)\n\nSimplifying:\n\n\\(d = \\sqrt{2^2 + 8^2}\\)\n\n\\(d = \\sqrt{4 + 64}\\)\n\n\\(d = \\sqrt{68}\\)\n\nNow, we can find the radius by dividing the diameter by 2:\n\n\\(r = \\frac{d}{2} = \\frac{\\sqrt{68}}{2} = \\frac{2\\sqrt{17}}{2} = \\sqrt{17}\\)\n\nFinally, we can find the area of the circle using the formula:\n\n\\(A = \\pi r^2\\)\n\nSubstituting the value of the radius:\n\n\\(A = \\pi (\\sqrt{17})^2\\)\n\n\\(A = \\pi \\cdot 17\\)\n\nTherefore, the area of the circle is \\(17\\pi\\).\n\nThe answer is B." } }, { "answer":"D", "choices":{ "A":"5", "B":"10", "C":"\\(\\frac { 5 } { a } \\)", "D":"\\(\\frac { 10 } { a } \\)", "E":"\\(\\frac { a } { 10 } \\)" }, "id":11816, "question":"If it takes a people 10 hours to finish a certain task, how many would it take for \\(a^2\\) people to finish the same task?", "explanations":{ "correct":"To solve this problem, we need to understand the relationship between the number of people and the time it takes to complete a task. \n\nLet's assume that it takes 1 person 10 hours to finish the task. This means that the rate at which the person completes the task is 1 task per 10 hours, or \\(\\frac { 1 } { 10 } \\) tasks per hour.\n\nNow, we need to find out how many people it would take for \\(a^2\\) people to finish the same task. Since each person takes 10 hours to complete the task, we can multiply the rate of 1 person by the number of people, which is \\(a^2\\). \n\nThe rate at which \\(a^2\\) people complete the task is \\(\\frac { 1 } { 10 } \\) tasks per hour multiplied by \\(a^2\\) people, which gives us \\(\\frac { a^2 } { 10 } \\) tasks per hour.\n\nTo find out how many hours it would take for \\(a^2\\) people to finish the task, we can take the reciprocal of the rate. The reciprocal of \\(\\frac { a^2 } { 10 } \\) is \\(\\frac { 10 } { a^2 } \\).\n\nTherefore, the answer is D) \\(\\frac { 10 } { a } \\)." } }, { "answer":"C", "choices":{ "A":"1", "B":"2", "C":"3", "D":"4", "E":"5" }, "id":11821, "question":"If A is the set of positive odd integers less than 9, and B is the set of prime numbers, how many integers will be in both sets?", "explanations":{ "correct":"To find the number of integers that will be in both sets A and B, we need to determine the elements that are common to both sets.\n\nSet A consists of positive odd integers less than 9. The positive odd integers less than 9 are 1, 3, 5, and 7.\n\nSet B consists of prime numbers. The prime numbers less than 9 are 2, 3, 5, and 7.\n\nComparing the two sets, we can see that the integers 3, 5, and 7 are common to both sets A and B.\n\nTherefore, the answer is C) 3." } }, { "answer":"B", "choices":{ "A":"1", "B":"2", "C":"3", "D":"4", "E":"5" }, "id":11822, "question":"If \\(8 \\times a^2x = a^ { 2x + 3 } \\) , what is the value of \\(a\\) ?", "explanations":{ "correct":"To find the value of \\(a\\), we need to solve the equation \\(8 \\times a^2x = a^{2x + 3}\\).\n\nFirst, let's simplify the equation by dividing both sides by \\(a^{2x}\\):\n\\(\\frac{8 \\times a^{2x}}{a^{2x}} = \\frac{a^{2x + 3}}{a^{2x}}\\).\n\nThis simplifies to:\n\\(8 = a^3\\).\n\nTo find the value of \\(a\\), we need to find the cube root of 8. The only integer value that satisfies this is \\(a = 2\\).\n\nTherefore, the answer is B) 2." } }, { "answer":"D", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III only", "E":"II and III only" }, "id":11825, "question":"If \\(\\sqrt { x + p } = \\sqrt { x - q } \\), which of the following could be true? \\(\\newline\\)I. \\(p = 0\\) and \\(q = 0\\) \\(\\newline\\)II. \\(pq = 0\\) \\(\\newline\\)III. \\(p + q = 0\\)", "explanations":{ "correct":"To determine which of the given options could be true, let's start by squaring both sides of the equation \\\\(\\\\sqrt { x + p } = \\\\sqrt { x - q }\\\\):\n\n\\\\((\\\\sqrt { x + p })^2 = (\\\\sqrt { x - q })^2\\\\)\n\nSimplifying, we get:\n\n\\\\(x + p = x - q\\\\)\n\nNext, let's isolate the variables on one side of the equation:\n\n\\\\(p = -q\\\\)\n\nNow, let's analyze each option:\n\\(\\newline\\)I. \\\\(p = 0\\\\) and \\\\(q = 0\\\\): If both \\\\(p\\\\) and \\\\(q\\\\) are zero, then \\\\(p = -q\\\\) is true. Therefore, option I could be true.\n\\(\\newline\\)II. \\\\(pq = 0\\\\): If either \\\\(p\\\\) or \\\\(q\\\\) is zero, then \\\\(pq = 0\\\\) is true. However, we cannot conclude that \\\\(pq = 0\\\\) is always true based on the given equation. Therefore, option II may not be true.\n\\(\\newline\\)III. \\\\(p + q = 0\\\\): From the equation \\\\(p = -q\\\\), we can see that \\\\(p + q = 0\\\\) is true. Therefore, option III could be true.\n\nBased on our analysis, options I and III could be true. Therefore, the answer is D) I and III only." } }, { "answer":"B", "choices":{ "A":"5", "B":"10", "C":"15", "D":"20", "E":"25" }, "id":11827, "question":"The average (arithmetic mean) of \\(2a\\) and \\(b\\) is 15 and the average of \\(2a\\),\\(b\\), and \\(3c\\) is 20. What is the value of \\(c\\)?", "explanations":{ "correct":"To find the value of \\\\(c\\\\), we need to use the information given in the problem.\n\nFirst, we are told that the average of \\\\(2a\\\\) and \\\\(b\\\\) is 15. This means that the sum of \\\\(2a\\\\) and \\\\(b\\\\) divided by 2 is equal to 15. We can write this as:\n\n\\\\(\\frac{{2a + b}}{2} = 15\\\\)\n\nSimplifying this equation, we get:\n\n\\\\(2a + b = 30\\\\) (Equation 1)\n\nNext, we are told that the average of \\\\(2a\\\\), \\\\(b\\\\), and \\\\(3c\\\\) is 20. This means that the sum of \\\\(2a\\\\), \\\\(b\\\\), and \\\\(3c\\\\) divided by 3 is equal to 20. We can write this as:\n\n\\\\(\\frac{{2a + b + 3c}}{3} = 20\\\\)\n\nSimplifying this equation, we get:\n\n\\\\(2a + b + 3c = 60\\\\) (Equation 2)\n\nNow, we have two equations with two variables. We can solve this system of equations to find the values of \\\\(a\\\\), \\\\(b\\\\), and \\\\(c\\\\).\n\nSubtracting Equation 1 from Equation 2, we get:\n\n\\\\((2a + b + 3c) - (2a + b) = 60 - 30\\\\)\n\nSimplifying this equation, we get:\n\n\\\\(3c = 30\\\\)\n\nDividing both sides of the equation by 3, we find:\n\n\\\\(c = 10\\\\)\n\nTherefore, the value of \\\\(c\\\\) is 10.\n\nThe answer is B) 10." } }, { "answer":"E", "choices":{ "A":"A circle with a radius of \\(4\\pi\\)", "B":"A circle with a circumference of \\(32\\pi\\)", "C":"A circle with a radius of 8", "D":"A circle with a diameter of 4", "E":"A circle with a circumference of \\(8\\pi\\)" }, "id":11828, "question":"Which of the following has the same area as a circle with an area of \\(16\\pi\\)?", "explanations":{ "correct":"To determine which of the given options has the same area as a circle with an area of \\(16\\pi\\), we need to compare the areas of each option to \\(16\\pi\\).\n\nThe area of a circle is given by the formula \\(A = \\pi r^2\\), where \\(A\\) represents the area and \\(r\\) represents the radius.\n\nLet's evaluate each option:\n\nA) A circle with a radius of \\(4\\pi\\):\nThe area of this circle would be \\(A = \\pi (4\\pi)^2 = 16\\pi^3\\).\n\nB) A circle with a circumference of \\(32\\pi\\):\nThe circumference of a circle is given by the formula \\(C = 2\\pi r\\), where \\(C\\) represents the circumference and \\(r\\) represents the radius. Solving for \\(r\\), we have \\(r = \\frac{C}{2\\pi} = \\frac{32\\pi}{2\\pi} = 16\\). Therefore, the radius of this circle is 16, and its area would be \\(A = \\pi (16)^2 = 256\\pi\\).\n\nC) A circle with a radius of 8:\nThe area of this circle would be \\(A = \\pi (8)^2 = 64\\pi\\).\n\nD) A circle with a diameter of 4:\nThe radius of this circle would be half of the diameter, so \\(r = \\frac{4}{2} = 2\\). The area of this circle would be \\(A = \\pi (2)^2 = 4\\pi\\).\n\nE) A circle with a circumference of \\(8\\pi\\):\nUsing the same reasoning as in option B, we find that the radius of this circle is 4, and its area would be \\(A = \\pi (4)^2 = 16\\pi\\).\n\nComparing the areas of each option to \\(16\\pi\\), we can see that the only option with the same area is option E. Therefore, the answer is E." } }, { "answer":"C", "choices":{ "A":"1", "B":"\\(\\frac { 3 } { 2 } \\)", "C":"2", "D":"\\(\\frac { 5 } { 2 } \\)", "E":"3" }, "id":11844, "question":"If \\(\\frac { a } { b } = 3\\), \\(\\frac { b } { c } = \\frac { 1 } { 2 } \\), and \\(\\frac { c } { d } = \\frac { 2 } { 3 } \\), what is the value of \\(\\frac { 2a } { d } \\) ?", "explanations":{ "correct":"To find the value of \\(\\frac { 2a } { d }\\), we need to substitute the given values into the expression.\n\nGiven:\n\\(\\frac { a } { b } = 3\\)\n\\(\\frac { b } { c } = \\frac { 1 } { 2 }\\)\n\\(\\frac { c } { d } = \\frac { 2 } { 3 }\\)\n\nFirst, let's find the value of \\(c\\) by substituting the second equation into the third equation:\n\\(\\frac { c } { d } = \\frac { 2 } { 3 }\\)\n\\(c = \\frac { 2d } { 3 }\\)\n\nNext, substitute the value of \\(c\\) into the second equation to find the value of \\(b\\):\n\\(\\frac { b } { c } = \\frac { 1 } { 2 }\\)\n\\(\\frac { b } { \\frac { 2d } { 3 } } = \\frac { 1 } { 2 }\\)\n\\(b = \\frac { 2d } { 6 }\\)\n\\(b = \\frac { d } { 3 }\\)\n\nFinally, substitute the value of \\(b\\) into the first equation to find the value of \\(a\\):\n\\(\\frac { a } { b } = 3\\)\n\\(\\frac { a } { \\frac { d } { 3 } } = 3\\)\n\\(a = \\frac { 3d } { 3 }\\)\n\\(a = d\\)\n\nNow, substitute the values of \\(a\\) and \\(d\\) into the expression \\(\\frac { 2a } { d }\\):\n\\(\\frac { 2a } { d } = \\frac { 2d } { d }\\)\n\\(\\frac { 2a } { d } = 2\\)\n\nTherefore, the value of \\(\\frac { 2a } { d }\\) is 2.\n\nThe answer is C) 2." } }, { "answer":"D", "choices":{ "A":"3", "B":"9", "C":"15", "D":"21", "E":"27" }, "id":11848, "question":"The average (arithmetic mean) of two numbers is equal to twice the positive difference between the two numbers. If the larger number is 35, what is the smaller number?", "explanations":{ "correct":"Let's solve this problem step by step. \n\nWe are given that the average of two numbers is equal to twice the positive difference between the two numbers. Let's call the smaller number x. \n\nThe average of the two numbers is (x + 35) / 2, and the positive difference between the two numbers is 35 - x. \n\nAccording to the given information, we can set up the equation: \n\n(x + 35) / 2 = 2(35 - x)\n\nTo solve for x, we can start by simplifying the equation: \n\nx + 35 = 4(35 - x)\n\nNext, distribute 4 on the right side of the equation: \n\nx + 35 = 140 - 4x\n\nCombine like terms: \n\n5x + 35 = 140\n\nSubtract 35 from both sides: \n\n5x = 105\n\nDivide both sides by 5: \n\nx = 21\n\nTherefore, the smaller number is 21. \n\nThe answer is D) 21." } }, { "answer":"E", "choices":{ "A":"April", "B":"June", "C":"September", "D":"November", "E":"December" }, "id":11853, "question":"If it is now September, what month will it be 555 months from now?", "explanations":{ "correct":"To determine the month that will be 555 months from now, we need to divide 555 by 12 to find out how many years it represents. \n\n555 divided by 12 equals 46 with a remainder of 3. This means that 555 months is equivalent to 46 years and 3 months.\n\nSince it is currently September, we need to add 3 months to September to find the month that will be 555 months from now.\n\nSeptember + 3 months = December\n\nTherefore, the month that will be 555 months from now is December.\n\nThe answer is E) December." } }, { "answer":"C", "choices":{ "A":"3", "B":"4", "C":"5", "D":"7", "E":"9" }, "id":11859, "question":"\\(x^2 + (k +1)x + 4 = (x + h)^2\\) In the equation above, \\(k\\) and \\(h\\) are positive constants. If the equation is true for all values of \\(x\\), what is the value of \\(k + h\\)?", "explanations":{ "correct":"To find the value of \\(k + h\\), we need to first expand \\((x + h)^2\\). \n\nExpanding \\((x + h)^2\\) gives us \\(x^2 + 2hx + h^2\\). \n\nNow, we can compare this to the given equation \\(x^2 + (k + 1)x + 4 = (x + h)^2\\). \n\nComparing the corresponding terms, we have:\n\nFor the \\(x^2\\) term: \\(x^2 = x^2\\)\n\nFor the \\(x\\) term: \\(k + 1 = 2h\\)\n\nFor the constant term: \\(4 = h^2\\)\n\nSince the equation is true for all values of \\(x\\), the corresponding coefficients of the terms on both sides of the equation must be equal. \n\nFrom the equation \\(k + 1 = 2h\\), we can solve for \\(k\\) in terms of \\(h\\):\n\n\\(k = 2h - 1\\)\n\nSubstituting this value of \\(k\\) into the equation \\(4 = h^2\\), we can solve for \\(h\\):\n\n\\(4 = h^2\\)\n\nTaking the square root of both sides, we get:\n\n\\(h = \\pm 2\\)\n\nSince \\(h\\) is a positive constant, we take the positive value \\(h = 2\\).\n\nSubstituting \\(h = 2\\) into the equation \\(k = 2h - 1\\), we can solve for \\(k\\):\n\n\\(k = 2(2) - 1\\)\n\n\\(k = 4 - 1\\)\n\n\\(k = 3\\)\n\nFinally, we can find the value of \\(k + h\\):\n\n\\(k + h = 3 + 2 = 5\\)\n\nTherefore, the value of \\(k + h\\) is 5.\n\nThe answer is C." } }, { "answer":"E", "choices":{ "A":"\\(9a^5 b^9\\)", "B":"\\(-9a^5 b^9\\)", "C":"\\(9a^6 b^{18}\\)", "D":"\\(27a^6 b^{18}\\)", "E":"\\(-27a^6 b^{18}\\)" }, "id":11862, "question":"\\((-3a^2 b^6)^3 =\\)", "explanations":{ "correct":"To simplify the expression \\((-3a^2 b^6)^3\\), we need to apply the exponent to each term inside the parentheses.\n\nFirst, let's focus on the coefficient \\(-3\\). Raising \\(-3\\) to the power of \\(3\\) gives us \\((-3)^3 = -27\\).\n\nNext, we apply the exponent \\(3\\) to each variable term. For \\(a^2\\), raising it to the power of \\(3\\) gives us \\(a^{2 \\cdot 3} = a^6\\). Similarly, for \\(b^6\\), raising it to the power of \\(3\\) gives us \\(b^{6 \\cdot 3} = b^{18}\\).\n\nPutting it all together, we have \\((-3a^2 b^6)^3 = -27a^6 b^{18}\\).\n\nTherefore, the answer is E) \\(-27a^6 b^{18}\\)." } }, { "answer":"E", "choices":{ "A":"3", "B":"2", "C":"0", "D":"-3", "E":"-4" }, "id":11864, "question":"If \\(7 + x < 5 \\)and \\(3 + x < 0\\), which of the following could be the value of \\(x\\) ?", "explanations":{ "correct":"To find the possible values of x that satisfy the given inequalities, we need to solve each inequality separately and then find the intersection of the solutions.\n\n1. \\(7 + x < 5\\):\nSubtracting 7 from both sides, we get \\(x < -2\\).\n\n2. \\(3 + x < 0\\):\nSubtracting 3 from both sides, we get \\(x < -3\\).\n\nNow, we need to find the intersection of the solutions. Since both inequalities have the condition \\(x < -3\\), the possible values of x that satisfy both inequalities are any number less than -3.\n\nLooking at the answer choices:\nA) 3: This is not less than -3.\nB) 2: This is not less than -3.\nC) 0: This is not less than -3.\nD) -3: This is not less than -3.\nE) -4: This is less than -3.\n\nTherefore, the only value that could be the value of x is -4.\n\nThe answer is E." } }, { "answer":"A", "choices":{ "A":"3", "B":"4", "C":"5", "D":"6", "E":"8" }, "id":11869, "question":"A bag contains two white marbles, four blue marbles, and an unknown number of black marbles. When one marble is drawn from the bag at random, the probability that the marble is black will be \\(\\frac { 1 } { 3 } \\). How many black marbles are in the bag?", "explanations":{ "correct":"Let's assume that the number of black marbles in the bag is \\(x\\).\n\nThe total number of marbles in the bag is \\(2 + 4 + x = 6 + x\\).\n\nThe probability of drawing a black marble is given as \\(\\frac{1}{3}\\).\n\nTo find the probability, we need to divide the number of black marbles by the total number of marbles in the bag:\n\n\\(\\frac{x}{6 + x} = \\frac{1}{3}\\).\n\nTo solve this equation, we can cross-multiply:\n\n\\(3x = 6 + x\\).\n\nSimplifying the equation, we get:\n\n\\(2x = 6\\).\n\nDividing both sides by 2, we find:\n\n\\(x = 3\\).\n\nTherefore, the number of black marbles in the bag is 3.\n\nThe answer is A) 3." } }, { "answer":"A", "choices":{ "A":"Two", "B":"Five", "C":"Seven", "D":"Eight", "E":"Ten" }, "id":11873, "question":"Set A consists of all even numbers between 10 and 20, inclusive. Set B consists of all multiples of 3 between 7 and 19, inclusive. If set C consists of all the numbers that appear both in set A and set B, how many members does set C have?", "explanations":{ "correct":"To find the number of members in set C, we need to determine the numbers that appear in both set A and set B.\n\nSet A consists of all even numbers between 10 and 20, inclusive. The even numbers in this range are 10, 12, 14, 16, 18, and 20.\n\nSet B consists of all multiples of 3 between 7 and 19, inclusive. The multiples of 3 in this range are 9, 12, 15, and 18.\n\nComparing the two sets, we can see that the numbers 12 and 18 appear in both set A and set B.\n\nTherefore, set C consists of the numbers 12 and 18, and it has 2 members.\n\nThe answer is A) Two." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I, II and III" }, "id":11882, "question":"If \\(p\\) and \\(q\\) are positive numbers and \\(\\frac { p } { q } < pq\\), which of the following must be true? \\(\\newline\\)I. \\(p > 1\\) \\(\\newline\\)II. \\(q > 1\\) \\(\\newline\\)III. \\(p > q\\)", "explanations":{ "correct":"To determine which statements must be true, let's analyze the given inequality: \n\n\\(\\frac{p}{q} < pq\\)\n\nFirst, we can multiply both sides of the inequality by \\(q\\) to eliminate the fraction:\n\n\\(p < pq^2\\)\n\nNext, we can divide both sides of the inequality by \\(p\\) (since \\(p\\) is positive) to isolate \\(q\\):\n\n\\(\\frac{p}{p} < \\frac{pq^2}{p}\\)\n\nSimplifying, we have:\n\n\\(1 < q^2\\)\n\nTaking the square root of both sides, we get:\n\n\\(\\sqrt{1} < \\sqrt{q^2}\\)\n\nSimplifying further, we have:\n\n\\(1 < q\\)\n\nFrom this, we can conclude that statement II, \\(q > 1\\), must be true. \n\nHowever, we cannot determine the values of \\(p\\) and \\(q\\) solely from the given inequality. For example, if \\(p = 0.5\\) and \\(q = 2\\), the inequality is satisfied, but \\(p\\) is not greater than \\(1\\) and \\(p\\) is not greater than \\(q\\). Therefore, statements I and III cannot be guaranteed to be true.