niryuu/sni-each-converted huggingface dataset:
If \(5x - x = 2x + x - 5\), then \(x =\)
(A)\(-10\) (B)\(-5\) (C)\(-1\) (D)5 (E)10
Output: B
If \(j, k, l, m,\) and \(n\) are consecutive integers and \(j < k < l < m < n\), then what is the value of \((j + n) - (k + m)\) ?
(A)0 (B)1 (C)2 (D)4 (E)It cannot be determined from the information given.
Output: A
If \(7d < 4r\) and \(4r < 8p\), which of the following must be true?
(A)\(7d < 8p\) (B)\(8p < 7d\) (C)\(p < d\) (D)\(r = 2p\) (E)\(2r = p\)
Output: A
If \(a^2 - 16 = b^2\), and \(2a = 10\), which of the following could be a value for b?
(A)\(-1\) (B)0 (C)1 (D)2 (E)3
Output: E
If \(5c + 3 = 3c + 5\), what is the value of \(c\)?
(A)-1 (B)0 (C)1 (D)3 (E)5
Output: C
What is \(p\) percent of \(\frac { k } { 2 } \)?
(A)\(0.5 pk\) (B)\(0.05 pk\) (C)\(0.005 pk\) (D)\(0.0005 pk\) (E)\(50 pk\)
Output: C
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?
(A)\(\frac{n + 16}{4}\) (B)\(\frac{n + 4}{4}\) (C)\(\frac{n + 16}{4n}\) (D)\(\frac{n + 4}{4n}\) (E)\(\frac{16 - n}{4}\)
Output: C
If \(|frac { \sqrt { x } } { 2 } | = 2\sqrt { 2 } \), what is the value of \(x\) ?
(A)4 (B)16 (C)\(16 \sqrt { 2 } \) (D)32 (E)64
Output: D
If \(\sqrt{x} = 3\), then \(x^2 - 1 =\)
(A)2 (B)8 (C)80 (D)81 (E)82
Output: C
In a sequence of numbers, the first number is\( k\) and each number after the first is 1 more than 2 times the preceding number. If the fourth term is 47, what is the value of \(k\)?
(A)4 (B)5 (C)6 (D)7 (E)8
Output: B
If \(\frac { a } { b } \) is an even number, where \(a\) and \(b\) are integers, which of the following must also be an even number?
(A)\(\frac { b } { a } \) (B)\(a + b\) (C)\(a - b\) (D)\(a\) (E)\(b\)
Output: D
If \(a\) is an odd integer, which of the following is an even integer?
(A)\(a + 2\) (B)\(a^2\) (C)\(a^3\) (D)\(a^2 - a\) (E)\((a + 2)^2\)
Output: D
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 \(B\) ?
(A)0 (B)5 (C)11 (D)62 (E)67
Output: B
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?
(A)\(\frac { 19 } { 50 } \) (B)\(\frac { 39 } { 100 } \) (C)\(\frac { 2 } { 5 } \) (D)\(\frac { 31 } { 50 } \) (E)\(\frac { 9 } { 10 } \)
Output: A
If \(a < 0\) and \((a - 4)^2 = 36\), what is \(a\) ?
(A)-32 (B)-10 (C)-6 (D)-4 (E)-2
Output: E
If the average (arithmetic mean) of 9 different positive integers is 15 and the median is 20, what is a possible value of the greatest of the 9 integers?
(A)99 (B)75 (C)39 (D)30 (E)28
Output: C
If \(|x| < 5\) and \(|x| > 2\), how many integers satisfy both of the two inequalities?
(A)1 (B)2 (C)3 (D)4 (E)More than 4
Output: D
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?
(A)1.014 (B)10.14 (C)1,014 (D)10,140 (E)101,400
Output: A
If \(6s - 2t = 10\) and \(s - 5t = 14\), what is the value of \(5s + 3t\) ?
(A)24 (B)4 (C)\(\frac { 5 } { 7 } \) (D)-4 (E)-7
Output: D
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?
(A)\(1\frac{1}{2}\) (B)2 (C)3 (D)4 (E)\(4\frac{1}{2}\)
Output: D
If \((p - 1)^2 = 64\) and \((q - 1)^2 = 36\), what is the greatest possible value of \((p - q)\)?
