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Practice · GRE · Quantitative Reasoning

GRE Quant practice questions

87 questions with worked solutions. Quantitative comparison, multiple choice and numeric entry across arithmetic, algebra, geometry and data analysis.

22 easy · 47 medium · 18 hard · Percentile levels 60th to 99th · what the levels mean

Practise timed: 15 questions · 26 min All GRE topics

1 Medium ≈ 75th percentile
Compare the two quantities.

x>0x>0

Quantity Axx+1\dfrac{x}{x+1}
Quantity Bx+1x+2\dfrac{x+1}{x+2}
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
Cross-multiply (all denominators positive): compare x(x+2)=x2+2xx(x+2)=x^2+2x with (x+1)2=x2+2x+1(x+1)^2=x^2+2x+1. B is larger by 1(x+1)(x+2)\tfrac{1}{(x+1)(x+2)} for every positive xx.
2 Medium ≈ 85th percentile
Compare the two quantities.

x2<xx^2<x

Quantity Ax3x^3
Quantity Bx2x^2
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
x2<xx^2<x holds only for 0<x<10<x<1. On that interval, multiplying x<1x<1 by x2>0x^2>0 gives x3<x2x^3<x^2.
3 Easy ≈ 70th percentile
Compare the two quantities.
Quantity AThe number of positive divisors of 60
Quantity BThe number of positive divisors of 64
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
60=22⋅3⋅560=2^2\cdot3\cdot5 has (2+1)(1+1)(1+1)=12(2+1)(1+1)(1+1)=12 divisors. 64=2664=2^6 has 7. The larger number does not have more divisors.
4 Easy ≈ 70th percentile
Compare the two quantities.

nn is a positive integer.

Quantity A(−1)n+(−1)n+1(-1)^n+(-1)^{n+1}
Quantity B00
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
The two powers always have opposite signs, one +1+1 and one −1-1, so A is 0 for every nn.
5 Hard ≈ 95th percentile
Compare the two quantities.
Quantity A17+19\sqrt{17}+\sqrt{19}
Quantity B2182\sqrt{18}
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
Square both (both positive): A2=36+2323^2=36+2\sqrt{323}, B2=72^2=72. Compare 323\sqrt{323} with 18, i.e. 323 with 324. B is larger, just. Concavity of  \sqrt{\ } says the same thing: the average of two roots is below the root of the average.
6 Hard ≈ 90th percentile
Compare the two quantities.

A list of five numbers has an average (arithmetic mean) of 10 and a median of 12.

Quantity AThe largest number in the list
Quantity B1414
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: D. The relationship cannot be determined from the information given.
The list 7, 7, 12, 12, 12 has mean 10, median 12 and largest number 12, so A < B. The list 1, 1, 12, 16, 20 also fits, with largest number 20, so A > B. Cannot be determined.
7 Easy ≈ 65th percentile
Compare the two quantities.
Quantity A0.1% of 10% of 2,000
Quantity B20% of 1
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
A: 0.001×0.1×2000=0.20.001\times0.1\times2000=0.2. B: 0.20.2. Equal.
8 Hard ≈ 95th percentile
Compare the two quantities.

xx and yy are integers, x>yx>y and xy=12xy=12.

Quantity Ax−yx-y
Quantity B11
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: D. The relationship cannot be determined from the information given.
The possible pairs are (12,1),(6,2),(4,3)(12,1),(6,2),(4,3) and (−1,−12),(−2,−6),(−3,−4)(-1,-12),(-2,-6),(-3,-4), giving x−y=11,4,1x-y=11,4,1. A can equal B (at 4,34,3) or exceed it. Cannot be determined. "A is at least B" is not one of the choices.
9 Medium ≈ 75th percentile
A fair coin is tossed 4 times. Compare the two quantities.
Quantity AThe probability of exactly 2 heads
Quantity BThe probability of at least 3 heads
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
A: (42)/16=616\binom42/16=\tfrac6{16}. B: ((43)+(44))/16=516\left(\binom43+\binom44\right)/16=\tfrac5{16}. A is larger.
10 Hard ≈ 99th percentile
Compare the two quantities.
Quantity A2302^{30}
Quantity B3193^{19}
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
230=(210)3=10243≈1.074×1092^{30}=(2^{10})^3=1024^3\approx1.074\times10^9. 319=3⋅(36)3=3⋅7293≈3×3.874×108≈1.162×1093^{19}=3\cdot(3^6)^3=3\cdot729^3\approx3\times3.874\times10^8\approx1.162\times10^9. B is larger. With logs: 30log⁡2≈9.0330\log2\approx9.03 against 19log⁡3≈9.0719\log3\approx9.07.
11 Medium ≈ 80th percentile
Compare the two quantities.