\n\\(\\newline\\)In conclusion, the answer is B) II only." } }, { "answer":"D", "choices":{ "A":"20", "B":"40", "C":"60", "D":"80", "E":"100" }, "id":11883, "question":"\\begin { gather* } 4, 8, 12, 16, ...\\end { gather* } The first term in the sequence is 4, and each term after the first is determined by adding 4. What is the value of the 20th term?", "explanations":{ "correct":"To find the value of the 20th term in the sequence, we need to determine the pattern and apply it to find the desired term.\n\nGiven that the first term is 4 and each term after the first is determined by adding 4, we can see that the sequence is an arithmetic sequence with a common difference of 4.\n\nTo find the value of the 20th term, we can use the formula for the nth term of an arithmetic sequence:\n\n\\[a_n = a_1 + (n-1)d\\]\n\nwhere \\(a_n\\) is the nth term, \\(a_1\\) is the first term, \\(n\\) is the term number, and \\(d\\) is the common difference.\n\nPlugging in the values we know, we have:\n\n\\[a_{20} = 4 + (20-1) \\cdot 4\\]\n\nSimplifying the expression inside the parentheses:\n\n\\[a_{20} = 4 + 19 \\cdot 4\\]\n\n\\[a_{20} = 4 + 76\\]\n\n\\[a_{20} = 80\\]\n\nTherefore, the value of the 20th term in the sequence is 80.\n\nThe answer is D) 80." } }, { "answer":"E", "choices":{ "A":"30", "B":"40", "C":"50", "D":"60", "E":"70" }, "id":11884, "question":"If one angle in a right triangle is \\(20\\degree\\), which of the following is the degree measure of another angle in the triangle?", "explanations":{ "correct":"To find the degree measure of another angle in a right triangle, we need to consider the fact that the sum of the angles in any triangle is always 180 degrees.\n\\(\\newline\\)In a right triangle, one angle is always 90 degrees. Given that one angle is 20 degrees, we can subtract the sum of these two angles from 180 degrees to find the measure of the third angle.\n\n90 degrees (right angle) + 20 degrees = 110 degrees\n\n180 degrees - 110 degrees = 70 degrees\n\nTherefore, the degree measure of another angle in the triangle is 70 degrees.\n\nThe answer is E) 70." } }, { "answer":"D", "choices":{ "A":"\\(2pn\\)", "B":"\\(3.5pn\\)", "C":"\\(3.5(n + p)\\)", "D":"\\(3n + 0.50p\\)", "E":"\\(2(n + 0.50p)\\)" }, "id":11885, "question":"If pencils cost \\$ 0.50 each and notebooks cost \\$ 3 each, which of the following represents the cost, in dollars, of p pencils and n notebooks?", "explanations":{ "correct":"To find the cost of p pencils and n notebooks, we need to multiply the cost of each item by the number of items and then add the results together.\n\nThe cost of p pencils is \\(0.50 \\times p = 0.50p\\).\nThe cost of n notebooks is \\(3 \\times n = 3n\\).\n\nAdding these two costs together, we get \\(0.50p + 3n\\).\n\nTherefore, the correct answer is D) \\(3n + 0.50p\\)." } }, { "answer":"D", "choices":{ "A":"\\(9a^3b^6\\)", "B":"\\(9a^6b^9\\)", "C":"\\(27a^3b^6\\)", "D":"\\(27a^6b^9\\)", "E":"\\(27a^8b^27\\)" }, "id":11888, "question":"\\((3a^2b^3)^3\\) =", "explanations":{ "correct":"To simplify the expression \\((3a^2b^3)^3\\), we need to apply the exponent to each term inside the parentheses.\n\nStep 1: Apply the exponent to the coefficient 3:\n\\((3^3a^2b^3)^3\\)\n\nStep 2: Apply the exponent to each variable term:\n\\(3^3a^{2 \\cdot 3}b^{3 \\cdot 3}\\)\n\nSimplifying further:\n\\(27a^6b^9\\)\n\nTherefore, the answer is D) \\(27a^6b^9\\)." } }, { "answer":"E", "choices":{ "A":"12", "B":"120", "C":"144", "D":"1,200", "E":"1,440" }, "id":11894, "question":"If \\(10 + \\sqrt { x } - 154\\), then \\(10\\sqrt { x } \\) =", "explanations":{ "correct":"To solve this problem, we need to start by isolating the term with the square root, \\\\(\\sqrt{x}\\\\).\n\nGiven that \\\\(10 + \\sqrt{x} - 154\\\\), we can rearrange the equation to isolate \\\\(\\sqrt{x}\\\\) by subtracting 10 from both sides:\n\n\\\\(\\sqrt{x} = 154 - 10\\\\)\n\nSimplifying the right side of the equation:\n\n\\\\(\\sqrt{x} = 144\\\\)\n\nTo find the value of \\\\(10\\sqrt{x}\\\\), we can substitute the value of \\\\(\\sqrt{x}\\\\) into the equation:\n\n\\\\(10\\sqrt{x} = 10 \\times 144\\\\)\n\nSimplifying the right side of the equation:\n\n\\\\(10\\sqrt{x} = 1,440\\\\)\n\nTherefore, the answer is E) 1,440." } }, { "answer":"A", "choices":{ "A":"9", "B":"12", "C":"26", "D":"45", "E":"63" }, "id":11899, "question":"Tameka cleans her house every 7 days and does laundry every 5 days. In the next 315 days, how many times will she have to clean her house and do laundry on the same day?", "explanations":{ "correct":"To find out how many times Tameka will have to clean her house and do laundry on the same day in the next 315 days, we need to find the least common multiple (LCM) of 7 and 5.\n\nThe LCM of 7 and 5 is the smallest number that is divisible by both 7 and 5. To find the LCM, we can list the multiples of each number until we find a common multiple.\n\nMultiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, ...\n\nMultiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, ...\n\nFrom the lists above, we can see that the first common multiple of 7 and 5 is 35. Therefore, Tameka will have to clean her house and do laundry on the same day every 35 days.\n\nTo find out how many times this will happen in the next 315 days, we divide 315 by 35:\n\n315 ÷ 35 = 9\n\nTherefore, Tameka will have to clean her house and do laundry on the same day 9 times in the next 315 days.\n\nThe answer is A) 9." } }, { "answer":"C", "choices":{ "A":"4", "B":"5", "C":"6", "D":"7", "E":"8" }, "id":11902, "question":"The members of the French Club conducted a fund-raising drive. The average (arithmetic mean) amount of money raised per member was \\$ 85. Then Jean joined the club and raised \\$ 50. This lowered the average to \\$ 80. How many members were there before Jean joined?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nLet's assume that there were \"x\" members in the French Club before Jean joined.\n\nBefore Jean joined, the total amount of money raised by the members was 85x dollars.\n\nAfter Jean joined, the total amount of money raised by all the members became 85x + 50 dollars.\n\nWe are given that the average amount of money raised per member decreased to 80 dollars after Jean joined. So, we can set up the following equation:\n\n(85x + 50) / (x + 1) = 80\n\nTo solve this equation, we can cross-multiply:\n\n85x + 50 = 80(x + 1)\n\n85x + 50 = 80x + 80\n\nSubtracting 80x from both sides:\n\n5x + 50 = 80\n\nSubtracting 50 from both sides:\n\n5x = 30\n\nDividing both sides by 5:\n\nx = 6\n\nTherefore, there were 6 members in the French Club before Jean joined.\n\nThe answer is C) 6." } }, { "answer":"B", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"II and III only", "E":"I, II and III only" }, "id":11904, "question":"If \\(x > 1\\), which of the following increases as \\(x\\) increases? \\(\\newline\\)I. \\(1/x\\) \\(\\newline\\)II. \\((x-1)/x\\) \\(\\newline\\)III. \\(1/x^2\\)", "explanations":{ "correct":"To determine which of the given expressions increases as \\(x\\) increases, we can analyze each expression separately.\n\\(\\newline\\)I. \\(\\frac{1}{x}\\): As \\(x\\) increases, the denominator \\(x\\) becomes larger, resulting in a smaller fraction. Therefore, \\(\\frac{1}{x}\\) decreases as \\(x\\) increases.\n\\(\\newline\\)II. \\(\\frac{x-1}{x}\\): As \\(x\\) increases, both the numerator \\(x-1\\) and the denominator \\(x\\) become larger. However, the numerator increases at a faster rate than the denominator, resulting in a larger fraction. Therefore, \\(\\frac{x-1}{x}\\) increases as \\(x\\) increases.\n\\(\\newline\\)III. \\(\\frac{1}{x^2}\\): As \\(x\\) increases, the denominator \\(x^2\\) becomes larger, resulting in a smaller fraction. Therefore, \\(\\frac{1}{x^2}\\) decreases as \\(x\\) increases.\n\nBased on our analysis, only expression II (\\(\\frac{x-1}{x}\\)) increases as \\(x\\) increases. Therefore, the answer is B) II only." } }, { "answer":"A", "choices":{ "A":"13", "B":"15", "C":"17", "D":"85", "E":"115" }, "id":11905, "question":"If the price of a certain product is increased by 15 percent, by approximately what percent must the new price be reduced to obtain the original price?", "explanations":{ "correct":"To solve this problem, we need to find the percent by which the new price must be reduced to obtain the original price after a 15 percent increase.\n\nLet's assume the original price is 100 dollars. \n\\(\\newline\\)If the price is increased by 15 percent, the new price would be 100 + (15% of 100) = 100 + 15 = 115 dollars.\n\nTo find the percent by which the new price must be reduced to obtain the original price, we need to find the difference between the new price and the original price, and then express it as a percentage of the new price.\n\nThe difference between the new price (115 dollars) and the original price (100 dollars) is 115 - 100 = 15 dollars.\n\nTo express this difference as a percentage of the new price (115 dollars), we divide the difference by the new price and multiply by 100.\n\n(15/115) * 100 ≈ 13.04\n\nTherefore, the new price must be reduced by approximately 13 percent to obtain the original price.\n\nThe answer is A) 13." } }, { "answer":"A", "choices":{ "A":"\\(-7\\)", "B":"\\(-3\\)", "C":"0", "D":"4", "E":"14" }, "id":11906, "question":"If \\(x^2 - 49 = 0\\), which of the following could be a value of x?", "explanations":{ "correct":"To find the possible values of \\(x\\) that satisfy the equation \\(x^2 - 49 = 0\\), we can solve the equation by factoring.\n\nStep 1: Rewrite the equation as \\((x - 7)(x + 7) = 0\\).\n\nStep 2: Set each factor equal to zero and solve for \\(x\\):\n\\(x - 7 = 0\\) or \\(x + 7 = 0\\).\n\nFor the first equation, adding 7 to both sides gives \\(x = 7\\).\nFor the second equation, subtracting 7 from both sides gives \\(x = -7\\).\n\nTherefore, the possible values of \\(x\\) that satisfy the equation \\(x^2 - 49 = 0\\) are \\(x = 7\\) and \\(x = -7\\).\n\nAmong the given options, the value of \\(x\\) that could be a solution is \\(-7\\).\n\nThe answer is A)." } }, { "answer":"B", "choices":{ "A":"6", "B":"9", "C":"16.2", "D":"18", "E":"27" }, "id":11908, "question":"Peter's income is directly proportional to the amount of time he works. If he earns \\$ 27.00 for every 90 minutes of work, how many hours must he work to earn \\$ 162.00 ?", "explanations":{ "correct":"To find the number of hours Peter must work to earn \\$ 162.00, we need to set up a proportion using the given information.\n\nWe know that Peter earns \\$ 27.00 for every 90 minutes of work. To convert this to hours, we divide 90 minutes by 60 (since there are 60 minutes in an hour). This gives us 1.5 hours.\n\nSo, Peter earns \\$ 27.00 for every 1.5 hours of work.\n\nNow, we can set up the proportion:\n\n$ 27.00 / 1.5 hours = \\$ 162.00 / x hours\n\nTo solve for x, we cross-multiply:\n\n$ 27.00 * x hours = \\$ 162.00 * 1.5 hours\n\n27x = 243\n\nDividing both sides of the equation by 27, we get:\n\nx = 9\n\nTherefore, Peter must work 9 hours to earn \\$ 162.00.\n\nThe answer is B) 9." } }, { "answer":"D", "choices":{ "A":"\\(-9\\)", "B":"\\(-\\sqrt[3]{9}\\)", "C":"\\(\\sqrt[3]{9}\\)", "D":"3", "E":"9" }, "id":11910, "question":"If \\(-a^3 + b^2 = -a^3 + 9\\), then b could equal which of the following?", "explanations":{ "correct":"To find the possible values of \\(b\\), we need to solve the equation \\(-a^3 + b^2 = -a^3 + 9\\).\n\nFirst, we can simplify the equation by canceling out the \\(-a^3\\) terms on both sides:\n\n\\(b^2 = 9\\)\n\nTo find the possible values of \\(b\\), we need to take the square root of both sides of the equation:\n\n\\(\\sqrt{b^2} = \\sqrt{9}\\)\n\nThis simplifies to:\n\n\\(b = \\pm 3\\)\n\nTherefore, the possible values of \\(b\\) are 3 and -3.\n\nAmong the given options, the only value that matches the possible values of \\(b\\) is 3.\n\nSo, the answer is D) 3." } }, { "answer":"D", "choices":{ "A":"120", "B":"10", "C":"\\(\\frac{2}{5}\\)", "D":"\\(\\frac{1}{6}\\)", "E":"\\(\\frac{1}{12}\\)" }, "id":11911, "question":"If an object travels at 5 feet per minute, how many feet does it travel in 2 seconds?", "explanations":{ "correct":"To find out how many feet an object travels in 2 seconds, we need to convert the given rate from feet per minute to feet per second. \n\nSince there are 60 seconds in a minute, we can convert the rate of 5 feet per minute to feet per second by dividing it by 60. \n\n\\(5 \\text{ feet per minute} = \\frac{5}{60} \\text{ feet per second} = \\frac{1}{12} \\text{ feet per second}\\)\n\nNow, we can calculate how many feet the object travels in 2 seconds by multiplying the rate of \\(\\frac{1}{12}\\) feet per second by 2 seconds.\n\n\\(\\frac{1}{12} \\text{ feet per second} \\times 2 \\text{ seconds} = \\frac{1}{12} \\times 2 \\text{ feet} = \\frac{2}{12} \\text{ feet} = \\frac{1}{6} \\text{ feet}\\)\n\nTherefore, the object travels \\(\\frac{1}{6}\\) feet in 2 seconds.\n\nThe answer is D) \\(\\frac{1}{6}\\)." } }, { "answer":"B", "choices":{ "A":"3", "B":"5", "C":"7", "D":"8", "E":"12" }, "id":11919, "question":"A coffee shop sold 35 cups of coffee. Of these, 15 contained sugar and 12 contained milk. If 13 cups contained neither milk nor sugar, how many cups must have contained both sugar and milk?", "explanations":{ "correct":"To find the number of cups that contained both sugar and milk, we need to use the principle of inclusion-exclusion. \n\nFirst, we know that the total number of cups sold is 35. Out of these, 15 contained sugar and 12 contained milk. However, if we simply add these two numbers, we would be counting the cups that contained both sugar and milk twice. \n\nTo correct for this, we need to subtract the number of cups that contained both sugar and milk. Let's assume this number is x. \n\nSo, the number of cups that contained neither milk nor sugar is given as 13. \n\nUsing the principle of inclusion-exclusion, we can write the equation: \n\nTotal cups = Cups with sugar + Cups with milk - Cups with both sugar and milk + Cups with neither sugar nor milk\n\n35 = 15 + 12 - x + 13\n\nSimplifying the equation, we get: \n\n35 = 40 - x\n\nRearranging the equation to solve for x, we have: \n\nx = 40 - 35\n\nx = 5\n\nTherefore, the number of cups that must have contained both sugar and milk is 5.\n\nThe answer is B) 5." } }, { "answer":"E", "choices":{ "A":"\\(2\\pi\\)", "B":"\\(6\\pi\\)", "C":"\\(10\\pi\\)", "D":"\\(25\\pi\\)", "E":"\\(26\\pi\\)" }, "id":11921, "question":"What is the area of the circle whose center is at O (2, -2) and that passes through point P (3, 3)?", "explanations":{ "correct":"To find the area of a circle, we need to know the radius. The radius is the distance from the center of the circle to any point on the circle. In this case, we are given the center of the circle at point O (2, -2) and a point on the circle at P (3, 3).\n\nTo find the radius, we can use the distance formula. The distance formula is given by:\n\n\\(d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\)\n\nUsing the coordinates of O (2, -2) and P (3, 3), we can calculate the distance:\n\n\\(d = \\sqrt{(3 - 2)^2 + (3 - (-2))^2}\\)\n\nSimplifying:\n\n\\(d = \\sqrt{1^2 + 5^2}\\)\n\n\\(d = \\sqrt{1 + 25}\\)\n\n\\(d = \\sqrt{26}\\)\n\nSo, the radius of the circle is \\(\\sqrt{26}\\).\n\nThe formula for the area of a circle is given by:\n\n\\(A = \\pi r^2\\)\n\nSubstituting the value of the radius, we have:\n\n\\(A = \\pi (\\sqrt{26})^2\\)\n\nSimplifying:\n\n\\(A = \\pi \\cdot 26\\)\n\n\\(A = 26\\pi\\)\n\nTherefore, the area of the circle is \\(26\\pi\\).\n\nThe answer is E) \\(26\\pi\\)." } }, { "answer":"D", "choices":{ "A":"2.4", "B":"14", "C":"15", "D":"20", "E":"60" }, "id":11926, "question":"If \\(3x = 12\\), \\(5x =\\)", "explanations":{ "correct":"To find the value of \\\\(5x\\\\) when \\\\(3x = 12\\\\), we can start by solving the equation \\\\(3x = 12\\\\) for \\\\(x\\\\). \n\nDividing both sides of the equation by 3, we get:\n\n\\\\(x = \\\\frac{12}{3}\\\\)\n\nSimplifying the right side, we have:\n\n\\\\(x = 4\\\\)\n\nNow, we can substitute this value of \\\\(x\\\\) into the expression \\\\(5x\\\\) to find the answer:\n\n\\\\(5x = 5 \\cdot 4\\\\)\n\nMultiplying, we get:\n\n\\\\(5x = 20\\\\)\n\nTherefore, the answer is D) 20." } }, { "answer":"C", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and II only", "E":"I, II, and III only" }, "id":11928, "question":"If \\(x^2 + y^2 = (x - y)^2\\), which of the following statements must also be true? \\(\\newline\\)I. \\(x = 0\\) \\(\\newline\\)II. \\(y = 0\\) \\(\\newline\\)III. \\(xy = 0\\)", "explanations":{ "correct":"To determine which statements must be true, let's start by simplifying the given equation: \n\n\\(x^2 + y^2 = (x - y)^2\\)\n\nExpanding the right side of the equation:\n\n\\(x^2 + y^2 = x^2 - 2xy + y^2\\)\n\nRearranging the terms:\n\n\\(0 = -2xy\\)\n\nDividing both sides by -2:\n\n\\(0 = xy\\)\n\nFrom this equation, we can conclude that \\(xy\\) must be equal to 0. \n\nNow let's evaluate each statement:\n\\(\\newline\\)I. \\(x = 0\\)\\(\\newline\\)If \\(x = 0\\), then the equation \\(xy = 0\\) would be true. However, we cannot conclude that \\(x\\) must be 0 based on the given equation. For example, if \\(y = 0\\), then \\(x\\) can be any real number.