(A)16 (B)14 (C)12 (D)10 (E)8
Output: B
\(f(x) = x^2 - 5\) If \(y = f(x)\), which of the following represents all possible values of y?
(A)\(y < 5\) (B)\(y \geq -5\) (C)\(y < -5\) (D)\(y > 5\) (E)All real numbers
Output: B
If a and b are positive integers and \(a^2 - b^2 = 24\), which of the following could be the smallest value of \(a\)?
(A)1 (B)2 (C)4 (D)5 (E)6
Output: D
For all positive integers r and q, where \(r > q\), let the operation \& be defined by \(r \& q = \frac{3^{r + q}}{3^{r - q}}\). for how many positive integers r is r \& 1 equal to 9?
(A)0 (B)1 (C)2 (D)4 (E)More than 4
Output: E
If \(\frac { m + m } { m \times m } = 8\) and \(m \neq 0\), what is the value of \(m\)?
(A)\(\frac { 1 } { 4 } \) (B)\(\frac { 1 } { 2 } \) (C)\(\frac { 2 } { 3 } \) (D)\(\frac { 3 } { 4 } \) (E)\(\frac { 4 } { 5 } \)
Output: A
\begin { gather* } A = \{ n|n = multiple of 3 \} \\ B = \{ n|n = multiple of 8 \} \\ C = \{ n|n = not divisible by 16 \} \end { gather* }  If n is a member of all of the sets \(A\), \(B\), and \(C\) above, which of the following could be the value of \(n\)?
(A)28 (B)36 (C)48 (D)72 (E)96
Output: D
\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?
(A)16 (B)24 (C)32 (D)48 (E)96
Output: C
For all numbers a and b, let a \(\Phi b = a^2b + ab^2\). What is the value of \((1 \Phi 2) \Phi 3\)?
(A)6 (B)12 (C)54 (D)108 (E)162
Output: E
If \(f(x) = x^{-3} + x^{\frac{3}{2}}\), what is the value of \(f(3)\)?
(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)\(27 + 81\sqrt{3}\)
Output: C
If \(\frac { z } { 3 } - \frac { 5 } { 6 } = |\frac { z } { 3 } - \frac { 5 } { 6 } |\), what is the least possible integer value of \(z\) ?
(A)-3 (B)0 (C)2 (D)3 (E)6
Output: D
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?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: C
If \(8 \times a^2x = a^ { 2x + 3 } \) , what is the value of \(a\) ?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: B
If \(\sqrt { x + p } = \sqrt { x - q } \), which of the following could be true? I. \(p = 0\) and \(q = 0\) II. \(pq = 0\) III. \(p + q = 0\)
(A)I only (B)II only (C)III only (D)I and III only (E)II and III only
Output: D
If \(3a + 5b - 4c = 12\) and \(a + 3b - 2c = -4\), what is the value of \(a - b\)?
(A)4 (B)8 (C)16 (D)20 (E)The answer can not be determined from this information.
Output: D
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 \(c\)?
(A)5 (B)10 (C)15 (D)20 (E)25
Output: B
Set X has \(x + 2b\) members, and set Y has \(y + 3b\) members. If they have exactly b members in common, how many elements are in the union of X and Y but not in the intersection of X and Y?
(A)\(x + y\) (B)\(x + y + 3b\) (C)\(x + y + 4b\) (D)\(x + y + 5b\) (E)\(x + y + 6b\)
Output: B
If \(a + b < 0\) and \(c < 0\), which of the following must be true?
(A)\(a + b < c\) (B)\(\frac { a } { c } + \frac { b } { c } < 0\) (C)\(ac + bc > 0\) (D)\(a + b + c > 0\) (E)\(a + b - c > 0\)
Output: C
For how many ordered pairs of positive integers \((a, b)\) is \(5a + 7b < 20\) ?
(A)One (B)Two (C)Three (D)Four (E)More than four
Output: C
If \(k = (\frac { 1 } { 64 } )^ { \frac { 1 } { m } } \), where \(k\) is a positive integer, which of the following could be the value of \(m\) ?
(A)3 (B)2 (C)1 (D)0 (E)-1
Output: E
If \(\frac { a } { b } = 3\), \(\frac { b } { c } = \frac { 1 } { 2 } \), and \(\frac { c } { d } = \frac { 2 } { 3 } \), what is the value of \(\frac { 2a } { d } \) ?