0<a<b0<a<b

Quantity Aa+1b+1\dfrac{a+1}{b+1}
Quantity Bab\dfrac ab
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
a+1b+1−ab=b−ab(b+1)>0\dfrac{a+1}{b+1}-\dfrac ab=\dfrac{b-a}{b(b+1)}>0. Adding the same amount to top and bottom of a fraction below 1 moves it towards 1.
12 Easy ≈ 65th percentile
Compare the two quantities.
Quantity AThe percent increase from 40 to 50
Quantity BThe percent decrease from 50 to 40
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
A: 1040=25%\tfrac{10}{40}=25\%. B: 1050=20%\tfrac{10}{50}=20\%. The same change is a larger percentage of the smaller base.
13 Easy ≈ 65th percentile
If 3x=813^x=81 and 2y=322^y=32, what is the value of xyx^y?
  1. 20
  2. 125
  3. 625
  4. 1024
  5. 3125
Show answer
Answer: D. 1024
x=4x=4 and y=5y=5, so xy=45=1024x^y=4^5=1024.
14 Medium ≈ 80th percentile
A jar holds red, blue and green marbles in the ratio 2 : 3 : 5. After 12 green marbles are removed, the numbers of red and green marbles are equal. How many marbles were in the jar originally?
  1. 20
  2. 30
  3. 40
  4. 50
  5. 60
Show answer
Answer: C. 40
Let the counts be 2k,3k,5k2k,3k,5k. Then 2k=5k−122k=5k-12, so k=4k=4 and the jar held 10k=4010k=40.
15 Hard ≈ 95th percentile
How many integers from 100 to 999 inclusive have digits that sum to 5?
  1. 12
  2. 15
  3. 18
  4. 21
  5. 24
Show answer
Answer: B. 15
Solve a+b+c=5a+b+c=5 with a≥1a\ge1. Put a′=a−1a'=a-1: a′+b+c=4a'+b+c=4 with all parts ≥0\ge0, which has (62)=15\binom{6}{2}=15 solutions (stars and bars). No digit can exceed 9, so none are lost.
16 Easy ≈ 65th percentile
The average of aa, bb and cc is 20, and the average of aa and bb is 17. What is cc?
  1. 17
  2. 20
  3. 23
  4. 26
  5. 29
Show answer
Answer: D. 26
a+b+c=60a+b+c=60 and a+b=34a+b=34, so c=26c=26.
17 Easy ≈ 70th percentile
The length of a rectangle is increased by 20% and its width is decreased by 20%. What happens to its area?
  1. It decreases by 4%
  2. It decreases by 2%
  3. It does not change
  4. It increases by 2%
  5. It increases by 4%
Show answer
Answer: A. It decreases by 4%
1.2×0.8=0.961.2\times0.8=0.96: a 4% decrease. Equal percentage moves up and down never cancel.
18 Easy ≈ 70th percentile
A circle is inscribed in a square of side 10. What is the area of the region inside the square but outside the circle?
  1. 100−10π100-10\pi
  2. 100−25π100-25\pi
  3. 100−50π100-50\pi
  4. 25π25\pi
  5. 75
Show answer
Answer: B. 100−25π100-25\pi
The circle's diameter equals the side, so its radius is 5 and its area 25π25\pi. The region is 100−25π≈21.5100-25\pi\approx21.5.
19 Medium ≈ 85th percentile
In a sequence, each term after the first is 3 more than twice the term before it. If the fourth term is 69, what is the first term?
  1. 4
  2. 5
  3. 6
  4. 7
  5. 9
Show answer
Answer: C. 6
Run the rule backwards, tn−1=tn−32t_{n-1}=\tfrac{t_n-3}2: 69→33→15→669\to33\to15\to6.
20 Easy ≈ 70th percentile
Which of the following describes all values of xx for which x2−5x+6<0x^2-5x+6<0?
  1. x<2x<2 or x>3x>3
  2. 2<x<32<x<3
  3. x>2x>2
  4. x<3x<3
  5. −3<x<−2-3<x<-2
Show answer
Answer: B. 2<x<32<x<3
x2−5x+6=(x−2)(x−3)x^2-5x+6=(x-2)(x-3), negative exactly when the factors have opposite signs: 2<x<32<x<3.