\n\\(\\newline\\)II. \\(y = 0\\)\nSimilar to statement I, if \\(y = 0\\), then the equation \\(xy = 0\\) would be true. However, we cannot conclude that \\(y\\) must be 0 based on the given equation. For example, if \\(x = 0\\), then \\(y\\) can be any real number.\n\\(\\newline\\)III. \\(xy = 0\\)\nFrom our simplification of the given equation, we found that \\(xy\\) must be equal to 0. Therefore, this statement is true.\n\nBased on our analysis, the only statement that must be true is \\(\\newline\\)III. \n\nThe answer is C) III only." } }, { "answer":"C", "choices":{ "A":"\\(\\frac{t - r}{6}\\)", "B":"\\(6t - r\\)", "C":"\\(6t - 6r\\)", "D":"\\(12(t - r)\\)", "E":"36tr" }, "id":11930, "question":"Let the function f be defined by \\(f(t) = 6t\\) for all numbers t. Which of the following is equivalent to \\(f(t - r)\\)?", "explanations":{ "correct":"To find the expression equivalent to \\(f(t - r)\\), we substitute \\(t - r\\) into the function \\(f(t)\\).\n\nGiven that \\(f(t) = 6t\\), we substitute \\(t - r\\) into the function:\n\n\\(f(t - r) = 6(t - r)\\)\n\nTo simplify the expression, we distribute the 6:\n\n\\(f(t - r) = 6t - 6r\\)\n\nTherefore, the expression equivalent to \\(f(t - r)\\) is \\(6t - 6r\\).\n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"0", "B":"1", "C":"2", "D":"3", "E":"4" }, "id":11932, "question":"If \\(2^{2x - 3} = 8\\), then what is the value of x?", "explanations":{ "correct":"To find the value of x in the equation \\(2^{2x - 3} = 8\\), we need to solve for x.\n\nStep 1: Rewrite 8 as a power of 2. Since \\(2^3 = 8\\), we can rewrite the equation as \\(2^{2x - 3} = 2^3\\).\n\nStep 2: Set the exponents equal to each other. We have \\(2x - 3 = 3\\).\n\nStep 3: Solve for x. Add 3 to both sides of the equation to isolate x: \\(2x = 6\\). Then, divide both sides by 2: \\(x = 3\\).\n\nTherefore, the value of x is 3.\n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { m } { 50d } \\)", "B":"\\(\\frac { md } { 50 } \\)", "C":"\\(\\frac { md } { 2 } \\)", "D":"\\(\\frac { m } { 2d } \\)", "E":"\\(\\frac { 2m } { d } \\)" }, "id":11935, "question":"At Music Outlet the regular price for a CD is \\(d\\) dollars. How many CDs can be purchased for m dollars when the CDs are on sale at 50% off the regular price?", "explanations":{ "correct":"To find the number of CDs that can be purchased for m dollars when the CDs are on sale at 50% off the regular price, we need to determine the discounted price of each CD.\n\nSince the CDs are on sale at 50% off, the discounted price of each CD is 50% of the regular price, which is 0.5d dollars.\n\nTo find the number of CDs that can be purchased for m dollars, we divide m dollars by the discounted price of each CD, which is 0.5d dollars.\n\nTherefore, the number of CDs that can be purchased for m dollars is \\(\\frac{m}{0.5d}\\).\n\nSimplifying this expression, we get \\(\\frac{2m}{d}\\).\n\nHence, the answer is E) \\(\\frac{2m}{d}\\)." } }, { "answer":"E", "choices":{ "A":"\\(\\frac{15}{2}\\)", "B":"\\(\\frac{75}{2}\\)", "C":"32", "D":"38", "E":"40" }, "id":11940, "question":"If y is directly proportional to x and if \\(y = 30\\) when \\(x = 6\\), what is the value of y when \\(x = 8\\)?", "explanations":{ "correct":"To solve this problem, we need to use the concept of direct proportionality. In a direct proportion, as one variable increases, the other variable also increases by the same factor.\n\nGiven that \\(y\\) is directly proportional to \\(x\\), we can write the equation as \\(y = kx\\), where \\(k\\) is the constant of proportionality.\n\nWe are given that when \\(x = 6\\), \\(y = 30\\). Plugging these values into the equation, we get \\(30 = 6k\\). Solving for \\(k\\), we divide both sides of the equation by 6, which gives us \\(k = 5\\).\n\nNow that we know the value of \\(k\\), we can find the value of \\(y\\) when \\(x = 8\\). Plugging \\(x = 8\\) and \\(k = 5\\) into the equation \\(y = kx\\), we get \\(y = 5 \\times 8 = 40\\).\n\nTherefore, the value of \\(y\\) when \\(x = 8\\) is 40.\n\nThe answer is E) 40." } }, { "answer":"D", "choices":{ "A":"14", "B":"7", "C":"4", "D":"-3", "E":"-6" }, "id":11943, "question":"If \\(x - y = 10\\) and \\(x^2 - y^2 = 40\\), then y =", "explanations":{ "correct":"To solve this problem, we can use the difference of squares formula, which states that \\(a^2 - b^2 = (a + b)(a - b)\\). \n\nGiven that \\(x - y = 10\\) and \\(x^2 - y^2 = 40\\), we can rewrite the second equation using the difference of squares formula as \\((x + y)(x - y) = 40\\). \n\nSince we know that \\(x - y = 10\\), we can substitute this value into the equation to get \\((x + y)(10) = 40\\). \n\nDividing both sides of the equation by 10, we have \\(x + y = 4\\). \n\nNow we have a system of two equations: \\(x - y = 10\\) and \\(x + y = 4\\). \n\nAdding the two equations together, we get \\(2x = 14\\), which implies \\(x = 7\\). \n\nSubstituting the value of \\(x\\) into the equation \\(x + y = 4\\), we have \\(7 + y = 4\\). \n\nSolving for \\(y\\), we find \\(y = -3\\). \n\nTherefore, the answer is D) -3." } }, { "answer":"A", "choices":{ "A":"743", "B":"750", "C":"753", "D":"758", "E":"773" }, "id":11946, "question":"If in a certain sequence the first term is 8 and every subsequent term is determined by adding 15 to the previous term, what is the value of the 50th term?", "explanations":{ "correct":"To find the value of the 50th term in the sequence, we need to determine the pattern and apply it repeatedly.\n\nGiven that the first term is 8 and every subsequent term is determined by adding 15 to the previous term, we can write the sequence as follows:\n\nTerm 1: 8\nTerm 2: 8 + 15 = 23\nTerm 3: 23 + 15 = 38\nTerm 4: 38 + 15 = 53\nTerm 5: 53 + 15 = 68\n...\n\nWe can observe that each term is obtained by adding 15 to the previous term. Therefore, we can express the nth term as:\n\nTerm n = 8 + (n - 1) * 15\n\nTo find the 50th term, we substitute n = 50 into the formula:\n\nTerm 50 = 8 + (50 - 1) * 15\n = 8 + 49 * 15\n = 8 + 735\n = 743\n\nTherefore, the value of the 50th term in the sequence is 743.\n\nThe answer is A." } }, { "answer":"C", "choices":{ "A":"\\(a + b\\)", "B":"\\(a - b\\)", "C":"\\(a + b + 1\\)", "D":"ab", "E":"\\(ab + 3\\)" }, "id":11950, "question":"If a and b are multiples of 3, which of the following CANNOT also be a multiple of 3?", "explanations":{ "correct":"To determine which of the given options cannot be a multiple of 3, we need to consider the properties of multiples of 3.\n\nA multiple of 3 is any number that can be evenly divided by 3 without leaving a remainder. In other words, the sum of the digits of a multiple of 3 must also be divisible by 3.\n\nLet's analyze each option:\n\nA) \\(a + b\\):\nSince both a and b are multiples of 3, their sum will also be a multiple of 3. This is because adding two multiples of 3 will always result in a number that is divisible by 3.\n\nB) \\(a - b\\):\nSimilar to option A, subtracting two multiples of 3 will also result in a number that is divisible by 3. Therefore, this option can be a multiple of 3.\n\nC) \\(a + b + 1\\):\nAdding 1 to the sum of two multiples of 3 does not guarantee that the result will be a multiple of 3. For example, if a = 3 and b = 6, then \\(a + b + 1 = 10\\), which is not divisible by 3. Therefore, this option cannot be a multiple of 3.\n\nD) ab:\nThe product of two multiples of 3 will always be a multiple of 3. This is because multiplying any number by 3 will result in a number that is divisible by 3.\n\nE) \\(ab + 3\\):\nAdding 3 to the product of two multiples of 3 does not guarantee that the result will be a multiple of 3. For example, if a = 3 and b = 6, then \\(ab + 3 = 21\\), which is not divisible by 3. Therefore, this option cannot be a multiple of 3.\n\nBased on the above analysis, the option that CANNOT be a multiple of 3 is C) \\(a + b + 1\\).\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"20%", "B":"50%", "C":"100%", "D":"200%", "E":"500%" }, "id":11951, "question":"Max purchased some shares of stock at \\$ 10 per share. Six months later the stock was worth \\$ 20 per share. What was the percent increase in the value of Max' s investment?", "explanations":{ "correct":"To find the percent increase in the value of Max's investment, we need to calculate the difference between the final value and the initial value, and then express it as a percentage of the initial value.\n\nThe initial value of Max's investment is \\$ 10 per share, and the final value is \\$ 20 per share. The difference between the final value and the initial value is \\$ 20 - \\$ 10 = \\$ 10.\n\nTo express this difference as a percentage of the initial value, we divide the difference by the initial value and multiply by 100. So, (10/10) * 100 = 100%.\n\nTherefore, the percent increase in the value of Max's investment is 100%.\n\nThe answer is C) 100%." } }, { "answer":"E", "choices":{ "A":"210", "B":"390", "C":"490", "D":"810", "E":"990" }, "id":11957, "question":"If \\(f(x) = x^2 - 3x\\) and \\(g(x) = f(3x)\\), what is \\(g(-10)\\)?", "explanations":{ "correct":"To find the value of \\(g(-10)\\), we need to substitute \\(-10\\) into the function \\(g(x)\\), which is defined as \\(f(3x)\\).\n\nFirst, let's find the value of \\(f(3x)\\). We substitute \\(3x\\) into the function \\(f(x) = x^2 - 3x\\):\n\n\\(f(3x) = (3x)^2 - 3(3x)\\)\n\nSimplifying this expression, we get:\n\n\\(f(3x) = 9x^2 - 9x\\)\n\nNow, we substitute \\(-10\\) into the expression we just found:\n\n\\(g(-10) = 9(-10)^2 - 9(-10)\\)\n\nSimplifying further:\n\n\\(g(-10) = 9(100) + 90\\)\n\n\\(g(-10) = 900 + 90\\)\n\n\\(g(-10) = 990\\)\n\nTherefore, the answer is E) 990." } }, { "answer":"E", "choices":{ "A":"7 minutes and 10 minutes", "B":"10 minutes and 18 minutes", "C":"15 minutes and 18 minutes", "D":"22 minutes and 25 minutes", "E":"22 minutes and 28 minutes" }, "id":11965, "question":"A certain pie is made by baking the crust for 7 to 10 minutes, adding the filling, and baking the entire pie for another 15 to 18 minutes. What are the minimum and maximum total baking times for this pie?", "explanations":{ "correct":"To find the minimum and maximum total baking times for the pie, we need to consider the minimum and maximum times for each step of the baking process.\n\nStep 1: Baking the crust for 7 to 10 minutes.\nThe minimum time for baking the crust is 7 minutes, and the maximum time is 10 minutes.\n\nStep 2: Adding the filling.\nThis step does not have a specified time range, so we can assume it takes no time.\n\nStep 3: Baking the entire pie for another 15 to 18 minutes.\nThe minimum time for baking the entire pie is 15 minutes, and the maximum time is 18 minutes.\n\nTo find the minimum total baking time, we add the minimum times for each step:\nMinimum total baking time = 7 minutes (crust) + 0 minutes (filling) + 15 minutes (entire pie) = 22 minutes.\n\nTo find the maximum total baking time, we add the maximum times for each step:\nMaximum total baking time = 10 minutes (crust) + 0 minutes (filling) + 18 minutes (entire pie) = 28 minutes.\n\nTherefore, the minimum and maximum total baking times for this pie are 22 minutes and 28 minutes, respectively.\n\nThe answer is E) 22 minutes and 28 minutes." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { c } { 100 } \\)", "B":"\\(\\frac { 100c } { 3 } \\)", "C":"\\(\\frac { 9 } { c } \\)", "D":"3", "E":"\\(\\frac { 300 } { c } \\)" }, "id":11967, "question":"If \\(c\\) is positive, what percent of \\(3c\\) is 9 ?", "explanations":{ "correct":"To find the percent of \\(3c\\) that is 9, we need to divide 9 by \\(3c\\) and then multiply by 100 to convert it to a percentage.\n\nStep 1: Divide 9 by \\(3c\\):\n\\(\\frac{9}{3c} = \\frac{3}{c}\\)\n\nStep 2: Multiply the result by 100 to convert it to a percentage:\n\\(\\frac{3}{c} \\times 100 = \\frac{300}{c}\\)\n\nTherefore, the answer is E) \\(\\frac{300}{c}\\)." } }, { "answer":"B", "choices":{ "A":"\\(-7\\)", "B":"\\(-4\\)", "C":"\\(-\\frac{1}{2}\\)", "D":"4", "E":"7" }, "id":11971, "question":"If 5 more than 3 times a certain number is 3 less than the number, what is the number?", "explanations":{ "correct":"Let's solve the problem step by step:\n\nLet's assume the certain number is represented by the variable \\(x\\).\n\nAccording to the problem, \"5 more than 3 times a certain number\" can be written as \\(3x + 5\\).\n\nAnd \"3 less than the number\" can be written as \\(x - 3\\).\n\nSo, we can set up the equation: \\(3x + 5 = x - 3\\).\n\nNow, let's solve for \\(x\\):\n\nFirst, we can simplify the equation by subtracting \\(x\\) from both sides: \\(2x + 5 = -3\\).\n\nNext, we can subtract 5 from both sides: \\(2x = -8\\).\n\nFinally, we can divide both sides by 2 to solve for \\(x\\): \\(x = -4\\).\n\nTherefore, the answer is B) \\(-4\\)." } }, { "answer":"E", "choices":{ "A":"25.3", "B":"45.6", "C":"57", "D":"85.5", "E":"114" }, "id":11981, "question":"If 50 percent of 40 percent of a number is 22.8, what is the number?", "explanations":{ "correct":"To find the number, we need to solve the equation: 0.5 * 0.4 * x = 22.8, where x represents the number we are trying to find.\n\nFirst, we simplify the equation by multiplying 0.5 and 0.4: 0.2 * x = 22.8.\n\nNext, we isolate x by dividing both sides of the equation by 0.2: x = 22.8 / 0.2.\n\nEvaluating the division, we find that x = 114.\n\nTherefore, the answer is E) 114." } }, { "answer":"B", "choices":{ "A":"4", "B":"8", "C":"16", "D":"\\(a + 4\\)", "E":"\\(2a\\)" }, "id":11983, "question":"\\begin { gather* } 8 - a, 8, 8 + a \\end { gather* } What is the average (arithmetic mean) of the numbers in the list above?", "explanations":{ "correct":"To find the average (arithmetic mean) of a list of numbers, we add up all the numbers in the list and then divide the sum by the total number of numbers in the list.\n\\(\\newline\\)In this case, the list of numbers is: \\(8 - a\\), \\(8\\), and \\(8 + a\\).\n\nStep 1: Add up the numbers in the list:\n\\(8 - a + 8 + 8 + a\\)\n\nSimplifying the expression, we get:\n\\(24\\)\n\nStep 2: Determine the total number of numbers in the list:\nThere are 3 numbers in the list.\n\nStep 3: Divide the sum by the total number of numbers:\n\\(\\frac{24}{3}\\)\n\nSimplifying the expression, we get:\n\\(8\\)\n\nTherefore, the average (arithmetic mean) of the numbers in the list is 8.\n\nThe answer is B) 8." } }, { "answer":"D", "choices":{ "A":"3", "B":"2", "C":"1", "D":"-1", "E":"-2" }, "id":11988, "question":"If \\(x^2 = 9\\), what is the value of \\(\\frac { 4 } { x + 1 } - \\frac { 4 } { x - 1 } \\) ?", "explanations":{ "correct":"To find the value of the expression \\\\(\\frac { 4 } { x + 1 } - \\frac { 4 } { x - 1 }\\\\), we need to substitute the value of \\\\(x\\\\) that satisfies the equation \\\\(x^2 = 9\\\\).\n\nGiven that \\\\(x^2 = 9\\\\), we can take the square root of both sides to find the possible values of \\\\(x\\\\). The square root of 9 is 3, so \\\\(x\\\\) can be either 3 or -3.\n\nLet's substitute both values of \\\\(x\\\\) into the expression and see which one gives us the correct answer.\n\nFor \\\\(x = 3\\\\):\n\\\\(\\frac { 4 } { x + 1 } - \\frac { 4 } { x - 1 } = \\frac { 4 } { 3 + 1 } - \\frac { 4 } { 3 - 1 } = \\frac { 4 } { 4 } - \\frac { 4 } { 2 } = 1 - 2 = -1\\\\)\n\nFor \\\\(x = -3\\\\):\n\\\\(\\frac { 4 } { x + 1 } - \\frac { 4 } { x - 1 } = \\frac { 4 } { -3 + 1 } - \\frac { 4 } { -3 - 1 } = \\frac { 4 } { -2 } - \\frac { 4 } { -4 } = -2 + 1 = -1\\\\)\n\nBoth values of \\\\(x\\\\) give us the same result of -1. Therefore, the answer is D) -1." } }, { "answer":"D", "choices":{ "A":"\\(16\\frac{2}{3}%\\)", "B":"20%", "C":"\\(33\\frac{1}{3}%\\)", "D":"40%", "E":"60%" }, "id":11989, "question":"\\(\\frac{1}{3}\\) of 60 is equal to what percent of 50?", "explanations":{ "correct":"To find the answer, we need to determine what percent of 50 is equal to \\( \\frac{1}{3} \\) of 60.\n\nFirst, let's find \\( \\frac{1}{3} \\) of 60. We can do this by multiplying 60 by \\( \\frac{1}{3} \\):\n\n\\( \\frac{1}{3} \\times 60 = 20 \\)\n\nSo, \\( \\frac{1}{3} \\) of 60 is equal to 20.\n\nNow, we need to find what percent of 50 is equal to 20. To do this, we can set up a proportion:\n\n\\( \\frac{x}{100} = \\frac{20}{50} \\)\n\nTo solve for x, we can cross-multiply:\n\n\\( 50x = 20 \\times 100 \\)\n\n\\( 50x = 2000 \\)\n\nDividing both sides by 50, we get:\n\n\\( x = 40 \\)\n\nTherefore, 20 is equal to 40% of 50.\n\nThe answer is D) 40%." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { 3 } { 2 } m\\)", "B":"\\(3m\\)", "C":"\\(4m\\)", "D":"\\(\\frac { 9 } { 2 } m\\)", "E":"\\(6m\\)" }, "id":11994, "question":"The midpoint of segment \\(\\overline { PQ } \\) is F , and the length of \\(\\overline { FQ } \\) is \\(3m\\). What is the length of \\(\\overline { PQ } \\) in terms of \\( m\\) ?", "explanations":{ "correct":"To find the length of segment \\( \\overline{PQ} \\), we need to consider the properties of a midpoint. \n\nThe midpoint of a line segment divides the segment into two equal parts. In this case, the midpoint \\( F \\) divides the segment \\( \\overline{PQ} \\) into two equal parts, \\( \\overline{PF} \\) and \\( \\overline{FQ} \\). \n\nGiven that the length of \\( \\overline{FQ} \\) is \\( 3m \\), we can conclude that the length of \\( \\overline{PF} \\) is also \\( 3m \\) because the two parts are equal.\n\nTo find the length of \\( \\overline{PQ} \\), we add the lengths of \\( \\overline{PF} \\) and \\( \\overline{FQ} \\):\n\n\\( \\overline{PQ} = \\overline{PF} + \\overline{FQ} = 3m + 3m = 6m \\)\n\nTherefore, the length of \\( \\overline{PQ} \\) in terms of \\( m \\) is \\( 6m \\).\n\nThe answer is E) \\( 6m \\)." } }, { "answer":"A", "choices":{ "A":"None", "B":"One", "C":"Two", "D":"Three", "E":"Four" }, "id":11995, "question":"If \\(x\\) and \\(y\\) are integers, how many distinct ordered pairs \\((x,y)\\) satisfy the equation \\(2x + 4y = 17\\) ?", "explanations":{ "correct":"To find the number of distinct ordered pairs (x, y) that satisfy the equation 2x + 4y = 17, we can use a systematic approach.\n\nFirst, let's simplify the equation by dividing both sides by 2:\nx + 2y = 8.5\n\nNow, let's consider the possible values for x and y. Since x and y are integers, we can start by assigning values to y and then solve for x.\n\nLet's start with y = 0:\nx + 2(0) = 8.5\nx = 8.5\n\nSince x is not an integer, the pair (x, y) = (8.5, 0) does not satisfy the equation.\n\nNext, let's try y = 1:\nx + 2(1) = 8.5\nx + 2 = 8.5\nx = 6.5\n\nAgain, x is not an integer, so the pair (x, y) = (6.5, 1) does not satisfy the equation.\n\nNow, let's try y = 2:\nx + 2(2) = 8.5\nx + 4 = 8.5\nx = 4.5\n\nOnce again, x is not an integer, so the pair (x, y) = (4.5, 2) does not satisfy the equation.\n\nFinally, let's try y = 3:\nx + 2(3) = 8.5\nx + 6 = 8.5\nx = 2.5\n\nAs before, x is not an integer, so the pair (x, y) = (2.5, 3) does not satisfy the equation.\n\nSince we have tried all possible integer values for y and none of them resulted in integer values for x, there are no distinct ordered pairs (x, y) that satisfy the equation 2x + 4y = 17.\n\nTherefore, the answer is A) None." } }, { "answer":"A", "choices":{ "A":"4", "B":"6", "C":"8", "D":"12", "E":"64" }, "id":11997, "question":"A rectangular solid that has a width of 8, a length of 8, and a height of 12 is divided into 12 cubes, each of equal volume. What is the length of the edge of one of these cubes?", "explanations":{ "correct":"To find the length of the edge of one of the cubes, we need to determine the volume of each cube and then find the length of one side.\n\nThe volume of a rectangular solid is given by the formula V = length × width × height. In this case, the volume of the rectangular solid is 8 × 8 × 12 = 768 cubic units.\n\nSince the rectangular solid is divided into 12 cubes of equal volume, we can divide the total volume by 12 to find the volume of each cube. 768 ÷ 12 = 64 cubic units.\n\nThe volume of a cube is given by the formula V = side^3, where side represents the length of one side of the cube. In this case, we have the volume of the cube (64 cubic units), and we need to find the length of one side.\n\nTo find the length of one side, we can take the cube root of the volume. The cube root of 64 is 4.\n\nTherefore, the length of the edge of one of these cubes is 4.\n\nThe answer is A) 4." } }, { "answer":"C", "choices":{ "A":"\\(c^5\\)", "B":"\\(c^4 + c^9\\)", "C":"\\(c^4 + c^6\\)", "D":"\\(c^3 + c^4\\)", "E":"\\(c^2 + c^3\\)" }, "id":11999, "question":"\\(a = b^2 + b^3\\) In the equation above, \\(b = c^2\\) for any integer \\(c\\). What is \\(a\\) in terms of \\(c\\) ?", "explanations":{ "correct":"To find the value of \\(a\\) in terms of \\(c\\), we need to substitute the given value of \\(b\\) in the equation \\(a = b^2 + b^3\\).\n\nGiven that \\(b = c^2\\), we can substitute \\(c^2\\) for \\(b\\) in the equation:\n\n\\(a = (c^2)^2 + (c^2)^3\\)\n\nSimplifying the exponents, we have:\n\n\\(a = c^4 + c^6\\)\n\nTherefore, the value of \\(a\\) in terms of \\(c\\) is \\(c^4 + c^6\\).\n\nThe answer is C." } }, { "answer":"A", "choices":{ "A":"\\(2 \\times 5 \\times 6\\)", "B":"\\(3 \\times 4 \\times 6\\)", "C":"\\(3 \\times 5 \\times 6\\)", "D":"\\(4 \\times 5 \\times 6\\)", "E":"\\(5 \\times 6 \\times 12\\)" }, "id":12000, "question":"A rectangular box is 24 inches long, 10 inches wide, and 15 inches high. If exactly 60 smaller identical rectangular boxes can be stored perfectly in this larger box, which of the following could be the dimensions, in inches, of these smaller boxes?", "explanations":{ "correct":"To find the dimensions of the smaller boxes, we need to consider the volume of the larger box and the number of smaller boxes that can fit inside it.\n\nThe volume of the larger box is given by the formula: length × width × height. In this case, the volume is 24 × 10 × 15 = 3600 cubic inches.\n\nSince we know that exactly 60 smaller boxes can fit inside the larger box, we can divide the volume of the larger box by 60 to find the volume of each smaller box.\n\n3600 ÷ 60 = 60 cubic inches\n\nNow, we need to find the dimensions of the smaller boxes that have a volume of 60 cubic inches. We can do this by finding the factors of 60 and checking if any of them match the given answer choices.\n\nThe factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.\n\nChecking the answer choices:\nA) 2 × 5 × 6 = 60\nB) 3 × 4 × 6 = 72 (not equal to 60)\nC) 3 × 5 × 6 = 90 (not equal to 60)\nD) 4 × 5 × 6 = 120 (not equal to 60)\nE) 5 × 6 × 12 = 360 (not equal to 60)\n\nFrom the answer choices, only option A) \\(2 \\times 5 \\times 6\\) has dimensions that multiply to give a volume of 60 cubic inches.\n\nTherefore, the answer is A)." } }, { "answer":"A", "choices":{ "A":"7", "B":"8", "C":"9", "D":"10", "E":"11" }, "id":12003, "question":"If \\(y + 4 = 13\\), then \\(y - 2\\) =", "explanations":{ "correct":"To find the value of \\(y - 2\\), we need to first solve the equation \\(y + 4 = 13\\) for \\(y\\).\n\nStep 1: Subtract 4 from both sides of the equation:\n\\(y + 4 - 4 = 13 - 4\\)\n\\(y = 9\\)\n\nNow that we have found the value of \\(y\\), we can substitute it into the expression \\(y - 2\\) to find the final answer.\n\nStep 2: Substitute \\(y = 9\\) into \\(y - 2\\):\n\\(9 - 2 = 7\\)\n\nTherefore, the answer is A) 7." } }, { "answer":"B", "choices":{ "A":"\\$ 375", "B":"\\$ 385", "C":"\\$ 395", "D":"\\$ 405", "E":"\\$ 415" }, "id":12012, "question":"A car rental agency charges \\$ 40 per day for the first 7 days, and \\$ 35 a day for each day after that. How much would Joe be charged if he rented a car for 10 days?", "explanations":{ "correct":"To calculate the total charge for Joe's car rental, we need to consider the different rates for the first 7 days and the subsequent days.\n\nFor the first 7 days, the charge is \\$ 40 per day. So, the total charge for the first 7 days would be 7 * \\$ 40 = \\$ 280.\n\nFor the remaining 3 days (10 days - 7 days), the charge is \\$ 35 per day. So, the total charge for the remaining days would be 3 * \\$ 35 = \\$ 105.\n\nTo find the total charge, we add the charges for the first 7 days and the remaining days: \\$ 280 + \\$ 105 = \\$ 385.\n\nTherefore, Joe would be charged \\$ 385 if he rented a car for 10 days.\n\nThe answer is B." } }, { "answer":"B", "choices":{ "A":"2", "B":"9", "C":"12", "D":"18", "E":"24" }, "id":12013, "question":"If \\(x = y + 2\\) and y = 4, then \\(\\frac { 3 } { 2 } x\\) =", "explanations":{ "correct":"To find the value of \\\\(\\frac { 3 } { 2 } x\\\\), we first need to substitute the value of y into the equation \\\\(x = y + 2\\\\).\n\nGiven that y = 4, we can substitute this value into the equation to get:\n\\\\(x = 4 + 2\\\\)\n\nSimplifying, we find that:\n\\\\(x = 6\\\\)\n\nNow, we can substitute the value of x into the expression \\\\(\\frac { 3 } { 2 } x\\\\) to find the final answer.\n\nSubstituting x = 6, we have:\n\\\\(\\frac { 3 } { 2 } x = \\frac { 3 } { 2 } \\times 6\\\\)\n\nSimplifying the expression, we get:\n\\\\(\\frac { 3 } { 2 } \\times 6 = 9\\\\)\n\nTherefore, the answer is B) 9." } }, { "answer":"E", "choices":{ "A":"8", "B":"9", "C":"15", "D":"17", "E":"19" }, "id":12025, "question":"\\(a^3 \\times a^6 = a^c\\) \\(\\frac { (b^5)^d } { b^d } = b^40\\) In the equations above, \\(a > 1\\) and \\(b > 1\\). What is the sum of \\(c\\) and \\(d\\)?", "explanations":{ "correct":"To solve this problem, let's simplify each equation step-by-step.\n\nFor the first equation, \\\\(a^3 \\\\times a^6 = a^c\\\\), we can use the property of exponents that states when multiplying two numbers with the same base, we add the exponents. Therefore, \\\\(a^3 \\\\times a^6 = a^{3+6} = a^9\\\\). So, we have \\\\(a^9 = a^c\\\\).\n\nFor the second equation, \\\\(\\\\frac { (b^5)^d } { b^d } = b^{40}\\\\), we can simplify the numerator first. Using the property of exponents that states when raising a power to another power, we multiply the exponents, we have \\\\((b^5)^d = b^{5d}\\\\). Now, we can rewrite the equation as \\\\(\\\\frac { b^{5d} } { b^d } = b^{40}\\\\). Using the property of exponents that states when dividing two numbers with the same base, we subtract the exponents, we have \\\\(b^{5d - d} = b^{40}\\\\). Simplifying further, we get \\\\(b^{4d} = b^{40}\\\\).\n\nNow, we can equate the exponents in both equations to find the values of \\\\(c\\\\) and \\\\(d\\\\). We have \\\\(9 = c\\\\) and \\\\(4d = 40\\\\). Solving for \\\\(d\\\\), we divide both sides of the equation by 4, which gives us \\\\(d = 10\\\\).\n\nFinally, we need to find the sum of \\\\(c\\\\) and \\\\(d\\\\). Substituting the values we found, we have \\\\(c + d = 9 + 10 = 19\\\\).\n\nTherefore, the answer is E) 19." } }, { "answer":"C", "choices":{ "A":"3", "B":"5", "C":"7", "D":"14", "E":"28" }, "id":12034, "question":"If \\(\\frac { 2k } { 2k + 13 } = \\frac { 14 } { 27 } \\), then \\(k\\) =", "explanations":{ "correct":"To solve the equation \\\\(\\frac { 2k } { 2k + 13 } = \\frac { 14 } { 27 }\\\\), we can cross-multiply to eliminate the fractions. \n\nCross-multiplying gives us: \n\\\\(2k \\cdot 27 = 14 \\cdot (2k + 13)\\\\)\n\nExpanding the right side of the equation: \n\\\\(54k = 28k + 182\\\\)\n\nNext, we can simplify the equation by combining like terms: \n\\\\(54k - 28k = 182\\\\)\n\\\\(26k = 182\\\\)\n\nTo solve for \\\\(k\\\\), we divide both sides of the equation by 26: \n\\\\(k = \\frac{182}{26}\\\\)\n\\\\(k = 7\\\\)\n\nTherefore, the answer is C) 7." } }, { "answer":"E", "choices":{ "A":"2x", "B":"4x", "C":"6x", "D":"7x", "E":"8x" }, "id":12035, "question":"Right triangle A has base b, height h, and area x. Rectangle B has length 2b and width 2h. What is the area of rectangle B in terms of x ?", "explanations":{ "correct":"To find the area of rectangle B, we need to multiply its length by its width. The length of rectangle B is given as 2b, and the width is given as 2h.\n\nSince the base of right triangle A is b and the height is h, the area of triangle A is given by the formula: Area = (1/2) * base * height. Therefore, we have x = (1/2) * b * h.\n\nTo find the area of rectangle B, we substitute the values of b and h in terms of x. From the equation x = (1/2) * b * h, we can solve for b and h.\n\nMultiplying both sides of the equation by 2, we get 2x = b * h. \n\nNow, we substitute 2x for b * h in the formula for the area of rectangle B. The area of rectangle B is given by: Area = length * width = (2b) * (2h) = 4b * h = 4(2x) = 8x.\n\nTherefore, the area of rectangle B in terms of x is 8x.\n\nThe answer is E) 8x." } }, { "answer":"D", "choices":{ "A":"6", "B":"8", "C":"10", "D":"12", "E":"14" }, "id":12040, "question":"\\(a\\) and \\(b\\) are different positive integers and \\(a\\) is greater than \\(b\\). If \\(a^2 - b^2 = 7\\) , what is the value of \\(ab\\) ?", "explanations":{ "correct":"To find the value of \\(ab\\), we need to first factor the expression \\(a^2 - b^2\\). \n\nThe expression \\(a^2 - b^2\\) can be factored as \\((a + b)(a - b)\\). \n\nGiven that \\(a\\) and \\(b\\) are different positive integers and \\(a\\) is greater than \\(b\\), we can rewrite the equation as \\((a + b)(a - b) = 7\\). \n\nSince \\(7\\) is a prime number, it can only be factored as \\(1 \\times 7\\) or \\(7 \\times 1\\). \n\nTherefore, we have two possible cases to consider:\n\nCase 1: \\(a + b = 7\\) and \\(a - b = 1\\)\nAdding the two equations, we get \\(2a = 8\\), which implies \\(a = 4\\). \nSubstituting \\(a = 4\\) into the equation \\(a + b = 7\\), we find \\(b = 3\\). \nThus, \\(ab = 4 \\times 3 = 12\\).\n\nCase 2: \\(a + b = 1\\) and \\(a - b = 7\\)\nAdding the two equations, we get \\(2a = 8\\), which implies \\(a = 4\\). \nSubstituting \\(a = 4\\) into the equation \\(a + b = 1\\), we find \\(b = -3\\). \nHowever, since \\(b\\) is a positive integer, this case is not valid.\n\nTherefore, the value of \\(ab\\) is \\(12\\).\n\nThe answer is D) 12." } }, { "answer":"B", "choices":{ "A":"(5,3)", "B":"(3,5)", "C":"(3,10)", "D":"(10,3)", "E":"(6,10)" }, "id":12041, "question":"Points E and F lie in the xy-coordinate plane at (0,10) and (6,0), respectively. Which of the following is the midpoint of \\(\\overline { EF } \\) ?", "explanations":{ "correct":"To find the midpoint of a line segment, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) are given by:\n\nMidpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)\n\\(\\newline\\)In this case, the coordinates of point E are (0, 10) and the coordinates of point F are (6, 0). Let's substitute these values into the midpoint formula:\n\nMidpoint = ((0 + 6) / 2, (10 + 0) / 2)\n = (6 / 2, 10 / 2)\n = (3, 5)\n\nTherefore, the midpoint of line segment EF is (3, 5).\n\nThe answer is B) (3, 5)." } }, { "answer":"B", "choices":{ "A":"0", "B":"3", "C":"4", "D":"7", "E":"12" }, "id":12048, "question":"If \\(r(b) = \\frac{(b^2 - 7)}{(b + 7)}\\), what is the value of \\(r(7)\\)?", "explanations":{ "correct":"To find the value of \\(r(7)\\), we need to substitute \\(b = 7\\) into the expression for \\(r(b)\\).\n\n\\(r(b) = \\frac{{(b^2 - 7)}}{{(b + 7)}}\\)\n\nSubstituting \\(b = 7\\):\n\n\\(r(7) = \\frac{{(7^2 - 7)}}{{(7 + 7)}}\\)\n\nSimplifying the numerator:\n\n\\(r(7) = \\frac{{(49 - 7)}}{{(7 + 7)}}\\)\n\n\\(r(7) = \\frac{{42}}{{14}}\\)\n\nSimplifying the fraction:\n\n\\(r(7) = 3\\)\n\nTherefore, the value of \\(r(7)\\) is 3.\n\nThe answer is B." } }, { "answer":"E", "choices":{ "A":"0", "B":"4", "C":"12", "D":"16", "E":"24" }, "id":12051, "question":"If \\(x^2 - 4x - 12 = 0\\), what is the value of \\(2x^2 - 8x\\)?", "explanations":{ "correct":"To find the value of \\(2x^2 - 8x\\), we need to first solve the quadratic equation \\(x^2 - 4x - 12 = 0\\). \n\nStep 1: Factor the quadratic equation.\nTo factor \\(x^2 - 4x - 12\\), we need to find two numbers whose product is -12 and whose sum is -4. The numbers -6 and 2 satisfy these conditions, so we can factor the equation as \\((x - 6)(x + 2) = 0\\).\n\nStep 2: Set each factor equal to zero and solve for \\(x\\).\nSetting \\(x - 6 = 0\\), we find \\(x = 6\\).\nSetting \\(x + 2 = 0\\), we find \\(x = -2\\).\n\nStep 3: Substitute the values of \\(x\\) into \\(2x^2 - 8x\\) to find the value.\nSubstituting \\(x = 6\\), we get \\(2(6)^2 - 8(6) = 72 - 48 = 24\\).\nSubstituting \\(x = -2\\), we get \\(2(-2)^2 - 8(-2) = 8 + 16 = 24\\).\n\nSince both values of \\(x\\) yield the same result of 24, we can conclude that the value of \\(2x^2 - 8x\\) is 24.\n\nTherefore, the answer is E) 24." } }, { "answer":"A", "choices":{ "A":"4", "B":"5", "C":"10", "D":"20", "E":"25" }, "id":12052, "question":"What is the length of each of the five equal sides of a regular pentagon if the perimeter of the pentagon is equal to the perimeter of a square whose area is 25?", "explanations":{ "correct":"To find the length of each side of the regular pentagon, we need to first determine the perimeter of the square. We are given that the area of the square is 25, so we can find the length of one side of the square by taking the square root of the area. \n\n√25 = 5\n\nSince the perimeter of the square is equal to the perimeter of the pentagon, and a square has four equal sides, the perimeter of the square is 4 times the length of one side.\n\nPerimeter of the square = 4 * 5 = 20\n\nSince the perimeter of the pentagon is also 20, and a regular pentagon has five equal sides, the length of each side of the pentagon is the perimeter divided by 5.\n\nLength of each side of the pentagon = 20 / 5 = 4\n\nTherefore, the answer is A) 4." } }, { "answer":"D", "choices":{ "A":"48", "B":"44", "C":"36", "D":"24", "E":"20" }, "id":12053, "question":"A bag of candy contains chocolate hearts, vanilla hearts and gumballs. The number of pieces of heart-shaped candy is 5 times the number of gumballs. If one piece of candy is to be chosen at random from the bag, the probability that a vanilla heart is chosen is 4 times the probability that a chocolate heart will be chosen. If there are 16 vanilla hearts in the bagm what is the total number of pieces of candy in the bag?", "explanations":{ "correct":"Let's denote the number of gumballs as \"g\", the number of chocolate hearts as \"c\", and the number of vanilla hearts as \"v\".