(A)1 (B)\(\frac { 3 } { 2 } \) (C)2 (D)\(\frac { 5 } { 2 } \) (E)3
Output: C
If \(\frac { 1 } { p } + \frac { 1 } { q } = t\), and \(pq = w\), then \(p + q = \)
(A)\(\frac { w } { t } \) (B)\(\frac { t } { w } \) (C)\(tw\) (D)\(\frac { tw } { 2 } \) (E)\(t(t + w)\)
Output: C
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?
(A)3 (B)9 (C)15 (D)21 (E)27
Output: D
If a is a non-negative integer, and b is a positive multiple of 4, how many possible distinct values of ab are less than 10?
(A)None (B)One (C)Two (D)Three (E)Five
Output: D
\(x^2 + (k +1)x + 4 = (x + h)^2\) 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 \(k + h\)?
(A)3 (B)4 (C)5 (D)7 (E)9
Output: C
\((-3a^2 b^6)^3 =\)
(A)\(9a^5 b^9\) (B)\(-9a^5 b^9\) (C)\(9a^6 b^{18}\) (D)\(27a^6 b^{18}\) (E)\(-27a^6 b^{18}\)
Output: E
If \(7 + x < 5 \)and \(3 + x < 0\), which of the following could be the value of \(x\) ?
(A)3 (B)2 (C)0 (D)-3 (E)-4
Output: E
10\% more than 10\% less than \(x\) is what percent of \(10x\)?
(A)9\% (B)9.9\% (C)10\% (D)99\% (E)100\%
Output: B
When the sum of the odd integers from 5 to 27, inclusive, is subtracted from the sum of the even integers from 8 to 30, inclusive, the result is
(A)1 (B)3 (C)27 (D)30 (E)36
Output: E
The members of set A are the single digit prime numbers. Two numbers are selected at random from set A. What is the probability that the sum of the two selected numbers is even?
(A)0 (B)\(\frac { 1 } { 6 } \) (C)\(\frac { 1 } { 3 } \) (D)\(\frac { 1 } { 2 } \) (E)1
Output: D
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?
(A)Two (B)Five (C)Seven (D)Eight (E)Ten
Output: A
If \(p\) and \(q\) are positive numbers and \(\frac { p } { q } < pq\), which of the following must be true? I. \(p > 1\) II. \(q > 1\) III. \(p > q\)
(A)I only (B)II only (C)III only (D)I and II only (E)I, II and III
Output: B
\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?
(A)20 (B)40 (C)60 (D)80 (E)100
Output: D
\((3a^2b^3)^3\) =
(A)\(9a^3b^6\) (B)\(9a^6b^9\) (C)\(27a^3b^6\) (D)\(27a^6b^9\) (E)\(27a^8b^27\)
Output: D
If \(10 + \sqrt { x } - 154\), then \(10\sqrt { x } \) =
(A)12 (B)120 (C)144 (D)1,200 (E)1,440
Output: E
If \(a\), \(b\), and \(c\) are positive numbers such that \(3a = 4b = 5c\), and if \(a + b = kc\), what is the value of \(k\)?
(A)\(\frac { 12 } { 35 } \) (B)\(\frac { 5 } { 7 } \) (C)\(\frac { 10 } { 7 } \) (D)\(\frac { 7 } { 5 } \) (E)\(\frac { 35 } { 12 } \)
Output: E
What positive number n satisfies the equation \((16)(16)(16)n = \frac { (64)(64) } { n } \)?
(A)\(\frac { 1 } { 4 } \) (B)1 (C)4 (D)8 (E)16
Output: B
If \(x > 1\), which of the following increases as \(x\) increases? I. \(\frac { 1 } { x } \) II. \(\frac { x-1 } { x } \) III. \(\frac { 1 } { x^2 } \)
(A)I only (B)II only (C)III only (D)II and III only (E)I, II and III only
Output: B
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?
(A)13 (B)15 (C)17 (D)85 (E)115
Output: A
If \(x^2 - 49 = 0\), which of the following could be a value of x?
(A)\(-7\) (B)\(-3\) (C)0 (D)4 (E)14
Output: A
If \(-a^3 + b^2 = -a^3 + 9\), then b could equal which of the following?