21 Easy ≈ 65th percentile
The legs of a right triangle are in the ratio 3 : 4 and its hypotenuse is 20. What is its area?
  1. 48
  2. 60
  3. 72
  4. 96
  5. 192
Show answer
Answer: D. 96
A 3-4-5 triangle scaled by 4: legs 12 and 16. Area 12⋅12⋅16=96\tfrac12\cdot12\cdot16=96.
22 Medium ≈ 80th percentile
The data set 2, 4, 4, 4, 5, 5, 7, 9 has a standard deviation of 2. Adding which one of these numbers to the set reduces the standard deviation the most?
  1. 4
  2. 5
  3. 6
  4. 9
  5. 12
Show answer
Answer: B. 5
The mean is 408=5\tfrac{40}{8}=5. A new value equal to the mean adds zero squared deviation while increasing the count, so it shrinks the spread the most. 4 and 6 also reduce it, but by less.
23 Medium ≈ 75th percentile
Pipe A fills an empty tank in 12 hours. Pipe B empties a full tank in 18 hours. If both are opened when the tank is empty, how many hours does it take to fill?
  1. 6
  2. 7.2
  3. 30
  4. 36
  5. 72
Show answer
Answer: D. 36
Net rate 112−118=3−236=136\tfrac1{12}-\tfrac1{18}=\tfrac{3-2}{36}=\tfrac1{36} tank per hour: 36 hours. 7.2 is what you get if both pipes fill.
24 Hard ≈ 99th percentile
In how many ways can 5 people be seated around a circular table if two particular people must not sit next to each other? (Arrangements that differ only by rotation are the same.)
  1. 12
  2. 18
  3. 24
  4. 36
  5. 48
Show answer
Answer: A. 12
Circular arrangements of 5: (5−1)!=24(5-1)!=24. With the two together, treat them as one block: (4−1)!⋅2=12(4-1)!\cdot2=12. Apart: 24−12=1224-12=12.
25 Medium ≈ 75th percentile
What is the least positive integer nn for which 150n150n is a perfect square?
Show answer
Answer: 66
150=2⋅3⋅52150=2\cdot3\cdot5^2. The exponents of 2 and 3 are odd, so multiply by 2⋅3=62\cdot3=6: 900=302900=30^2.
26 Easy ≈ 70th percentile
After successive discounts of 10% and 20%, an item costs $144. What was its original price, in dollars?
Show answer
Answer: $200
0.9×0.8=0.720.9\times0.8=0.72 of the original is $144, so the original is 144/0.72=200144/0.72=200.
27 Medium ≈ 85th percentile
On any given day the probability of rain is 0.3, independently of other days. What is the probability that it rains on exactly 2 of the next 3 days? Give a decimal.
Show answer
Answer: 0.1890.189
Choose which 2 days: (32)=3\binom32=3. Each pattern has probability 0.32⋅0.7=0.0630.3^2\cdot0.7=0.063. Total 0.1890.189.
28 Medium ≈ 75th percentile
What is the sum of all the integers from −20-20 to 2525 inclusive?
Show answer
Answer: 115115
−20-20 to 2020 cancels in pairs. What remains is 21+22+23+24+25=11521+22+23+24+25=115.
29 Easy ≈ 70th percentile
A number is increased by 25%, and the result is then decreased by 20%, giving 150. What was the original number?
Show answer
Answer: 150150
1.25×0.8=11.25\times0.8=1, so the two changes cancel exactly and the original number is 150.
30 Medium ≈ 85th percentile
How many positive integers less than 200 are multiples of 4 or of 6?
Show answer
Answer: 6666
Multiples of 4: 49. Of 6: 33. Of both, i.e. of 12: 16. Inclusion–exclusion: 49+33−16=6649+33-16=66.
31 Medium ≈ 75th percentile
Compare the two quantities.