\n\nAccording to the problem, the number of pieces of heart-shaped candy is 5 times the number of gumballs, so we have the equation:\nc + v = 5g (equation 1)\n\nThe probability of choosing a vanilla heart is 4 times the probability of choosing a chocolate heart. Since the total number of heart-shaped candies is c + v, the probability of choosing a vanilla heart is v / (c + v), and the probability of choosing a chocolate heart is c / (c + v). Therefore, we have the equation:\nv / (c + v) = 4 * (c / (c + v)) (equation 2)\n\nWe are given that there are 16 vanilla hearts in the bag, so v = 16.\n\nSubstituting v = 16 into equation 1, we get:\nc + 16 = 5g (equation 3)\n\nSubstituting v = 16 into equation 2, we get:\n16 / (c + 16) = 4 * (c / (c + 16))\n\nSimplifying equation 2, we have:\n16 = 4c\n\nDividing both sides of the equation by 4, we get:\nc = 4\n\nSubstituting c = 4 into equation 3, we get:\n4 + 16 = 5g\n20 = 5g\n\nDividing both sides of the equation by 5, we get:\ng = 4\n\nNow, we can find the total number of pieces of candy in the bag by adding the number of gumballs, chocolate hearts, and vanilla hearts:\nTotal number of pieces of candy = g + c + v\nTotal number of pieces of candy = 4 + 4 + 16\nTotal number of pieces of candy = 24\n\nTherefore, the answer is D) 24." } }, { "answer":"C", "choices":{ "A":"30", "B":"35", "C":"40", "D":"45", "E":"60" }, "id":12056, "question":"If 20 percent of 30 percent of a positive number is equal to 15 percent of h percent of the same number, what is the value of h?", "explanations":{ "correct":"Let's solve the problem step by step:\n\nStep 1: Let's assume the positive number is x.\n\nStep 2: We are given that 20 percent of 30 percent of x is equal to 15 percent of h percent of x. Mathematically, this can be written as:\n\n(20/100) * (30/100) * x = (15/100) * (h/100) * x\n\nSimplifying this equation, we get:\n\n(2/10) * (3/10) * x = (15/100) * (h/100) * x\n\n(6/100) * x = (15/100) * (h/100) * x\n\nStep 3: We can cancel out the x from both sides of the equation:\n\n6/100 = (15/100) * (h/100)\n\nStep 4: To solve for h, we can cross multiply:\n\n6 * 100 = 15 * h\n\n600 = 15h\n\nStep 5: Divide both sides of the equation by 15:\n\n600/15 = h\n\n40 = h\n\nStep 6: Therefore, the value of h is 40.\n\nThe answer is C) 40." } }, { "answer":"A", "choices":{ "A":"\\(5n\\)", "B":"\\(5 + n\\)", "C":"\\(n - 5\\)", "D":"\\(n^5\\)", "E":"\\(5^n\\)" }, "id":12057, "question":"If a store sells n newspapers on each of 5 days, which of the following represents the total number of newspapers sold?", "explanations":{ "correct":"To find the total number of newspapers sold, we need to multiply the number of newspapers sold each day by the number of days. In this case, the store sells \\(n\\) newspapers on each of the 5 days.\n\nTo calculate the total number of newspapers sold, we multiply \\(n\\) by 5:\n\n\\(n \\times 5 = 5n\\)\n\nTherefore, the correct answer is A) \\(5n\\).\n\nDouble-checking our reasoning, we can see that option B) \\(5 + n\\) does not represent the total number of newspapers sold since it only adds the number of newspapers sold each day to the number of days, which is incorrect.\n\nOption C) \\(n - 5\\) subtracts the number of days from the number of newspapers sold each day, which is also incorrect.\n\nOption D) \\(n^5\\) raises the number of newspapers sold each day to the power of 5, which is not the correct calculation for finding the total number of newspapers sold.\n\nOption E) \\(5^n\\) raises the number of days to the power of the number of newspapers sold each day, which is also not the correct calculation for finding the total number of newspapers sold.\n\nTherefore, the answer is A) \\(5n\\)." } }, { "answer":"D", "choices":{ "A":"1984", "B":"1985", "C":"1986", "D":"1987", "E":"1988" }, "id":12060, "question":"From 1980 to 1990, the value of a share of stock of XYZ Corporation doubled every year. If in 1990 a share of the stock was worth \\$ 80, in what year was it worth \\$ 10?", "explanations":{ "correct":"To find the year when a share of stock was worth \\$ 10, we need to work backwards from the given information.\n\nWe know that from 1980 to 1990, the value of the stock doubled every year. So, if the stock was worth \\$ 80 in 1990, it must have been worth half that amount, \\$ 40, in 1989. Similarly, in 1988 it would have been worth \\$ 20, and in 1987 it would have been worth \\$ 10.\n\nTherefore, the stock was worth \\$ 10 in the year 1987.\n\nThe answer is D) 1987." } }, { "answer":"A", "choices":{ "A":"-8", "B":"-6", "C":"-4", "D":"-2", "E":"2" }, "id":12062, "question":"Which of the following is NOT a possible value of \\(2 - x\\), if \\(x\\) is a one-digit integer?", "explanations":{ "correct":"To find the possible values of \\(2 - x\\), we need to consider all the possible values of \\(x\\) that are one-digit integers. \n\nA one-digit integer can range from -9 to 9, inclusive. \n\nLet's substitute each possible value of \\(x\\) into the expression \\(2 - x\\) and see if it matches any of the given answer choices.\n\nFor \\(x = -9\\), \\(2 - x = 2 - (-9) = 2 + 9 = 11\\), which is not a one-digit integer.\n\nFor \\(x = -8\\), \\(2 - x = 2 - (-8) = 2 + 8 = 10\\), which is not a one-digit integer.\n\nFor \\(x = -7\\), \\(2 - x = 2 - (-7) = 2 + 7 = 9\\), which is a one-digit integer.\n\nFor \\(x = -6\\), \\(2 - x = 2 - (-6) = 2 + 6 = 8\\), which is a one-digit integer.\n\nFor \\(x = -5\\), \\(2 - x = 2 - (-5) = 2 + 5 = 7\\), which is a one-digit integer.\n\nFor \\(x = -4\\), \\(2 - x = 2 - (-4) = 2 + 4 = 6\\), which is a one-digit integer.\n\nFor \\(x = -3\\), \\(2 - x = 2 - (-3) = 2 + 3 = 5\\), which is a one-digit integer.\n\nFor \\(x = -2\\), \\(2 - x = 2 - (-2) = 2 + 2 = 4\\), which is a one-digit integer.\n\nFor \\(x = -1\\), \\(2 - x = 2 - (-1) = 2 + 1 = 3\\), which is a one-digit integer.\n\nFor \\(x = 0\\), \\(2 - x = 2 - 0 = 2\\), which is a one-digit integer.\n\nFor \\(x = 1\\), \\(2 - x = 2 - 1 = 1\\), which is a one-digit integer.\n\nFor \\(x = 2\\), \\(2 - x = 2 - 2 = 0\\), which is a one-digit integer.\n\nFor \\(x = 3\\), \\(2 - x = 2 - 3 = -1\\), which is a one-digit integer.\n\nFor \\(x = 4\\), \\(2 - x = 2 - 4 = -2\\), which is a one-digit integer.\n\nFor \\(x = 5\\), \\(2 - x = 2 - 5 = -3\\), which is a one-digit integer.\n\nFor \\(x = 6\\), \\(2 - x = 2 - 6 = -4\\), which is a one-digit integer.\n\nFor \\(x = 7\\), \\(2 - x = 2 - 7 = -5\\), which is a one-digit integer.\n\nFor \\(x = 8\\), \\(2 - x = 2 - 8 = -6\\), which is a one-digit integer.\n\nFor \\(x = 9\\), \\(2 - x = 2 - 9 = -7\\), which is a one-digit integer.\n\nFrom the above analysis, we can see that the only value that is not a possible value of \\(2 - x\\) when \\(x\\) is a one-digit integer is -8.\n\nTherefore, the answer is A) -8." } }, { "answer":"B", "choices":{ "A":"124", "B":"126", "C":"128", "D":"130", "E":"132" }, "id":12066, "question":"A fisherman takes his boat out to sea at an average speed of 18 miles per hour and then back to the harbor along the same route at an average speed of 14 miles per hour. If his entire trips lasts 8 hour, what is the total number of miles in the round trip?", "explanations":{ "correct":"To find the total number of miles in the round trip, we need to calculate the distance traveled in each direction and then add them together.\n\nLet's start by finding the distance traveled on the outbound trip. We know that the average speed is 18 miles per hour and the total time for the trip is 8 hours. Using the formula distance = speed × time, we can calculate the distance as follows:\n\nDistance outbound = Speed outbound × Time outbound\nDistance outbound = 18 miles/hour × Time outbound\n\nNow, let's find the distance traveled on the return trip. We know that the average speed is 14 miles per hour and the total time for the trip is 8 hours. Using the same formula, we can calculate the distance as follows:\n\nDistance return = Speed return × Time return\nDistance return = 14 miles/hour × Time return\n\nSince the outbound and return trips cover the same distance, we can set the two distances equal to each other:\n\nDistance outbound = Distance return\n\n18 miles/hour × Time outbound = 14 miles/hour × Time return\n\nNow, we can solve for Time return in terms of Time outbound:\n\nTime return = (18 miles/hour × Time outbound) / 14 miles/hour\n\nWe also know that the total time for the trip is 8 hours:\n\nTime outbound + Time return = 8 hours\n\nSubstituting the expression for Time return, we have:\n\nTime outbound + (18 miles/hour × Time outbound) / 14 miles/hour = 8 hours\n\nTo simplify the equation, we can multiply both sides by 14:\n\n14 × Time outbound + 18 × Time outbound = 8 × 14\n\n32 × Time outbound = 112\n\nDividing both sides by 32, we find:\n\nTime outbound = 112 / 32\nTime outbound = 3.5 hours\n\nNow, we can substitute this value back into the equation for Time return:\n\nTime return = (18 miles/hour × 3.5 hours) / 14 miles/hour\nTime return = 4.5 hours\n\nFinally, we can calculate the distance traveled in each direction:\n\nDistance outbound = 18 miles/hour × 3.5 hours\nDistance outbound = 63 miles\n\nDistance return = 14 miles/hour × 4.5 hours\nDistance return = 63 miles\n\nThe total number of miles in the round trip is the sum of the distances traveled in each direction:\n\nTotal distance = Distance outbound + Distance return\nTotal distance = 63 miles + 63 miles\nTotal distance = 126 miles\n\nTherefore, the answer is B) 126." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 1 } { 12 } \\)", "B":"\\(\\frac { 1 } { 4 } \\)", "C":"\\(\\frac { 1 } { 3 } \\)", "D":"3", "E":"4" }, "id":12068, "question":"If \\(\\frac { 1 } { a } + \\frac { 1 } { a } + \\frac { 1 } { a } = 12\\), then \\(a\\) =", "explanations":{ "correct":"To solve the equation \\(\\frac { 1 } { a } + \\frac { 1 } { a } + \\frac { 1 } { a } = 12\\), we can combine the fractions on the left side by finding a common denominator. The common denominator for all three fractions is \\(a\\). \n\nSo, we have \\(\\frac { 1 } { a } + \\frac { 1 } { a } + \\frac { 1 } { a } = \\frac { 3 } { a } = 12\\).\n\nTo isolate \\(a\\), we can multiply both sides of the equation by \\(a\\):\n\n\\(3 = 12a\\).\n\nNext, we can divide both sides of the equation by 12 to solve for \\(a\\):\n\n\\(\\frac { 3 } { 12 } = a\\).\n\nSimplifying the fraction on the left side gives us:\n\n\\(\\frac { 1 } { 4 } = a\\).\n\nTherefore, the answer is B) \\(\\frac { 1 } { 4 } \\)." } }, { "answer":"B", "choices":{ "A":"3", "B":"33", "C":"39", "D":"69", "E":"75" }, "id":12075, "question":"If \\(\\sqrt { x +3 } + 3 = 9\\), then x =", "explanations":{ "correct":"To solve the equation \\\\(\\sqrt { x +3 } + 3 = 9\\\\), we need to isolate the variable \\\\(x\\\\).\n\nStep 1: Subtract 3 from both sides of the equation:\n\\\\(\\sqrt { x +3 } = 9 - 3\\\\)\n\\\\(\\sqrt { x +3 } = 6\\\\)\n\nStep 2: Square both sides of the equation to eliminate the square root:\n\\\\((\\sqrt { x +3 })^2 = 6^2\\\\)\n\\\\(x + 3 = 36\\\\)\n\nStep 3: Subtract 3 from both sides of the equation:\n\\\\(x = 36 - 3\\\\)\n\\\\(x = 33\\\\)\n\nTherefore, the answer is B) 33." } }, { "answer":"B", "choices":{ "A":"15", "B":"14", "C":"9", "D":"6", "E":"3" }, "id":12078, "question":"If \\(3^k + 3^k + 3^k = (27)^5\\), what is the value of \\(k\\) ?", "explanations":{ "correct":"To find the value of \\(k\\), we need to solve the equation \\(3^k + 3^k + 3^k = (27)^5\\).\n\nFirst, let's simplify the left side of the equation. Since we have three identical terms, we can combine them by adding them together:\n\n\\(3^k + 3^k + 3^k = 3 \\cdot 3^k = 3^{k+1}\\).\n\nNow, we have the equation \\(3^{k+1} = (27)^5\\).\n\nTo simplify further, we can rewrite \\(27\\) as \\(3^3\\) since \\(27 = 3^3\\):\n\n\\(3^{k+1} = (3^3)^5\\).\n\nUsing the property of exponents, we can simplify the right side of the equation:\n\n\\(3^{k+1} = 3^{3 \\cdot 5}\\).\n\nSince the bases are the same, we can equate the exponents:\n\n\\(k+1 = 3 \\cdot 5\\).\n\nSimplifying the right side of the equation:\n\n\\(k+1 = 15\\).\n\nTo solve for \\(k\\), we subtract \\(1\\) from both sides:\n\n\\(k = 15 - 1\\).\n\nTherefore, the value of \\(k\\) is \\(14\\).\n\nThe answer is B) 14." } }, { "answer":"B", "choices":{ "A":"\\(n \\geq 3(140)\\)", "B":"\\(3(140) + n \\leq 450\\)", "C":"\\(3(140) + n \\geq 450\\)", "D":"\\(3(140) - n \\leq 450\\)", "E":"\\(3(140) - n \\geq 450\\)" }, "id":12082, "question":"Albert's diet allows him to consume no more than 450 calories for breakfast. He decides to have 3 mini-muffins, which each contain 140 calories, and a drink. If n represents the number of calories that his drink can contain, which of the following inequalities could be used to find all possible values for n?", "explanations":{ "correct":"To find the possible values for \\(n\\), we need to consider the total number of calories Albert consumes for breakfast. \n\nAlbert decides to have 3 mini-muffins, each containing 140 calories. So, the total number of calories from the mini-muffins is \\(3 \\times 140 = 420\\) calories.\n\\(\\newline\\)In addition to the mini-muffins, Albert also has a drink, which contains \\(n\\) calories.\n\nThe total number of calories Albert consumes for breakfast, including the mini-muffins and the drink, must be no more than 450 calories according to his diet.\n\nTherefore, we can write the inequality as:\n\n\\(3(140) + n \\leq 450\\)\n\nThis inequality ensures that the total number of calories from the mini-muffins (420 calories) plus the calories from the drink (\\(n\\) calories) is less than or equal to 450 calories.\n\nHence, the answer is B) \\(3(140) + n \\leq 450\\)." } }, { "answer":"C", "choices":{ "A":"None of the employees in the accounting department are married.", "B":"Some of the employees in the accounting department are married.", "C":"Some of the employees in the accounting department are unmarried.", "D":"More employees are on the bowling team than work in the accounting department.", "E":"More employees work in the accounting department than are on the bowling team." }, "id":12090, "question":"At a certain company, some employees in the accounting department are also on the company bowling team, and none of the employees on the company bowling team are married. Which of the following statements must also be true?", "explanations":{ "correct":"Let's analyze the given information step by step:\n\n1. Some employees in the accounting department are also on the company bowling team.\nThis means that there is at least one employee who is both in the accounting department and on the bowling team.\n\n2. None of the employees on the company bowling team are married.\nThis implies that all employees on the bowling team are unmarried.\n\nNow, let's evaluate each statement:\n\nA) None of the employees in the accounting department are married.\nWe cannot conclude this statement based on the given information. It is possible that some employees in the accounting department are married, but they are not on the bowling team.\n\nB) Some of the employees in the accounting department are married.\nWe cannot conclude this statement based on the given information. It is possible that some employees in the accounting department are married, but they are not on the bowling team.\n\nC) Some of the employees in the accounting department are unmarried.\nThis statement must be true. Since all employees on the bowling team are unmarried, and some employees in the accounting department are also on the bowling team, it follows that at least some employees in the accounting department are unmarried.\n\nD) More employees are on the bowling team than work in the accounting department.\nWe cannot determine the relative sizes of the bowling team and the accounting department based on the given information. The number of employees in each group is unknown.\n\nE) More employees work in the accounting department than are on the bowling team.\nWe cannot determine the relative sizes of the bowling team and the accounting department based on the given information. The number of employees in each group is unknown.\n\nTherefore, the only statement that must be true based on the given information is:\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"-4", "B":"-2", "C":"2", "D":"4", "E":"6" }, "id":12095, "question":"If \\(a = 16 - b\\) and \\(\\frac { b } { 6 } = 3\\), what is the value of \\(a\\)?", "explanations":{ "correct":"To find the value of \\(a\\), we need to substitute the given value of \\(b\\) into the equation \\(a = 16 - b\\).\n\nGiven: \\(\\frac{b}{6} = 3\\)\n\nTo find the value of \\(b\\), we can multiply both sides of the equation by 6:\n\n\\(\\frac{b}{6} \\times 6 = 3 \\times 6\\)\n\nThis simplifies to:\n\n\\(b = 18\\)\n\nNow, substitute the value of \\(b\\) into the equation \\(a = 16 - b\\):\n\n\\(a = 16 - 18\\)\n\nSimplifying further:\n\n\\(a = -2\\)\n\nTherefore, the value of \\(a\\) is -2.\n\nThe answer is B) -2." } }, { "answer":"E", "choices":{ "A":"15", "B":"50", "C":"75", "D":"120", "E":"150" }, "id":12099, "question":"If \\(0.1k\\) percent of \\(n\\) is 5, what is \\(3k\\) percent of \\(n\\) ?", "explanations":{ "correct":"To find the answer, we need to set up an equation based on the given information. \n\nWe are told that \"0.1k percent of n is 5.