(A)\(-9\) (B)\(-\sqrt[3]{9}\) (C)\(\sqrt[3]{9}\) (D)3 (E)9
Output: D
If the sum of 4 numbers is between 53 and 57, then the average (arithmetic mean) of the 4 numbers could be which of the following?
(A)\(11 \frac{1}{2}\) (B)12 (C)\(12 \frac{1}{2}\) (D)13 (E)14
Output: E
If \(3x = 12\), \(5x =\)
(A)2.4 (B)14 (C)15 (D)20 (E)60
Output: D
If 3 times y is decreased by 4, then which of the following must be true?
(A)\(4(3 - y)\) (B)\(3(4 - y)\) (C)\(3(y - 4)\) (D)\(4 - 3y\) (E)\(3y - 4\)
Output: E
If \(x^2 + y^2 = (x - y)^2\), which of the following statements must also be true? I. \(x = 0\) II. \(y = 0\) III. \(xy = 0\)
(A)I only (B)II only (C)III only (D)I and II only (E)I, II, and III only
Output: C
Let the function f be defined by \(f(t) = 6t\) for all numbers t. Which of the following is equivalent to \(f(t - r)\)?
(A)\(\frac{t - r}{6}\) (B)\(6t - r\) (C)\(6t - 6r\) (D)\(12(t - r)\) (E)36tr
Output: C
If \(2^{2x - 3} = 8\), then what is the value of x?
(A)0 (B)1 (C)2 (D)3 (E)4
Output: D
The least integer of a set of consecutive even integers is \(-30\). If the sum of these integers is 66, how many integers are in the set?
(A)12 (B)14 (C)30 (D)33 (E)35
Output: D
If y is directly proportional to x and if \(y = 30\) when \(x = 6\), what is the value of y when \(x = 8\)?
(A)\(\frac{15}{2}\) (B)\(\frac{75}{2}\) (C)32 (D)38 (E)40
Output: E
If \((x + y)(x - y) + (4 - y)(2 - y) = 0\), then what is the value of \(y\) when \(x\) equals 8 ?
(A)-12 (B)\(-9 \frac { 1 } { 3 } \) (C)6 (D)\(9 \frac { 1 } { 3 } \) (E)12
Output: E
If \(x - y = 10\) and \(x^2 - y^2 = 40\), then y =
(A)14 (B)7 (C)4 (D)-3 (E)-6
Output: D
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?
(A)743 (B)750 (C)753 (D)758 (E)773
Output: A
If a and b are multiples of 3, which of the following CANNOT also be a multiple of 3?
(A)\(a + b\) (B)\(a - b\) (C)\(a + b + 1\) (D)ab (E)\(ab + 3\)
Output: C
If \(f(x) = x^2 - 3x\) and \(g(x) = f(3x)\), what is \(g(-10)\)?
(A)210 (B)390 (C)490 (D)810 (E)990
Output: E
If \(c\) is positive, what percent of \(3c\) is 9 ?
(A)\(\frac { c } { 100 } \) (B)\(\frac { 100c } { 3 } \) (C)\(\frac { 9 } { c } \) (D)3 (E)\(\frac { 300 } { c } \)
Output: E
If 5 more than 3 times a certain number is 3 less than the number, what is the number?
(A)\(-7\) (B)\(-4\) (C)\(-\frac{1}{2}\) (D)4 (E)7
Output: B
If 50 percent of 40 percent of a number is 22.8, what is the number?
(A)25.3 (B)45.6 (C)57 (D)85.5 (E)114
Output: E
\begin { gather* } 8 - a, 8, 8 + a \end { gather* } What is the average (arithmetic mean) of the numbers in the list above?
(A)4 (B)8 (C)16 (D)\(a + 4\) (E)\(2a\)
Output: B
If \(x^2 = 9\), what is the value of \(\frac { 4 } { x + 1 } - \frac { 4 } { x - 1 } \) ?
(A)3 (B)2 (C)1 (D)-1 (E)-2
Output: D
\(\frac{1}{3}\) of 60 is equal to what percent of 50?
(A)\(16\frac{2}{3}\%\) (B)20\% (C)\(33\frac{1}{3}\%\) (D)40\% (E)60\%
Output: D
If \(x\) and \(y\) are integers, how many distinct ordered pairs \((x,y)\) satisfy the equation \(2x + 4y = 17\) ?