0<x<10<x<1

Quantity Ax2x^2
Quantity Bx3x^3
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
Multiplying x<1x<1 by x2>0x^2>0 gives x3<x2x^3<x^2, so A is greater throughout the interval.
32 Medium ≈ 80th percentile
Compare the two quantities.

nn is a positive integer.

Quantity A(n+1)!n!\dfrac{(n+1)!}{n!}
Quantity Bn+1n+1
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
(n+1)!=(n+1)⋅n!(n+1)!=(n+1)\cdot n!, so the quotient is exactly n+1n+1.
33 Easy ≈ 70th percentile
Compare the two quantities.
Quantity A2102^{10}
Quantity B10310^3
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
210=1024>10002^{10}=1024>1000. Worth remembering: 2102^{10} is a little more than 10310^3.
34 Hard ≈ 90th percentile
Compare the two quantities.

x+y=10x+y=10 and xy=21xy=21

Quantity Ax2+y2x^2+y^2
Quantity B6060
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
x2+y2=(x+y)2−2xy=100−42=58x^2+y^2=(x+y)^2-2xy=100-42=58, which is less than 60.
35 Medium ≈ 80th percentile
Compare the two quantities.
Quantity AThe standard deviation of 2, 4, 6, 8, 10
Quantity BThe standard deviation of 12, 14, 16, 18, 20
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
The second list is the first shifted by 10. Shifting moves the mean but not the spread, so the standard deviations are equal.
36 Medium ≈ 85th percentile
Compare the two quantities.

A circle has radius rr and a square has side rr, with r>0r>0.

Quantity AThe area of the circle
Quantity BThree times the area of the square
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
πr2\pi r^2 against 3r23r^2: since π>3\pi>3, the circle is larger.
37 Hard ≈ 95th percentile
Compare the two quantities.

nn is an integer greater than 1.