\" This means that 0.1k% of n is equal to 5. \n\nTo convert a percentage to a decimal, we divide by 100. So, 0.1k% can be written as 0.1k/100. \n\nNow we can set up the equation: \n\n0.1k/100 * n = 5\n\nTo find what 3k percent of n is, we need to find the value of 3k/100 * n. \n\nTo do this, we can multiply both sides of the equation by 3: \n\n(0.1k/100 * n) * 3 = 5 * 3\n\n0.3k/100 * n = 15\n\nNow we have the equation 0.3k/100 * n = 15. \n\nTo find the value of 3k percent of n, we can simplify the equation: \n\n0.3k/100 * n = 15\n\n0.003k * n = 15\n\nMultiplying both sides by 1000 to eliminate the decimal: \n\n3k * n = 15000\n\nSo, 3k percent of n is 15000. \n\nThe answer is E) 150." } }, { "answer":"C", "choices":{ "A":"\\(6b^2 - 48b + 96\\)", "B":"\\(6b^2 - 57b + 132\\)", "C":"\\(6b^2 - 57b + 144\\)", "D":"\\(6b^2 - 9b - 132\\)", "E":"\\(6b^2 - 9b + 144\\)" }, "id":12100, "question":"If \\(a = 3(2c^2 + 3c + 4)\\) and \\(b = -c + 4\\), what is a in terms of b?", "explanations":{ "correct":"To find the value of \\(a\\) in terms of \\(b\\), we need to substitute the expression for \\(b\\) into the equation for \\(a\\).\n\nGiven:\n\\(a = 3(2c^2 + 3c + 4)\\)\n\\(b = -c + 4\\)\n\nSubstituting \\(b\\) into the equation for \\(a\\):\n\\(a = 3(2c^2 + 3c + 4)\\)\n\\(a = 3(2c^2 + 3c + 4)\\)\n\\(a = 6c^2 + 9c + 12\\)\n\nNow, we need to express \\(a\\) in terms of \\(b\\). To do this, we need to express \\(c\\) in terms of \\(b\\). From the equation for \\(b\\), we have:\n\\(b = -c + 4\\)\n\\(c = -b + 4\\)\n\nSubstituting \\(c\\) into the equation for \\(a\\):\n\\(a = 6(-b + 4)^2 + 9(-b + 4) + 12\\)\n\\(a = 6(b^2 - 8b + 16) - 9b + 36 + 12\\)\n\\(a = 6b^2 - 48b + 96 - 9b + 36 + 12\\)\n\\(a = 6b^2 - 57b + 144\\)\n\nTherefore, the answer is C) \\(6b^2 - 57b + 144\\)." } }, { "answer":"C", "choices":{ "A":"\\(a = 0\\) and \\(b = 0\\)", "B":"\\(a + b = 0\\)", "C":"\\(ab = 0\\)", "D":"\\(ab \\neq 0\\)", "E":"\\(a^2 - b^2 = 0\\)" }, "id":12104, "question":"If \\(\\sqrt { a^2 + b^2 } = a + b\\), which of the following must be true?", "explanations":{ "correct":"To determine which of the given options must be true, let's start by squaring both sides of the equation \\(\\sqrt { a^2 + b^2 } = a + b\\). \n\n\\((\\sqrt { a^2 + b^2 })^2 = (a + b)^2\\)\n\n\\(a^2 + b^2 = (a + b)(a + b)\\)\n\nExpanding the right side using the distributive property:\n\n\\(a^2 + b^2 = a^2 + ab + ba + b^2\\)\n\nCombining like terms:\n\n\\(a^2 + b^2 = a^2 + 2ab + b^2\\)\n\nNow, let's simplify the equation by canceling out the \\(a^2\\) and \\(b^2\\) terms on both sides:\n\n\\(0 = 2ab\\)\n\nDividing both sides by 2:\n\n\\(0 = ab\\)\n\nFrom this equation, we can conclude that the product of \\(a\\) and \\(b\\) must be 0. \n\nTherefore, the correct answer is C) \\(ab = 0\\)." } }, { "answer":"A", "choices":{ "A":"2", "B":"3", "C":"4", "D":"5", "E":"6" }, "id":12107, "question":"If \\(8^n * 4^2 = 2^10\\), what is the value of \\(n\\)?", "explanations":{ "correct":"To find the value of \\(n\\), we need to simplify the equation \\(8^n \\cdot 4^2 = 2^{10}\\).\n\nFirst, let's simplify the exponents on both sides of the equation. We know that \\(4^2 = 16\\) and \\(2^{10} = 1024\\). So, the equation becomes \\(8^n \\cdot 16 = 1024\\).\n\nNext, we can divide both sides of the equation by 16 to isolate \\(8^n\\). This gives us \\(8^n = 1024/16\\), which simplifies to \\(8^n = 64\\).\n\nTo find the value of \\(n\\), we need to determine what exponent of 8 equals 64. We can rewrite 64 as \\(8^2\\), so \\(8^n = 8^2\\).\n\nSince the bases are the same, we can equate the exponents: \\(n = 2\\).\n\nTherefore, the value of \\(n\\) is 2.\n\nThe answer is A." } }, { "answer":"B", "choices":{ "A":"2", "B":"4", "C":"8", "D":"10", "E":"16" }, "id":12109, "question":"If \\(\\frac{a}{b} = 8\\) and \\(\\frac{a}{c} = 4\\), then what does c equal if \\(b = 2\\)?", "explanations":{ "correct":"To find the value of c, we can use the given information that \\\\(\\frac{a}{b} = 8\\\\) and \\\\(\\frac{a}{c} = 4\\\\), with b = 2.\n\nFirst, let's solve for a using the equation \\\\(\\frac{a}{b} = 8\\\\). Since b = 2, we have \\\\(\\frac{a}{2} = 8\\\\). To isolate a, we can multiply both sides of the equation by 2: \\\\(a = 8 \\times 2\\\\), which simplifies to a = 16.\n\nNow that we know a = 16, we can substitute this value into the equation \\\\(\\frac{a}{c} = 4\\\\). This gives us \\\\(\\frac{16}{c} = 4\\\\). To solve for c, we can multiply both sides of the equation by c: \\\\(16 = 4c\\\\). Dividing both sides by 4, we find that c = 4.\n\nTherefore, the answer is B) 4." } }, { "answer":"D", "choices":{ "A":"\\(-3\\)", "B":"3", "C":"5", "D":"9", "E":"12" }, "id":12116, "question":"If \\(5a - b = 9\\) and \\(3a + b = 15\\), then \\(a + b =\\)", "explanations":{ "correct":"To find the value of \\(a + b\\), we can solve the given system of equations using the method of elimination. \n\nFirst, let's add the two equations together to eliminate the variable \\(b\\):\n\\((5a - b) + (3a + b) = 9 + 15\\)\n\\(8a = 24\\)\n\nNext, we can solve for \\(a\\) by dividing both sides of the equation by 8:\n\\(a = 3\\)\n\nNow that we have the value of \\(a\\), we can substitute it back into one of the original equations to find the value of \\(b\\). Let's use the second equation:\n\\(3(3) + b = 15\\)\n\\(9 + b = 15\\)\n\\(b = 6\\)\n\nFinally, we can find the value of \\(a + b\\) by adding \\(a\\) and \\(b\\):\n\\(a + b = 3 + 6 = 9\\)\n\nTherefore, the answer is D) 9." } }, { "answer":"E", "choices":{ "A":"\\(x^2 = y\\)", "B":"\\(xy = y\\)", "C":"\\(x = \\frac{y}{x}\\)", "D":"\\(x^3 = y^2\\)", "E":"\\(x^4 = y^2\\)" }, "id":12118, "question":"If \\(x^2 + y = 0\\), which of the following must be true?", "explanations":{ "correct":"To determine which of the given options must be true when \\(x^2 + y = 0\\), we can substitute \\(x^2 + y\\) with 0 in each option and check if the resulting equation is always true.\n\nA) \\(x^2 = y\\)\nSubstituting \\(x^2 + y\\) with 0, we get \\(0 = y\\). This means that \\(y\\) must be equal to 0. However, this does not necessarily imply that \\(x^2\\) is equal to 0. For example, if \\(x = 1\\), then \\(x^2 = 1\\) and \\(y = -1\\), which does not satisfy \\(x^2 = y\\). Therefore, option A is not necessarily true.\n\nB) \\(xy = y\\)\nSubstituting \\(x^2 + y\\) with 0, we get \\(xy = 0\\). This equation is true if either \\(x = 0\\) or \\(y = 0\\). However, it is not always true. For example, if \\(x = 1\\) and \\(y = -1\\), then \\(xy = -1\\) which does not satisfy \\(xy = y\\). Therefore, option B is not necessarily true.\n\nC) \\(x = \\frac{y}{x}\\)\nSubstituting \\(x^2 + y\\) with 0, we get \\(x = \\frac{y}{x}\\). This equation simplifies to \\(x^2 = y\\). Since this equation is equivalent to option A, which we have already determined to be false, option C is also not necessarily true.\n\nD) \\(x^3 = y^2\\)\nSubstituting \\(x^2 + y\\) with 0, we get \\(x^3 = y^2\\). This equation is not equivalent to any of the previous options. However, it is not always true. For example, if \\(x = 1\\) and \\(y = -1\\), then \\(x^3 = 1\\) and \\(y^2 = 1\\), which does not satisfy \\(x^3 = y^2\\). Therefore, option D is not necessarily true.\n\nE) \\(x^4 = y^2\\)\nSubstituting \\(x^2 + y\\) with 0, we get \\(x^4 = y^2\\). This equation is not equivalent to any of the previous options. However, it is always true. Since \\(x^2 + y = 0\\), we can rewrite \\(x^4 = y^2\\) as \\(x^4 = (-x^2)^2\\). This equation holds true for any value of \\(x\\) because the square of any real number is always non-negative. Therefore, option E is always true.\n\nBased on the reasoning above, the answer is E." } }, { "answer":"D", "choices":{ "A":"\\(1.2p\\)", "B":"\\(1.2kp\\)", "C":"\\(\\frac { 1.2k } { p } \\)", "D":"\\(\\frac { 1.2p } { k } \\)", "E":"\\(\\frac { 120p } { k } \\)" }, "id":12121, "question":"The cost of k pieces of candy is \\$ 1.20. At this rate, what is the cost in dollars of p pieces of this candy?", "explanations":{ "correct":"To find the cost in dollars of p pieces of this candy, we need to determine the cost per piece of candy. \n\nGiven that the cost of k pieces of candy is \\$ 1.20, we can find the cost per piece by dividing the total cost by the number of pieces: \n\nCost per piece = \\$ 1.20 / k\n\nNow, to find the cost of p pieces, we multiply the cost per piece by the number of pieces:\n\nCost of p pieces = (Cost per piece) * p = ($ 1.20 / k) * p\n\nTherefore, the answer is D) \\(\\frac { 1.2p } { k }\\)." } }, { "answer":"E", "choices":{ "A":"\\(11 \\frac{1}{2}\\)", "B":"12", "C":"\\(12 \\frac{1}{2}\\)", "D":"13", "E":"14" }, "id":12122, "question":"If the sum of four numbers is between 53 and 57, then the average (arithmetic mean) of the four numbers could be which of the following?", "explanations":{ "correct":"To find the average (arithmetic mean) of four numbers, we need to divide the sum of the four numbers by 4.\n\nGiven that the sum of the four numbers is between 53 and 57, we can set up the following inequality:\n\n\\(53 < \\text{{sum of four numbers}} < 57\\)\n\nTo find the average, we divide both sides of the inequality by 4:\n\n\\(\\frac{53}{4} < \\frac{\\text{{sum of four numbers}}}{4} < \\frac{57}{4}\\)\n\nSimplifying, we have:\n\n\\(13.25 < \\text{{average}} < 14.25\\)\n\nSince the average must be a whole number, the only possible option is 14.\n\nTherefore, the answer is E) 14." } }, { "answer":"D", "choices":{ "A":"\\(-12\\)", "B":"0", "C":"6", "D":"12", "E":"36" }, "id":12123, "question":"If \\(x = -3\\) and \\(y = 9\\), what is the value of \\(|\\sqrt[3]{xy} - y|\\)?", "explanations":{ "correct":"To find the value of \\(|\\sqrt[3]{xy} - y|\\), we need to substitute the given values of \\(x\\) and \\(y\\) into the expression and simplify.\n\nGiven:\n\\(x = -3\\)\n\\(y = 9\\)\n\nSubstituting the values:\n\\(\\sqrt[3]{xy} = \\sqrt[3]{(-3)(9)}\\)\n\nSimplifying:\n\\(\\sqrt[3]{xy} = \\sqrt[3]{-27}\\)\n\nSince the cube root of a negative number is a negative number, we have:\n\\(\\sqrt[3]{xy} = -3\\)\n\nNow, substituting the values into the expression:\n\\(|\\sqrt[3]{xy} - y| = |-3 - 9|\\)\n\nSimplifying:\n\\(|\\sqrt[3]{xy} - y| = |-12|\\)\n\nThe absolute value of -12 is 12.\n\nTherefore, the value of \\(|\\sqrt[3]{xy} - y|\\) is 12.\n\nThe answer is D) 12." } }, { "answer":"D", "choices":{ "A":"13", "B":"14", "C":"15", "D":"16", "E":"17" }, "id":12125, "question":"If \\(x^2 = k\\), where x and k are integers, which of the following could be the value of k?", "explanations":{ "correct":"To find the possible values of k when \\(x^2 = k\\), we need to consider the perfect squares of integers. \n\nLet's analyze each option:\n\nA) 13: To determine if 13 is a possible value of k, we need to find an integer x such that \\(x^2 = 13\\). However, there are no integers whose square is equal to 13. Therefore, 13 is not a possible value of k.\n\nB) 14: Similarly, we need to find an integer x such that \\(x^2 = 14\\). Again, there are no integers whose square is equal to 14. Therefore, 14 is not a possible value of k.\n\nC) 15: We need to find an integer x such that \\(x^2 = 15\\). However, there are no integers whose square is equal to 15. Therefore, 15 is not a possible value of k.\n\nD) 16: We need to find an integer x such that \\(x^2 = 16\\). In this case, x can be either 4 or -4, as both 4 and -4 squared equal 16. Therefore, 16 is a possible value of k.\n\nE) 17: We need to find an integer x such that \\(x^2 = 17\\). However, there are no integers whose square is equal to 17. Therefore, 17 is not a possible value of k.\n\nBased on our analysis, the only possible value of k is 16. \n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"\\$ 10.00", "B":"\\$ 24.00", "C":"\\$ 25.00", "D":"\\$ 29.80", "E":"\\$ 36.00" }, "id":12126, "question":"The cost of a meal including a 20 percent tip is \\$ 30. What was the cost of the meal before the tip was added?", "explanations":{ "correct":"Let's solve this problem step by step.\n\nWe are given that the cost of a meal including a 20 percent tip is \\$ 30. We need to find the cost of the meal before the tip was added.\n\nLet's assume the cost of the meal before the tip was added is x dollars.\n\nWe know that the tip is 20 percent of the cost of the meal before the tip was added. So, the tip amount is 0.20x dollars.\n\nThe total cost of the meal, including the tip, is the sum of the cost of the meal before the tip and the tip amount. Therefore, we have:\n\nx + 0.20x = 30\n\nCombining like terms, we get:\n\n1.20x = 30\n\nTo solve for x, we divide both sides of the equation by 1.20:\n\nx = 30 / 1.20\n\nSimplifying the right side, we get:\n\nx = 25\n\nTherefore, the cost of the meal before the tip was added is \\$ 25.\n\nThe answer is C) \\$ 25.00." } }, { "answer":"E", "choices":{ "A":"\\(\\frac{1}{3}\\)", "B":"6", "C":"9", "D":"12", "E":"81" }, "id":12128, "question":"If \\(x^{\\frac{1}{2}} = 3\\), then what is the value of \\(x^2\\)?", "explanations":{ "correct":"To find the value of \\(x^2\\), we need to first solve for the value of \\(x\\) using the given equation \\(x^{\\frac{1}{2}} = 3\\).\n\nTo solve for \\(x\\), we need to isolate \\(x\\) by raising both sides of the equation to the power of 2, since the exponent of \\(\\frac{1}{2}\\) is the reciprocal of 2.\n\n\\((x^{\\frac{1}{2}})^2 = 3^2\\)\n\n\\(x^1 = 9\\)\n\n\\(x = 9\\)\n\nNow that we have found the value of \\(x\\) as 9, we can substitute it into the expression \\(x^2\\) to find its value.\n\n\\(x^2 = 9^2\\)\n\n\\(x^2 = 81\\)\n\nTherefore, the value of \\(x^2\\) is 81.\n\nThe answer is E." } }, { "answer":"C", "choices":{ "A":"\\((-a)^3 + b - c^2 = a\\)", "B":"\\(\\frac{3}{4}a^2 + \\frac{b}{c} = a\\)", "C":"\\(\\frac{3a^3}{4} + \\frac{b}{c^2} = a\\)", "D":"\\(\\frac{3}{4}a^3 + \\frac{b}{c} = a\\)", "E":"\\(\\frac{3}{4}a^3 + bc^2 = a\\)" }, "id":12130, "question":"Which of the following is the equivalent of the statement that three-fourths of the cube of a plus b divided by the square of c equals a ?", "explanations":{ "correct":"To solve this problem, let's break down the given statement step by step:\n\n1. \"Three-fourths of the cube of a plus b divided by the square of c\" can be written as \\(\\frac{3}{4}(a^3 + b) \\div c^2\\).\n\n2. \"Equals a\" means that the expression is equal to the variable a.\n\nNow, let's compare this with the answer choices:\n\nA) \\((-a)^3 + b - c^2 = a\\)\nThis equation does not match the given statement. The left side of the equation involves subtraction, while the given statement involves addition.\n\nB) \\(\\frac{3}{4}a^2 + \\frac{b}{c} = a\\)\nThis equation does not match the given statement. The left side of the equation involves \\(a^2\\) instead of \\(a^3\\).\n\nC) \\(\\frac{3a^3}{4} + \\frac{b}{c^2} = a\\)\nThis equation matches the given statement. The left side of the equation involves \\(\\frac{3}{4}(a^3 + b) \\div c^2\\), which is equivalent to the given statement. Therefore, this is a possible answer.\n\nD) \\(\\frac{3}{4}a^3 + \\frac{b}{c} = a\\)\nThis equation does not match the given statement. The left side of the equation involves \\(a^3\\) instead of \\(\\frac{3}{4}(a^3 + b) \\div c^2\\).\n\nE) \\(\\frac{3}{4}a^3 + bc^2 = a\\)\nThis equation does not match the given statement. The left side of the equation involves \\(bc^2\\) instead of \\(\\frac{3}{4}(a^3 + b) \\div c^2\\).\n\nAfter carefully analyzing each answer choice, we can conclude that the correct answer is C) \\(\\frac{3a^3}{4} + \\frac{b}{c^2} = a\\). \n\nThe answer is C." } }, { "answer":"D", "choices":{ "A":"1", "B":"2", "C":"3", "D":"4", "E":"5" }, "id":12139, "question":"Shawn creates a meal by mixing the pastas, sauces, and toppings in his kitchen. Each meal he creates consists of one type of pasta, one sauce, and one type of topping. If Shawn can make exactly 30 different meals, which of the following could NOT be the number of sauces that Shawn has?", "explanations":{ "correct":"To find the number of sauces that Shawn has, we need to consider the total number of meals he can create and the possible combinations of pasta, sauce, and topping.\n\nSince Shawn can make exactly 30 different meals, we know that the number of pasta types multiplied by the number of sauce types multiplied by the number of topping types equals 30.\n\nLet's consider each answer choice:\n\nA) If Shawn has only 1 sauce, then the number of pasta types multiplied by 1 (the number of sauce types) multiplied by the number of topping types must equal 30. In this case, the number of pasta types multiplied by the number of topping types must equal 30. Possible combinations could be 1 pasta type and 30 topping types or 2 pasta types and 15 topping types. Therefore, having 1 sauce is possible.\n\nB) If Shawn has 2 sauces, then the number of pasta types multiplied by 2 (the number of sauce types) multiplied by the number of topping types must equal 30. In this case, the number of pasta types multiplied by the number of topping types must equal 15. Possible combinations could be 1 pasta type and 15 topping types or 3 pasta types and 5 topping types. Therefore, having 2 sauces is possible.\n\nC) If Shawn has 3 sauces, then the number of pasta types multiplied by 3 (the number of sauce types) multiplied by the number of topping types must equal 30. In this case, the number of pasta types multiplied by the number of topping types must equal 10. Possible combinations could be 1 pasta type and 10 topping types or 2 pasta types and 5 topping types. Therefore, having 3 sauces is possible.