(A)None (B)One (C)Two (D)Three (E)Four
Output: A
\(a = b^2 + b^3\) In the equation above, \(b = c^2\) for any integer \(c\). What is \(a\) in terms of \(c\) ?
(A)\(c^5\) (B)\(c^4 + c^9\) (C)\(c^4 + c^6\) (D)\(c^3 + c^4\) (E)\(c^2 + c^3\)
Output: C
If \(y + 4 = 13\), then \(y - 2\) =
(A)7 (B)8 (C)9 (D)10 (E)11
Output: A
The sum of two consecutive positive integers is never divisible by
(A)2 (B)3 (C)5 (D)7 (E)211
Output: A
If \((k - a)x + k = 5\) for all values of \(x\), where \(a\) and \(k\) are constants, which of the following is equal to \(a\)?
(A)4 (B)5 (C)6 (D)12 (E)15
Output: B
If \(x = y + 2\) and y = 4, then \(\frac { 3 } { 2 } x\) =
(A)2 (B)9 (C)12 (D)18 (E)24
Output: B
If \(a = 3n + 4\) and \(b = 7 + 9n^2\), what is b in terms of a?
(A)\(a^2 + 8a + 23\) (B)\(a^2 - 8a + 23\) (C)\(9a^2 - 108a + 144\) (D)\(9a^2 - 108a + 148\) (E)\(9a^2 + 108a + 148\)
Output: B
\(a^3 \times a^6 = a^c\) \(\frac { (b^5)^d } { b^d } = b^40\) In the equations above, \(a > 1\) and \(b > 1\). What is the sum of \(c\) and \(d\)?
(A)8 (B)9 (C)15 (D)17 (E)19
Output: E
If \(\frac { 2k } { 2k + 13 } = \frac { 14 } { 27 } \), then \(k\) =
(A)3 (B)5 (C)7 (D)14 (E)28
Output: C
If \(8a < 3b\) and \(3b < 10c\), which of the following is true?
(A)\(8a < 10c\) (B)\(10c < 8a\) (C)\(c < a\) (D)\(8a = 10c\) (E)\(8a + 1 = 10c\)
Output: A
\(a\) and \(b\) are different positive integers and \(a\) is greater than \(b\). If \(a^2 - b^2 = 7\) , what is the value of \(ab\) ?
(A)6 (B)8 (C)10 (D)12 (E)14
Output: D
In a sequence of numbers, the first term is 6. Each term after the first is calculated by adding 2 to the previous term and then dividing by -1. What is the value of the 101st term subtracted from the 70th term?
(A)31 (B)14 (C)-2 (D)-14 (E)-31
Output: D
If \(|p| > 10 \), which of the following must be true? I. \(p > 10\) II. \(p^2 > 10\) III. \(p^3 > 10\)
(A)I only (B)II only (C)III only (D)I and II only (E)I, II, and III
Output: B
If \(r(b) = \frac{(b^2 - 7)}{(b + 7)}\), what is the value of \(r(7)\)?
(A)0 (B)3 (C)4 (D)7 (E)12
Output: B
Let \(FI(x)\) be defined as the sum of the digits of the positive integer \(x\). For example, \(FI(7) = 7\) and \(FI(123) = 6\). For how many two-digit values of \(x\) does \(FI(FI(x)) = 2\) ?
(A)8 (B)9 (C)10 (D)12 (E)13
Output: C
If \(x^2 - 4x - 12 = 0\), what is the value of \(2x^2 - 8x\)?
(A)0 (B)4 (C)12 (D)16 (E)24
Output: E
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?
(A)30 (B)35 (C)40 (D)45 (E)60
Output: C
Which of the following is NOT a possible value of \(2 - x\), if \(x\) is a one-digit integer?
(A)-8 (B)-6 (C)-4 (D)-2 (E)2
Output: A
If \(\frac { 1 } { a } + \frac { 1 } { a } + \frac { 1 } { a } = 12\), then \(a\) =
(A)\(\frac { 1 } { 12 } \) (B)\(\frac { 1 } { 4 } \) (C)\(\frac { 1 } { 3 } \) (D)3 (E)4
Output: B
Twice the sum of two consecutive integers is \(k\) less than five times the smaller integer, where \(k\) is a positive integer. What is the larger integer, in terms of \(k\) ?