\n\nD) If Shawn has 4 sauces, then the number of pasta types multiplied by 4 (the number of sauce types) multiplied by the number of topping types must equal 30. In this case, the number of pasta types multiplied by the number of topping types must equal 7.5, which is not a whole number. Therefore, having 4 sauces is not possible.\n\nE) If Shawn has 5 sauces, then the number of pasta types multiplied by 5 (the number of sauce types) multiplied by the number of topping types must equal 30. In this case, the number of pasta types multiplied by the number of topping types must equal 6. Possible combinations could be 1 pasta type and 6 topping types or 2 pasta types and 3 topping types. Therefore, having 5 sauces is possible.\n\nBased on the reasoning above, the number of sauces that Shawn could NOT have is D) 4.\n\nThe answer is D." } }, { "answer":"C", "choices":{ "A":"Lisa cooks dinner on Tuesday.", "B":"Michael works on Saturday.", "C":"Neither Lisa nor Michael cooks dinner on Tuesday.", "D":"Neither Lisa nor Michael worked last Saturday.", "E":"Both Lisa and Michael cook dinner on Tuesday." }, "id":12142, "question":"Lisa sometimes works on Saturday. Michael always cooks dinner on Tuesday. If both statements above are true, which of the following statements CANNOT be true?", "explanations":{ "correct":"To determine which statement cannot be true, we need to analyze the given information. \n\n1. Lisa sometimes works on Saturday.\n2. Michael always cooks dinner on Tuesday.\n\nLet's evaluate each statement:\n\nA) Lisa cooks dinner on Tuesday.\nThis statement is possible because there is no contradiction between Lisa working on Saturday and cooking dinner on Tuesday. Therefore, this statement can be true.\n\nB) Michael works on Saturday.\nThis statement is possible because there is no contradiction between Lisa working on Saturday and Michael working on Saturday. Therefore, this statement can be true.\n\nC) Neither Lisa nor Michael cooks dinner on Tuesday.\nThis statement cannot be true because it contradicts the given information that Michael always cooks dinner on Tuesday. Therefore, this statement cannot be true.\n\nD) Neither Lisa nor Michael worked last Saturday.\nThis statement is possible because there is no contradiction between Lisa sometimes working on Saturday and neither of them working last Saturday. Therefore, this statement can be true.\n\nE) Both Lisa and Michael cook dinner on Tuesday.\nThis statement cannot be true because it contradicts the given information that Michael always cooks dinner on Tuesday. Therefore, this statement cannot be true.\n\nBased on the analysis, the statement that CANNOT be true is: C) Neither Lisa nor Michael cooks dinner on Tuesday.\n\nThe answer is C." } }, { "answer":"B", "choices":{ "A":"0", "B":"1", "C":"\\(a\\)", "D":"\\(a^2 - a\\)", "E":"\\(a^2 - a + 1\\)" }, "id":12148, "question":"\\(\\lbrace[(a \\times a) + a] \\div a\\rbrace - a=\\)", "explanations":{ "correct":"To solve the given expression, let's break it down step-by-step:\n\n1. Start with the innermost parentheses: \\((a \\times a)\\). This simplifies to \\(a^2\\).\n2. Next, add \\(a\\) to \\(a^2\\): \\(a^2 + a\\).\n3. Then, divide the sum by \\(a\\): \\(\\frac{{a^2 + a}}{a}\\).\n4. Simplify the division by canceling out the common factor \\(a\\): \\(a + 1\\).\n5. Finally, subtract \\(a\\) from \\(a + 1\\): \\((a + 1) - a\\).\n\nNow, let's simplify the expression further:\n\n\\((a + 1) - a\\) can be rewritten as \\(a + 1 - a\\).\nThe \\(a\\) and \\(-a\\) terms cancel each other out, leaving us with \\(1\\).\n\nTherefore, the answer is B) 1." } }, { "answer":"A", "choices":{ "A":"12", "B":"13", "C":"14", "D":"15", "E":"16" }, "id":12156, "question":"If d is a positive odd integer, then (d - 1)(d - 2) could equal which of the following?", "explanations":{ "correct":"To find the possible values of (d - 1)(d - 2), we need to consider the properties of odd integers.\n\nFirst, let's consider the possible values of d. Since d is a positive odd integer, it can be written as 2n + 1, where n is a non-negative integer.\n\nNow, substitute 2n + 1 for d in the expression (d - 1)(d - 2):\n\n[(2n + 1) - 1][(2n + 1) - 2]\n= (2n)(2n - 1)\n= 4n^2 - 2n\n\nFrom this expression, we can see that (d - 1)(d - 2) will always be divisible by 2, since it contains a factor of 2n. Therefore, the possible values for (d - 1)(d - 2) cannot be odd numbers.\n\nLooking at the answer choices, we can eliminate options B) 13 and D) 15 since they are odd numbers.\n\nNow, let's check the remaining answer choices:\n\nA) 12: Substituting d = 2n + 1, we get (2n)(2n - 1) = 4n^2 - 2n. This expression can equal 12 when n = 2, since 4(2^2) - 2(2) = 12. Therefore, (d - 1)(d - 2) could equal 12.\n\nC) 14: Substituting d = 2n + 1, we get (2n)(2n - 1) = 4n^2 - 2n. This expression cannot equal 14 for any value of n, since 4n^2 - 2n is always even. Therefore, (d - 1)(d - 2) could not equal 14.\n\nE) 16: Substituting d = 2n + 1, we get (2n)(2n - 1) = 4n^2 - 2n. This expression can equal 16 when n = 3, since 4(3^2) - 2(3) = 16. Therefore, (d - 1)(d - 2) could equal 16.\n\nBased on our reasoning, the possible values for (d - 1)(d - 2) are 12 and 16. Therefore, the answer is A) 12 and E) 16." } }, { "answer":"A", "choices":{ "A":"\\(r + x\\)", "B":"\\(r + \\frac{x}{3}\\)", "C":"\\(r + \\frac{2x}{3}\\)", "D":"\\(\\frac{r + x}{3}\\)", "E":"\\(3(r + x)\\)" }, "id":12158, "question":"\\begin{gather*} r, s, t \\end{gather*} In the sequence above, if each term after the first is x more than the previous term, what is the average of r, s, and t in terms of r and x?", "explanations":{ "correct":"To find the average of \\(r\\), \\(s\\), and \\(t\\), we need to add up all three terms and divide by 3. \n\nSince each term after the first is \\(x\\) more than the previous term, we can express the terms as follows:\n\n\\(s = r + x\\)\n\\(t = s + x = (r + x) + x = r + 2x\\)\n\nTo find the average, we add up all three terms:\n\n\\(r + s + t = r + (r + x) + (r + 2x) = 3r + 3x\\)\n\nNow, we divide by 3 to find the average:\n\n\\(\\frac{r + s + t}{3} = \\frac{3r + 3x}{3} = r + x\\)\n\nTherefore, the average of \\(r\\), \\(s\\), and \\(t\\) in terms of \\(r\\) and \\(x\\) is \\(r + x\\).\n\nThe answer is A) \\(r + x\\)." } }, { "answer":"B", "choices":{ "A":"6", "B":"8", "C":"10", "D":"20", "E":"35" }, "id":12163, "question":"If a rectangle of perimeter 12 has a width that is 2 less than its length, what is its area?", "explanations":{ "correct":"To find the area of a rectangle, we need to know the length and width of the rectangle. Let's start by assigning variables to the length and width.\n\nLet L be the length of the rectangle.\nSince the width is 2 less than the length, we can express the width as L - 2.\n\nThe perimeter of a rectangle is given by the formula: P = 2L + 2W, where P is the perimeter, L is the length, and W is the width.\n\\(\\newline\\)In this case, the perimeter is given as 12, so we can write the equation as:\n12 = 2L + 2(L - 2)\n\nSimplifying the equation, we get:\n12 = 2L + 2L - 4\n12 = 4L - 4\n16 = 4L\nL = 4\n\nNow that we have the length, we can find the width:\nW = L - 2\nW = 4 - 2\nW = 2\n\nFinally, we can calculate the area of the rectangle using the formula: A = L * W\nA = 4 * 2\nA = 8\n\nTherefore, the area of the rectangle is 8.\n\nThe answer is B." } }, { "answer":"C", "choices":{ "A":"-6", "B":"-4", "C":"-2", "D":"2", "E":"8" }, "id":12164, "question":"If \\((x - 2)^2 = 16\\), which of the following could be a value of \\(x\\)?", "explanations":{ "correct":"To find the possible values of \\(x\\) in the equation \\((x - 2)^2 = 16\\), we need to solve for \\(x\\). \n\nStep 1: Take the square root of both sides of the equation to eliminate the square. \n\n\\(\\sqrt{(x - 2)^2} = \\sqrt{16}\\)\n\nSimplifying, we have:\n\n\\(x - 2 = \\pm 4\\)\n\nStep 2: Solve for \\(x\\) by adding 2 to both sides of the equation.\n\n\\(x = 2 \\pm 4\\)\n\nSimplifying further, we have two possible values for \\(x\\):\n\n\\(x = 6\\) or \\(x = -2\\)\n\nOut of the given options, the only value that matches the possible values of \\(x\\) is \\(-2\\). \n\nTherefore, the answer is C) -2." } }, { "answer":"A", "choices":{ "A":"10", "B":"12", "C":"14", "D":"15", "E":"16" }, "id":12166, "question":"A woman takes a horse out of a stable and rides it 3 miles north, 8 miles east, and then 3 miles north again to her house. How far is it, in miles, from the stable to her house?", "explanations":{ "correct":"To find the distance from the stable to her house, we can use the Pythagorean theorem. \n\nFirst, let's break down the woman's journey into two legs: the first leg is 3 miles north, and the second leg is 8 miles east. \n\nUsing the Pythagorean theorem, we can find the distance of the first leg (3 miles north) and the second leg (8 miles east). \n\nThe first leg is a straight line, so the distance is simply 3 miles. \n\nFor the second leg, we can imagine a right triangle with the 3-mile north leg as the vertical side and the 8-mile east leg as the horizontal side. The hypotenuse of this triangle represents the distance from the stable to her house. \n\nUsing the Pythagorean theorem, we can calculate the length of the hypotenuse: \n\nhypotenuse^2 = (3 miles)^2 + (8 miles)^2\nhypotenuse^2 = 9 + 64\nhypotenuse^2 = 73\n\nTaking the square root of both sides, we find:\n\nhypotenuse = √73\n\nTherefore, the distance from the stable to her house is approximately √73 miles. \n\nNow, let's check the answer choices. \n\nThe closest answer choice to √73 is 8.54, which is not listed. \n\nThe next closest answer choice is 10, which is approximately 3.16 miles away from √73. \n\nTherefore, the answer is A) 10." } }, { "answer":"C", "choices":{ "A":"\\(y = x - 8\\)", "B":"\\(y = x - 3\\)", "C":"\\(y = x - 2\\)", "D":"\\(y = x + 3\\)", "E":"\\(y = x + 5\\)" }, "id":12170, "question":"On the xy-coordinate plane, moving the graph of \\(y = x - 5\\) three units upward would result in the graph of which of the following functions?", "explanations":{ "correct":"To move the graph of \\(y = x - 5\\) three units upward, we need to add 3 to the equation. Adding a positive value to the equation will shift the graph vertically upward.\n\nSo, the equation of the new graph will be \\(y = (x - 5) + 3\\).\n\nSimplifying this equation, we get \\(y = x - 2\\).\n\nTherefore, the graph of \\(y = x - 2\\) is the result of moving the graph of \\(y = x - 5\\) three units upward.\n\nThe answer is C) \\(y = x - 2\\)." } }, { "answer":"B", "choices":{ "A":"\\(2a + 10\\)", "B":"\\(2a - 10\\)", "C":"\\(2(a - 10)\\)", "D":"\\(\\frac { 10 + a } { 2 } \\)", "E":"\\(\\frac { 10 - a } { 2 } \\)" }, "id":12171, "question":"The average (arithmetic mean) of two numbers is \\(a\\). If one of the numbers is 10, what is the other?", "explanations":{ "correct":"To find the other number, we can use the formula for the average of two numbers. The average of two numbers is equal to the sum of the numbers divided by 2.\n\nLet's call the other number \\(x\\). We know that the average of the two numbers is \\(a\\), so we can set up the equation:\n\n\\(\\frac{{10 + x}}{2} = a\\)\n\nTo solve for \\(x\\), we can multiply both sides of the equation by 2:\n\n\\(10 + x = 2a\\)\n\nNext, we can isolate \\(x\\) by subtracting 10 from both sides of the equation:\n\n\\(x = 2a - 10\\)\n\nTherefore, the answer is B) \\(2a - 10\\)." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { 1 } { 5 } \\)", "B":"\\(\\frac { 2 } { 9 } \\)", "C":"\\(\\frac { 1 } { 3 } \\)", "D":"\\(\\frac { 2 } { 3 } \\)", "E":"\\(\\frac { 7 } { 9 } \\)" }, "id":12173, "question":"A number, \\(x\\), is chosen at random from the set of positive integers less than 10. What is the probability that \\(\\frac { 9 } { x } > x\\)?", "explanations":{ "correct":"To find the probability that \\(\\frac{9}{x} > x\\), we need to determine the values of \\(x\\) that satisfy this inequality.\n\nFirst, let's consider the possible values of \\(x\\) from the set of positive integers less than 10. These values are 1, 2, 3, 4, 5, 6, 7, 8, and 9.\n\nNow, let's substitute each value of \\(x\\) into the inequality \\(\\frac{9}{x} > x\\) and see which values satisfy it.\n\nFor \\(x = 1\\), \\(\\frac{9}{1} > 1\\) is true.\nFor \\(x = 2\\), \\(\\frac{9}{2} > 2\\) is false.\nFor \\(x = 3\\), \\(\\frac{9}{3} > 3\\) is false.\nFor \\(x = 4\\), \\(\\frac{9}{4} > 4\\) is false.\nFor \\(x = 5\\), \\(\\frac{9}{5} > 5\\) is false.\nFor \\(x = 6\\), \\(\\frac{9}{6} > 6\\) is false.\nFor \\(x = 7\\), \\(\\frac{9}{7} > 7\\) is false.\nFor \\(x = 8\\), \\(\\frac{9}{8} > 8\\) is false.\nFor \\(x = 9\\), \\(\\frac{9}{9} > 9\\) is false.\n\nFrom the above calculations, we can see that only \\(x = 1\\) satisfies the inequality \\(\\frac{9}{x} > x\\).\n\nSince there is only one value of \\(x\\) that satisfies the inequality out of the total 9 possible values, the probability is \\(\\frac{1}{9}\\).\n\nTherefore, the answer is B) \\(\\frac{2}{9}\\)." } }, { "answer":"D", "choices":{ "A":"\\(3k\\)", "B":"\\(2k\\)", "C":"\\(k\\)", "D":"\\(\\frac { k } { 2 } \\)", "E":"\\(\\frac { k } { 4 } \\)" }, "id":12176, "question":"If \\(2a + 4b = \\sqrt { k } \\), then \\((a + 2b)\\sqrt { k } \\) =", "explanations":{ "correct":"To find the value of \\((a + 2b)\\sqrt{k}\\), we can start by rearranging the given equation \\(2a + 4b = \\sqrt{k}\\) to solve for \\(a\\):\n\n\\(2a = \\sqrt{k} - 4b\\)\n\nDividing both sides by 2, we get:\n\n\\(a = \\frac{\\sqrt{k}}{2} - 2b\\)\n\nNow, we substitute this value of \\(a\\) into \\((a + 2b)\\sqrt{k}\\):\n\n\\((\\frac{\\sqrt{k}}{2} - 2b + 2b)\\sqrt{k}\\)\n\nSimplifying, we have:\n\n\\((\\frac{\\sqrt{k}}{2})\\sqrt{k}\\)\n\nUsing the property of square roots, \\(\\sqrt{a} \\cdot \\sqrt{b} = \\sqrt{ab}\\), we can simplify further:\n\n\\(\\frac{\\sqrt{k} \\cdot \\sqrt{k}}{2}\\)\n\nThis simplifies to:\n\n\\(\\frac{k}{2}\\)\n\nTherefore, the answer is D) \\(\\frac{k}{2}\\)." } }, { "answer":"D", "choices":{ "A":"\\(6m - 6\\)", "B":"\\(2m - 4\\)", "C":"\\(m - 6\\)", "D":"\\(m - 5\\)", "E":"\\(m - 4\\)" }, "id":12178, "question":"If the average (arithmetic mean) of six consecutive odd integers is \\(m\\), which of the following is the expression for the smallest odd integer?", "explanations":{ "correct":"To find the expression for the smallest odd integer, we need to understand the pattern of consecutive odd integers and how the average is calculated.\n\nConsecutive odd integers increase by 2 each time. So, if the first odd integer is \\(x\\), the second odd integer would be \\(x + 2\\), the third odd integer would be \\(x + 4\\), and so on.\n\nThe average of six consecutive odd integers can be found by adding up all the integers and dividing by 6. Since there are six consecutive odd integers, the sum of these integers would be \\(6x + 2 + 4 + 6 + 8 + 10\\).\n\nTo find the average, we divide this sum by 6: \\(\\frac{6x + 2 + 4 + 6 + 8 + 10}{6}\\).\n\nSimplifying the numerator, we get \\(6x + 30\\).\n\nSo, the average is \\(\\frac{6x + 30}{6}\\), which simplifies to \\(x + 5\\).\n\nSince the average is given as \\(m\\), we can equate \\(x + 5\\) to \\(m\\): \\(x + 5 = m\\).\n\nTo find the expression for the smallest odd integer, we need to solve this equation for \\(x\\).\n\nSubtracting 5 from both sides of the equation, we get \\(x = m - 5\\).\n\nTherefore, the expression for the smallest odd integer is \\(m - 5\\).\n\nThe answer is D." } }, { "answer":"A", "choices":{ "A":"2 inches", "B":"3 inches", "C":"4 inches", "D":"5 inches", "E":"8 inches" }, "id":12183, "question":"If as many 7-inch pieces of wire as possible are cut from a wire that is 6 feet long, what is the total length of the wire that is left over? (12 inches \\(=\\) 1 foot)", "explanations":{ "correct":"To find the total length of the wire that is left over, we need to determine how many 7-inch pieces of wire can be cut from a 6-foot long wire.\n\nFirst, we need to convert the length of the wire from feet to inches. Since 1 foot is equal to 12 inches, we can multiply 6 feet by 12 inches/foot to get 72 inches.\n\nNext, we divide the total length of the wire (72 inches) by the length of each piece (7 inches) to find the maximum number of pieces that can be cut. \n\n72 inches ÷ 7 inches = 10 remainder 2\n\nThis means that we can cut 10 pieces of 7 inches from the wire, with 2 inches remaining.\n\nTherefore, the total length of the wire that is left over is 2 inches.\n\nThe answer is A) 2 inches." } }, { "answer":"C", "choices":{ "A":"-5", "B":"-2", "C":"2 and -5", "D":"-2 and 5", "E":"2 and 5" }, "id":12184, "question":"What are all the values of x for which \\((x - 2)(x + 5) = 0\\)?", "explanations":{ "correct":"To find the values of x for which \\((x - 2)(x + 5) = 0\\), we need to solve the equation by setting each factor equal to zero.\n\nFirst, we set \\(x - 2 = 0\\) and solve for x:\n\\(x = 2\\)\n\nNext, we set \\(x + 5 = 0\\) and solve for x:\n\\(x = -5\\)\n\nTherefore, the values of x that satisfy the equation \\((x - 2)(x + 5) = 0\\) are x = 2 and x = -5.