(A)\(k + 2\) (B)\(k + 3\) (C)\(k + 5\) (D)\(2k + 2\) (E)\(2k + 6\)
Output: B
If a and b are positive integers and the ratio of \(a + 2\) to \(b + 2\) is the same as the ratio of \(a + 4\) to \(b + 4\), which of the following must be true? I. \(a = b\) II. \(a = 2\) III. \(b = 2a\)
(A)none (B)I only (C)II only (D)III only (E)II and III only
Output: B
If \(g^{\frac{2}{3}} > 4\), which of the following must be true?
(A)\(g > 8\) (B)\(g > 2\) (C)\(g < 2\) (D)\(g < 8\) (E)None of the above
Output: E
If \(\sqrt { x +3 } + 3 = 9\), then x =
(A)3 (B)33 (C)39 (D)69 (E)75
Output: B
If a and b are positive integers, what is a percent of b percent of 200?
(A)\(\frac{ab}{100}\) (B)\(\frac{ab}{50}\) (C)ab (D)50ab (E)100ab
Output: B
If \(3^k + 3^k + 3^k = (27)^5\), what is the value of \(k\) ?
(A)15 (B)14 (C)9 (D)6 (E)3
Output: B
If x and y are different positive integers and \(3x + y = 17\), the difference between the largest possible value of y and the smallest possible value of y is
(A)16 (B)14 (C)12 (D)4 (E)2
Output: C
If \(4\sqrt { 8 } = n\sqrt { n } \), and if \(n\) and \(m\) are positive integers, which of the following cannot be the value of \(m + n\)?
(A)129 (B)34 (C)24 (D)12 (E)10
Output: C
If \(a = 16 - b\) and \(\frac { b } { 6 } = 3\), what is the value of \(a\)?
(A)-4 (B)-2 (C)2 (D)4 (E)6
Output: B
If \(0.1k\) percent of \(n\) is 5, what is \(3k\) percent of \(n\) ?
(A)15 (B)50 (C)75 (D)120 (E)150
Output: E
If \(-3 < k < 3\), which of the following must be true?
(A)\(k^2 < 3\) (B)\(k^2 > 3\) (C)\(|k| > 3\) (D)\(k^2 > 9\) (E)\(k^2 < 9\)
Output: E
If \(\sqrt { a^2 + b^2 } = a + b\), which of the following must be true?
(A)\(a = 0\) and \(b = 0\) (B)\(a + b = 0\) (C)\(ab = 0\) (D)\(ab \neq 0\) (E)\(a^2 - b^2 = 0\)
Output: C
If \(8^n * 4^2 = 2^10\), what is the value of \(n\)?
(A)2 (B)3 (C)4 (D)5 (E)6
Output: A
If \(\frac{a}{b} = 8\) and \(\frac{a}{c} = 4\), then what does c equal if \(b = 2\)?
(A)2 (B)4 (C)8 (D)10 (E)16
Output: B
If \(5a - b = 9\) and \(3a + b = 15\), then \(a + b =\)
(A)\(-3\) (B)3 (C)5 (D)9 (E)12
Output: D
If \(\frac{3}{x} + x = 7 + \frac{3}{7}\), then x can equal which of the following?
(A)\(\frac{1}{7}\) (B)\(\frac{2}{7}\) (C)1 (D)\(\frac{7}{3}\) (E)7
Output: E
If \(x^2 + y = 0\), which of the following must be true?
(A)\(x^2 = y\) (B)\(xy = y\) (C)\(x = \frac{y}{x}\) (D)\(x^3 = y^2\) (E)\(x^4 = y^2\)
Output: E
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?
(A)\(11 \frac{1}{2}\) (B)12 (C)\(12 \frac{1}{2}\) (D)13 (E)14
Output: E
If \(x = -3\) and \(y = 9\), what is the value of \(|\sqrt[3]{xy} - y|\)?
(A)\(-12\) (B)0 (C)6 (D)12 (E)36
Output: D
If \(x^2 = k\), where x and k are integers, which of the following could be the value of k?
(A)13 (B)14 (C)15 (D)16 (E)17
Output: D
If \(x^{\frac{1}{2}} = 3\), then what is the value of \(x^2\)?
(A)\(\frac{1}{3}\) (B)6 (C)9 (D)12 (E)81
Output: E
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 ?
(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)\(\frac{3}{4}a^3 + bc^2 = a\)
Output: C
\(p = y^8\) \(q = y^5\) If the above statements are true and \(y > 1\), which of the following is equal to \(y^9\)?
(A)\((p - q)^3\) (B)\((pq)^3\) (C)\((\frac { p } { q } )^3\) (D)\(p - 3q\) (E)\(p - q^3\)
Output: C
How many integers are there that are greater than 30, less than 40, and have exactly two factors (excluding the number itself and 1), that are both prime?
(A)2 (B)3 (C)4 (D)5 (E)6
Output: C
\(\lbrace[(a \times a) + a] \div a\rbrace - a=\)
(A)0 (B)1 (C)\(a\) (D)\(a^2 - a\) (E)\(a^2 - a + 1\)
Output: B
If \(a = 2b = 3c = 4d \) and \(abcd \neq 0\), what is the average (arithmetic mean) of \(a\), \(b\), and \(c\) in terms of \(d\) ?
(A)\(\frac { 26d } { 9 } \) (B)\(\frac { 25d } { 9 } \) (C)\(\frac { 8d } { 3 } \) (D)\(\frac { 23d } { 9 } \) (E)\(\frac { 22d } { 9 } \)
Output: E
What number increased by 4 is equal to 19 decreased by 4 ?
(A)10 (B)11 (C)15 (D)19 (E)27
Output: B
If d is a positive odd integer, then (d - 1)(d - 2) could equal which of the following?
(A)12 (B)13 (C)14 (D)15 (E)16
Output: A
\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?
(A)\(r + x\) (B)\(r + \frac{x}{3}\) (C)\(r + \frac{2x}{3}\) (D)\(\frac{r + x}{3}\) (E)\(3(r + x)\)
Output: A
If \((x - 2)^2 = 16\), which of the following could be a value of \(x\)?
(A)-6 (B)-4 (C)-2 (D)2 (E)8
Output: C
The average (arithmetic mean) of two numbers is \(a\). If one of the numbers is 10, what is the other?
(A)\(2a + 10\) (B)\(2a - 10\) (C)\(2(a - 10)\) (D)\(\frac { 10 + a } { 2 } \) (E)\(\frac { 10 - a } { 2 } \)
Output: B
Both \(c\) and \(d\) are positive numbers. If \(\frac { x - d } { cx } = \frac { 5 } { 2 } \), what is the value of \(x\) in terms of \(c\) and \(d\) ?
(A)\(10(5 - d) + c\) (B)\(10(5 + d) - c\) (C)\(5cd\) (D)\(-\frac { 2d } { 5c - 2 } \) (E)\(\frac { 2d } { 5c - 2 } \)
Output: D
A number, \(x\), is chosen at random from the set of positive integers less than 10. What is the probability that \(\frac { 9 } { x } > x\)?
(A)\(\frac { 1 } { 5 } \) (B)\(\frac { 2 } { 9 } \) (C)\(\frac { 1 } { 3 } \) (D)\(\frac { 2 } { 3 } \) (E)\(\frac { 7 } { 9 } \)
Output: B
If \(2a + 4b = \sqrt { k } \), then \((a + 2b)\sqrt { k } \) =
(A)\(3k\) (B)\(2k\) (C)\(k\) (D)\(\frac { k } { 2 } \) (E)\(\frac { k } { 4 } \)
Output: D
If \(\frac { a + 3 } { 3 } \) is an integer, what is the remainder when \(a\) is divided by 5 ?
(A)1 (B)2 (C)3 (D)4 (E)It cannot be determined from the information given
Output: B
If the average (arithmetic mean) of six consecutive odd integers is \(m\), which of the following is the expression for the smallest odd integer?
(A)\(6m - 6\) (B)\(2m - 4\) (C)\(m - 6\) (D)\(m - 5\) (E)\(m - 4\)
Output: D
If \(a\) and \(b\) are positive integers and \(8(2^ { 4a }) = \frac { 2^ { 2b } } { 2^3 } \), what is \(b\) in terms of \(a\)?