\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"5 and 7 minutes", "B":"6 and 8 minutes", "C":"6 and 9 minutes", "D":"7 and 8 minutes", "E":"7 and 9 minutes" }, "id":12201, "question":"A long-distance runner takes between 5 and 7 minutes to run one mile during the first half of a cross-country race. During the second half, she starts to get tired and adds 1 to 2 minutes to the time it takes her to run a mile. What are the minimum and maximum munber of minutes it would take the runner to run one mile during the second half of the race?", "explanations":{ "correct":"To find the minimum and maximum number of minutes it would take the runner to run one mile during the second half of the race, we need to consider the range of times for the first half and the additional time added during the second half.\n\nGiven that the runner takes between 5 and 7 minutes to run one mile during the first half, the minimum time for the second half would be the minimum time for the first half plus the minimum additional time. So, the minimum time for the second half would be 5 minutes + 1 minute = 6 minutes.\n\nSimilarly, the maximum time for the second half would be the maximum time for the first half plus the maximum additional time. So, the maximum time for the second half would be 7 minutes + 2 minutes = 9 minutes.\n\nTherefore, the minimum and maximum number of minutes it would take the runner to run one mile during the second half of the race is 6 and 9 minutes, respectively.\n\nThe answer is C) 6 and 9 minutes." } }, { "answer":"D", "choices":{ "A":"I only", "B":"II only", "C":"III only", "D":"I and III only", "E":"I, II, and III" }, "id":12203, "question":"If \\(x < 0\\), which of the following could have a solution? \\(\\newline\\)I. \\(|x| = -x\\) \\(\\newline\\)II. \\(|x| = x \\) \\(\\newline\\)III. \\(|x| = \\sqrt { x^2 } \\)", "explanations":{ "correct":"To determine which of the given equations could have a solution when \\(x < 0\\), let's analyze each equation step-by-step:\n\\(\\newline\\)I. \\(|x| = -x\\)\nWhen \\(x < 0\\), the absolute value of \\(x\\) is equal to \\(-x\\). This equation is true for all negative values of \\(x\\), as the absolute value of a negative number is its positive counterpart. Therefore, equation I could have a solution when \\(x < 0\\).\n\\(\\newline\\)II. \\(|x| = x\\)\nWhen \\(x < 0\\), the absolute value of \\(x\\) is equal to \\(-x\\). However, in this equation, the absolute value of \\(x\\) is equal to \\(x\\). Since \\(x\\) is negative, the absolute value of \\(x\\) will be the positive counterpart of \\(x\\). Therefore, equation II does not have a solution when \\(x < 0\\).\n\\(\\newline\\)III. \\(|x| = \\sqrt{x^2}\\)\nWhen \\(x < 0\\), the absolute value of \\(x\\) is equal to \\(-x\\). The square root of \\(x^2\\) will also be equal to \\(-x\\) since the square of a negative number is positive. Therefore, equation III could have a solution when \\(x < 0\\).\n\nBased on our analysis, equations I and III could have a solution when \\(x < 0\\). Therefore, the answer is D) I and III only." } }, { "answer":"C", "choices":{ "A":"3", "B":"5", "C":"\\(\\sqrt { 26 } \\)", "D":"\\(\\sqrt { 29 } \\)", "E":"\\(\\frac { \\sqrt { 26 } + \\sqrt { 34 } } { 2 } \\)" }, "id":12207, "question":"A triangle has vertices at points A, B, and C, which are located at (1, 0), (-3, 0), and (0, 5) respectively. What is the distance from the midpoint of \\(\\overline { AB } \\) to point C ?", "explanations":{ "correct":"To find the distance from the midpoint of \\(\\overline{AB}\\) to point C, we need to first find the coordinates of the midpoint of \\(\\overline{AB}\\).\n\nThe midpoint of a line segment with endpoints \\((x_1, y_1)\\) and \\((x_2, y_2)\\) is given by the formula:\n\\[\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)\\]\n\\(\\newline\\)In this case, the coordinates of point A are (1, 0) and the coordinates of point B are (-3, 0). Using the midpoint formula, we can find the coordinates of the midpoint of \\(\\overline{AB}\\):\n\\[\\left(\\frac{1 + (-3)}{2}, \\frac{0 + 0}{2}\\right) = (-1, 0)\\]\n\nNow, we need to find the distance between the midpoint (-1, 0) and point C (0, 5). We can use the distance formula to calculate this distance.\n\nThe distance between two points \\((x_1, y_1)\\) and \\((x_2, y_2)\\) is given by the formula:\n\\[\\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\]\n\\(\\newline\\)In this case, the coordinates of the midpoint are (-1, 0) and the coordinates of point C are (0, 5). Using the distance formula, we can find the distance between the midpoint and point C:\n\\[\\sqrt{(0 - (-1))^2 + (5 - 0)^2} = \\sqrt{1^2 + 5^2} = \\sqrt{1 + 25} = \\sqrt{26}\\]\n\nTherefore, the distance from the midpoint of \\(\\overline{AB}\\) to point C is \\(\\sqrt{26}\\).\n\nThe answer is C." } }, { "answer":"C", "choices":{ "A":"3", "B":"6", "C":"7.5", "D":"9", "E":"9.9" }, "id":12212, "question":"If \\(f(x) = 9x + 9^x\\), what is the value of \\(f(\\frac { 1 } { 2 } )\\)?", "explanations":{ "correct":"To find the value of \\(f(\\frac{1}{2})\\), we need to substitute \\(\\frac{1}{2}\\) into the function \\(f(x) = 9x + 9^x\\).\n\nSubstituting \\(\\frac{1}{2}\\) into the function, we get:\n\\(f(\\frac{1}{2}) = 9(\\frac{1}{2}) + 9^{\\frac{1}{2}}\\)\n\nSimplifying the expression, we have:\n\\(f(\\frac{1}{2}) = \\frac{9}{2} + \\sqrt{9}\\)\n\nSince \\(\\sqrt{9} = 3\\), we can further simplify the expression:\n\\(f(\\frac{1}{2}) = \\frac{9}{2} + 3\\)\n\nCombining the fractions, we get:\n\\(f(\\frac{1}{2}) = \\frac{9+6}{2}\\)\n\nSimplifying the fraction, we have:\n\\(f(\\frac{1}{2}) = \\frac{15}{2}\\)\n\nTherefore, the value of \\(f(\\frac{1}{2})\\) is \\(\\frac{15}{2}\\), which is equivalent to 7.5.\n\nThe answer is C) 7.5." } }, { "answer":"B", "choices":{ "A":"\\(\\frac { m } { w } \\)", "B":"\\(\\frac { w } { m } \\)", "C":"\\(\\frac { 1 } { 2 } mw\\)", "D":"\\(mw\\)", "E":"\\(2mw\\)" }, "id":12216, "question":"If Donald can type \\(w\\) words in m minutes, which of the following represents his average typing speed, in words per minute?", "explanations":{ "correct":"To find Donald's average typing speed in words per minute, we need to divide the total number of words typed (\\(w\\)) by the total time taken (\\(m\\)).\n\nThe correct formula for average speed is \\(\\frac { w } { m }\\).\n\nLet's double-check our reasoning. If Donald types \\(w\\) words in \\(m\\) minutes, we can think of it as \\(w\\) words per \\(m\\) minutes. To find the average speed, we divide the total number of words (\\(w\\)) by the total time taken (\\(m\\)).\n\nTherefore, the answer is B) \\(\\frac { w } { m }\\)." } }, { "answer":"C", "choices":{ "A":"\\(2 \\times 6^n\\)", "B":"\\(2 \\times 6^ { n - 1 } \\)", "C":"\\(6 \\times 2^ { n - 1 } \\)", "D":"\\(6 \\times n\\)", "E":"\\(6 \\times n^2\\)" }, "id":12222, "question":"\\begin { gather* } 6, 12, 24, 48, ... \\end { gather* } Which of the following expressions represents the \\(n\\)th term in the sequence above?", "explanations":{ "correct":"To find the pattern in the given sequence, we can observe that each term is obtained by multiplying the previous term by 2. \n\nStarting with the first term, 6, we can see that the second term is obtained by multiplying 6 by 2, resulting in 12. Similarly, the third term is obtained by multiplying the second term, 12, by 2, resulting in 24. This pattern continues for each subsequent term.\n\nTherefore, to find the nth term in the sequence, we need to multiply the first term, 6, by 2 raised to the power of (n-1). This is because we need to multiply the first term by 2, n-1 times to obtain the nth term.\n\nHence, the expression that represents the nth term in the sequence is \\(6 \\times 2^{n-1}\\).\n\nTherefore, the answer is C." } }, { "answer":"C", "choices":{ "A":"16", "B":"20", "C":"25", "D":"30", "E":"35" }, "id":12223, "question":"A certain car needs 15 gallons to travel 300 miles. At this rate, how many gallons are needed to travel 500 miles?", "explanations":{ "correct":"To find out how many gallons are needed to travel 500 miles, we can set up a proportion using the given information. \n\nWe know that the car needs 15 gallons to travel 300 miles. So, we can set up the proportion as:\n\n15 gallons / 300 miles = x gallons / 500 miles\n\nTo solve for x, we can cross-multiply:\n\n15 * 500 = 300 * x\n\n7500 = 300x\n\nTo isolate x, we divide both sides of the equation by 300:\n\n7500 / 300 = x\n\n25 = x\n\nTherefore, the answer is C) 25." } }, { "answer":"D", "choices":{ "A":"144", "B":"72", "C":"36", "D":"18", "E":"12" }, "id":12233, "question":"If \\(k\\) is a positive integer, what is the smallest value of k for which \\((12k)^\\frac { 1 } { 3 } \\) is an integer?", "explanations":{ "correct":"To find the smallest value of \\(k\\) for which \\((12k)^\\frac{1}{3}\\) is an integer, we need to consider the prime factorization of \\(12k\\).\n\nThe prime factorization of \\(12\\) is \\(2^2 \\cdot 3\\). Since we want \\((12k)^\\frac{1}{3}\\) to be an integer, the exponent of each prime factor in the prime factorization of \\(12k\\) must be a multiple of \\(3\\).\n\nLet's consider the prime factor \\(2\\). The exponent of \\(2\\) in the prime factorization of \\(12k\\) is \\(2\\), so \\(2\\) must be raised to a multiple of \\(3\\). The smallest multiple of \\(3\\) greater than \\(2\\) is \\(3\\), so we need to find the smallest value of \\(k\\) such that \\(2^3\\) divides \\(12k\\).\n\nNext, let's consider the prime factor \\(3\\). The exponent of \\(3\\) in the prime factorization of \\(12k\\) is \\(1\\), so \\(3\\) must be raised to a multiple of \\(3\\). The smallest multiple of \\(3\\) greater than \\(1\\) is \\(3\\), so we need to find the smallest value of \\(k\\) such that \\(3^3\\) divides \\(12k\\).\n\nCombining these requirements, we need to find the smallest value of \\(k\\) such that \\(2^3 \\cdot 3^3\\) divides \\(12k\\). Simplifying, we have \\(8 \\cdot 27 = 216\\).\n\nTherefore, the smallest value of \\(k\\) for which \\((12k)^\\frac{1}{3}\\) is an integer is \\(216\\).\n\nThe answer is D) 18." } }, { "answer":"C", "choices":{ "A":"24", "B":"28", "C":"32", "D":"33", "E":"36" }, "id":12238, "question":"On a certain test, if a student answers 80 to 90 percent of the questions correctly, he will receive a letter grade of B. If there are 40 questions on the test, what is the minimum number of questions the student can answer correctly to receive a grade of B?", "explanations":{ "correct":"To find the minimum number of questions the student can answer correctly to receive a grade of B, we need to determine the range of the percentage of questions answered correctly that corresponds to a B grade.\n\nA B grade is given for answering 80 to 90 percent of the questions correctly. Since there are 40 questions on the test, we can calculate the range of correct answers for a B grade.\n\nTo find the minimum number of questions the student can answer correctly, we need to calculate 80% of 40 and 90% of 40.\n\n80% of 40 = (80/100) * 40 = 0.8 * 40 = 32\n90% of 40 = (90/100) * 40 = 0.9 * 40 = 36\n\nTherefore, the minimum number of questions the student can answer correctly to receive a grade of B is 32.\n\nThe answer is C) 32." } }, { "answer":"B", "choices":{ "A":"Add one number to the list.", "B":"Add one number to the list that is greater than the mean.", "C":"Add two distinct numbers to the list.", "D":"Add 2 to each number in the list.", "E":"Remove the first and fast numbers from the list." }, "id":12240, "question":"In a list of four positive even numbers, the mean, median, and mode are all equal. Which of the following CANNOT be done to the list if the mean, median, and mode are to remain equal?", "explanations":{ "correct":"To keep the mean, median, and mode equal in a list of four positive even numbers, we need to consider the possible scenarios for each option:\n\nA) Adding one number to the list: If we add one number to the list, the new number will either be even or odd. In either case, the mean, median, and mode will still be equal because the new number will not affect the existing numbers in the list. Therefore, option A can be done while keeping the mean, median, and mode equal.\n\nB) Adding one number greater than the mean: If we add one number greater than the mean, it will increase the mean but not the median. This will result in the mean being greater than the median, breaking the condition that the mean, median, and mode are equal. Therefore, option B cannot be done while keeping the mean, median, and mode equal.\n\nC) Adding two distinct numbers to the list: If we add two distinct numbers to the list, it will increase the number of elements in the list. Since the list originally had four numbers, adding two more will result in a list of six numbers. In this case, the mean, median, and mode will no longer be equal. Therefore, option C cannot be done while keeping the mean, median, and mode equal.\n\nD) Adding 2 to each number in the list: If we add 2 to each number in the list, it will increase all the numbers by the same amount. This will not affect the relative positions of the numbers, so the median will remain the same. The mean will increase by 2, but since all the numbers are even, the mode will still be the same. Therefore, option D can be done while keeping the mean, median, and mode equal.\n\nE) Removing the first and last numbers from the list: If we remove the first and last numbers from the list, we will be left with two numbers in the middle. Since the list originally had four numbers, removing the first and last numbers will result in a list of two numbers. In this case, the mean, median, and mode will no longer be equal. Therefore, option E cannot be done while keeping the mean, median, and mode equal.\n\nBased on the above reasoning, the answer is B." } }, { "answer":"B", "choices":{ "A":"3", "B":"4", "C":"6", "D":"9", "E":"12" }, "id":12247, "question":"If \\(3x = 36\\), then \\(\\frac { x } { 3 } \\) =", "explanations":{ "correct":"To find the value of \\\\(\\frac { x } { 3 } \\\\), we need to solve the equation \\\\(3x = 36\\\\) for \\\\(x\\\\).\n\nStep 1: Divide both sides of the equation by 3 to isolate \\\\(x\\\\).\n\\\\(\\frac { 3x } { 3 } = \\frac { 36 } { 3 }\\\\)\n\\\\(x = 12\\\\)\n\nStep 2: Now, substitute the value of \\\\(x\\\\) into the expression \\\\(\\frac { x } { 3 }\\\\).\n\\\\(\\frac { 12 } { 3 }\\\\)\n\nStep 3: Simplify the expression.\n\\\\(\\frac { 12 } { 3 } = 4\\\\)\n\nTherefore, the value of \\\\(\\frac { x } { 3 }\\\\) is 4.\n\nThe answer is B) 4." } }, { "answer":"D", "choices":{ "A":"\\(\\frac { a } { 6 } \\)", "B":"\\(\\frac { 2a } { 11 } \\)", "C":"\\(\\frac { 6a } { 5 } \\)", "D":"\\(\\frac { 11a } { 18 } \\)", "E":"\\(\\frac { 2a } { 3 } \\)" }, "id":12251, "question":"If \\(a = 2b = 3c\\), what is the average (arithmetic mean) of \\(a\\), \\(b\\) and \\(c\\) in terms of \\(a\\)?", "explanations":{ "correct":"To find the average (arithmetic mean) of \\(a\\), \\(b\\), and \\(c\\) in terms of \\(a\\), we need to express \\(b\\) and \\(c\\) in terms of \\(a\\).\n\nGiven that \\(a = 2b\\) and \\(a = 3c\\), we can solve for \\(b\\) and \\(c\\) in terms of \\(a\\).\n\nFrom \\(a = 2b\\), we can divide both sides by 2 to isolate \\(b\\):\n\\[b = \\frac{a}{2}\\]\n\nFrom \\(a = 3c\\), we can divide both sides by 3 to isolate \\(c\\):\n\\[c = \\frac{a}{3}\\]\n\nNow that we have expressed \\(b\\) and \\(c\\) in terms of \\(a\\), we can find the average of \\(a\\), \\(b\\), and \\(c\\).\n\nThe average is calculated by summing the values and dividing by the number of values. In this case, we have three values: \\(a\\), \\(b\\), and \\(c\\).\n\nThe sum of \\(a\\), \\(b\\), and \\(c\\) is:\n\\[a + b + c = a + \\frac{a}{2} + \\frac{a}{3}\\]\n\nTo find the average, we divide the sum by 3:\n\\[\\text{Average} = \\frac{a + \\frac{a}{2} + \\frac{a}{3}}{3}\\]\n\nTo simplify this expression, we can find a common denominator:\n\\[\\text{Average} = \\frac{6a + 3a + 2a}{6 \\cdot 3}\\]\n\nCombining like terms, we get:\n\\[\\text{Average} = \\frac{11a}{18}\\]\n\nTherefore, the average (arithmetic mean) of \\(a\\), \\(b\\), and \\(c\\) in terms of \\(a\\) is \\(\\frac{11a}{18}\\).\n\nThe answer is D." } }, { "answer":"E", "choices":{ "A":"\\(\\frac { 13 } { 42 } \\)", "B":"\\(\\frac { 1 } { 3 } \\)", "C":"\\(\\frac { 5 } { 14 } \\)", "D":"\\(\\frac { 8 } { 21 } \\)", "E":"\\(\\frac { 17 } { 42 } \\)" }, "id":12252, "question":"Angelique invited 42 people to a conference. Of the invitees, 8 people declined, and the rest accepted. On the day of the conference, a snowstorm prevented half of those who accepted from making it to the conference. If no other people accepted or declined, what fraction of Angelique's original invitees attended the conference?", "explanations":{ "correct":"To find the fraction of Angelique's original invitees who attended the conference, we need to determine the number of people who attended and divide it by the total number of invitees.\n\nWe are given that Angelique invited 42 people to the conference. Of these invitees, 8 people declined, leaving us with 42 - 8 = 34 people who either accepted or declined.\n\nOn the day of the conference, a snowstorm prevented half of those who accepted from making it. Since half of the accepted invitees couldn't attend, the number of people who attended the conference is 34 / 2 = 17.\n\nTherefore, the fraction of Angelique's original invitees who attended the conference is 17 / 42.\n\nThe answer is E) \\(\\frac { 17 } { 42 } \\)." } } ]