(A)\(a + 1\) (B)\(a + 2\) (C)\(2a + 1\) (D)\(2a + 2\) (E)\(2a + 3\)
Output: E
What are all the values of x for which \((x - 2)(x + 5) = 0\)?
(A)-5 (B)-2 (C)2 and -5 (D)-2 and 5 (E)2 and 5
Output: C
If \(x\), \(y\), and \(z\) are all integers where \(-5 < x < y < z < 4\) , which of the following is the greatest possible value of \(xz\) ?
(A)2 (B)3 (C)4 (D)5 (E)8
Output: E
If x is a positive integer, all of the following are equal to \((4^3)^x\) EXCEPT
(A)\(4^ { 3x } \) (B)\(2^ { 4x } (4^x)\) (C)\(4^x (4^3x)\) (D)\(4^x (4^2x)\) (E)\(8^2x\)
Output: C
Let the function \(f\) be defined by \(f(x) = (g(x))^2 - 2g(x) - 2\). If \(f(2) = -3\), what is the value of \(g(2)\)?
(A)-3 (B)-20 (C)-1 (D)0 (E)1
Output: E
If \(x < 0\), which of the following could have a solution? I. \(|x| = -x\) II. \(|x| = x \) III. \(|x| = \sqrt { x^2 } \)
(A)I only (B)II only (C)III only (D)I and III only (E)I, II, and III
Output: D
If \(2 - 3 \sqrt { x } = 8\), what is the value of \(x\)?
(A)0 (B)2 (C)4 (D)9 (E)There is no value of x that satisfies the equation
Output: E
\begin { gather* } 66, 34, 18, m, 6, ... \end { gather* } The first five terms of a sequence are shown above. In this sequence, the first term is 66 and each term after it is 1 more than \(\frac { 1 } { 2 } \) the previous term. What is the value of \(m\) ?
(A)9 (B)10 (C)12 (D)13 (E)14
Output: B
If \(f(x) = 9x + 9^x\), what is the value of \(f(\frac { 1 } { 2 } )\)?
(A)3 (B)6 (C)7.5 (D)9 (E)9.9
Output: C
If a number is rounded to 16.8, which of the following could have been the original number?
(A)16 (B)16.704 (C)16.763 (D)16.873 (E)17
Output: C
\begin { gather* } 6, 12, 24, 48, ... \end { gather* } Which of the following expressions represents the \(n\)th term in the sequence above?
(A)\(2 \times 6^n\) (B)\(2 \times 6^ { n - 1 } \) (C)\(6 \times 2^ { n - 1 } \) (D)\(6 \times n\) (E)\(6 \times n^2\)
Output: C
In the function \(f(x) = (x^c)(x^\frac { 1 } { c } )\), \(c\) is a nonzero constant. If \(f(2) = 4\sqrt { 2 } \), what is the value of \(c\) ?
(A)2 (B)3 (C)4 (D)16 (E)32
Output: A
If \(k\) is a positive integer, what is the smallest value of k for which \((12k)^\frac { 1 } { 3 } \) is an integer?
(A)144 (B)72 (C)36 (D)18 (E)12
Output: D
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?
(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.
Output: B
If a is a nonzero integer and b is not an integer, which of the following could be an integer?
(A)\(a + b\) (B)\(a - b\) (C)ab (D)\(2a - b\) (E)\(\frac{b}{a}\)
Output: C
If \(3x = 36\), then \(\frac { x } { 3 } \) =
(A)3 (B)4 (C)6 (D)9 (E)12
Output: B
Subtract 6 from x. Divide this sum by 3. Add 4 to this quotient. Which of the following is the result obtained by performing the operations described above?
(A)\(\frac{x - 2}{3}\) (B)\(\frac{x + 2}{3}\) (C)\(\frac{x - 6}{3}\) (D)\(\frac{x + 6}{3}\) (E)\(\frac{6x + 4}{3}\)
Output: D
If \(a = 2b = 3c\), what is the average (arithmetic mean) of \(a\), \(b\) and \(c\) in terms of \(a\)?
(A)\(\frac { a } { 6 } \) (B)\(\frac { 2a } { 11 } \) (C)\(\frac { 6a } { 5 } \) (D)\(\frac { 11a } { 18 } \) (E)\(\frac { 2a } { 3 } \)
Output: D