Quantity A(12)n\left(\tfrac12\right)^{n}
Quantity B(13)n−1\left(\tfrac13\right)^{n-1}
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: D. The relationship cannot be determined from the information given.
At n=2n=2: 0.25<0.3330.25<0.333, so B is greater. At n=5n=5: 0.03125>0.01230.03125>0.0123, so A is greater. The relationship depends on nn.
38 Medium ≈ 75th percentile
Compare the two quantities.
Quantity A37\dfrac37
Quantity B49\dfrac49
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
Cross-multiply: 3⋅9=273\cdot9=27 against 4⋅7=284\cdot7=28. B is greater.
39 Easy ≈ 65th percentile
If f(x)=2x2−3x+1f(x)=2x^2-3x+1, what is f(−2)f(-2)?
  1. 3
  2. 5
  3. 11
  4. 15
  5. 19
Show answer
Answer: D. 15
2(4)−3(−2)+1=8+6+1=152(4)-3(-2)+1=8+6+1=15. The common slip is 2(−2)2=−82(-2)^2=-8.
40 Medium ≈ 80th percentile
A rectangle has perimeter 36 and area 72. What is the length of its longer side?
  1. 9
  2. 10
  3. 12
  4. 14
  5. 16
Show answer
Answer: C. 12
The sides sum to 18 and multiply to 72: 6 and 12.
41 Hard ≈ 90th percentile
How many distinct arrangements are there of the letters of LEVEL?
  1. 20
  2. 30
  3. 60
  4. 120
  5. 240
Show answer
Answer: B. 30
Five letters with L and E each repeated twice: 5!2! 2!=30\tfrac{5!}{2!\,2!}=30.
42 Medium ≈ 85th percentile
If 2x=32^x=3, what is 8x8^x?
  1. 9
  2. 18
  3. 24
  4. 27
  5. 81
Show answer
Answer: D. 27
8x=(23)x=(2x)3=278^x=(2^3)^x=(2^x)^3=27.
43 Medium ≈ 80th percentile
A line passes through (2,5)(2,5) and (6,13)(6,13). What is its yy-intercept?
  1. −1
  2. 0
  3. 1
  4. 2
  5. 3
Show answer
Answer: C. 1
The slope is 13−56−2=2\tfrac{13-5}{6-2}=2, so y=2x+cy=2x+c with 5=4+c5=4+c: c=1c=1.
44 Hard ≈ 95th percentile
A three-digit number is chosen at random. What is the probability that its digits are all different?
  1. 0.64
  2. 0.68
  3. 0.72
  4. 0.75
  5. 0.80
Show answer
Answer: C. 0.72
There are 900 three-digit numbers, and 9⋅9⋅8=6489\cdot9\cdot8=648 have distinct digits: 648/900=0.72648/900=0.72.
45 Easy ≈ 70th percentile
A cube has surface area 96. What is its volume?
  1. 16
  2. 24
  3. 48
  4. 64
  5. 96
Show answer
Answer: D. 64
Each of the six faces has area 16, so the side is 4 and the volume 43=644^3=64.
46 Medium ≈ 85th percentile
Three numbers are in the ratio 2 : 3 : 5 and add to 200. By how much does the largest exceed the smallest?
  1. 30
  2. 40
  3. 50
  4. 60
  5. 80
Show answer
Answer: D. 60
The parts total 10, so one part is 20: the numbers are 40, 60 and 100, and 100−40=60100-40=60.
47 Medium ≈ 75th percentile
What is the greatest common divisor of 84 and 126?
Show answer
Answer: 4242
84=22⋅3⋅784=2^2\cdot3\cdot7 and 126=2⋅32⋅7126=2\cdot3^2\cdot7. Take the lowest power of each shared prime: 2⋅3⋅7=422\cdot3\cdot7=42.
48 Medium ≈ 85th percentile
If 5 machines make 5 widgets in 5 minutes, how many minutes do 100 machines need to make 100 widgets?
Show answer
Answer: 55
Each machine takes 5 minutes per widget, whatever the number of machines. With one machine per widget, the answer stays 5. The trap answer is 100.
49 Medium ≈ 75th percentile
What is the sum of the interior angles of a twelve-sided polygon, in degrees?
Show answer
Answer: 18001800
(n−2)⋅180∘=10⋅180∘(n-2)\cdot180^\circ=10\cdot180^\circ.
50 Easy ≈ 65th percentile
Compare the two quantities.
Quantity AThe average (arithmetic mean) of 12\tfrac12 and 13\tfrac13
Quantity B512\tfrac5{12}
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
12(12+13)=12⋅56=512\tfrac12\left(\tfrac12+\tfrac13\right)=\tfrac12\cdot\tfrac56=\tfrac5{12}. Equal.
51 Medium ≈ 85th percentile
Compare the two quantities.

xx and yy are positive integers and 3x+2y=183x+2y=18.

Quantity Axx
Quantity Byy
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: D. The relationship cannot be determined from the information given.
2y=18−3x2y=18-3x must be positive and even, so xx is even and less than 6: x=2,y=6x=2,y=6 (B larger) or x=4,y=3x=4,y=3 (A larger). The relationship cannot be determined. List the integer solutions before comparing.
52 Medium ≈ 75th percentile
Compare the two quantities.
Quantity AThe number of prime numbers between 50 and 70
Quantity B44
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
The primes are 53, 59, 61 and 67. 51 = 3·17, 57 = 3·19 and 69 = 3·23 are the usual traps. Equal.
53 Hard ≈ 90th percentile
Compare the two quantities.

a>b>0a>b>0

Quantity Aa+b2\dfrac{a+b}2
Quantity Bab\sqrt{ab}
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
a+b2−ab=12(a−b)2\tfrac{a+b}2-\sqrt{ab}=\tfrac12(\sqrt a-\sqrt b)^2, which is positive because a≠ba\ne b. The arithmetic mean is greater. They would be equal only if a=ba=b, which the condition rules out.
54 Easy ≈ 70th percentile
Compare the two quantities.
Quantity A25% of 36% of 500
Quantity B9% of 520
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
25% of 36% is 9%, so A is 9% of 500. B is 9% of a larger number. No arithmetic is needed beyond that.
55 Hard ≈ 90th percentile
Compare the two quantities.
Quantity A1+12+14+18+⋯+12101+\tfrac12+\tfrac14+\tfrac18+\dots+\tfrac1{2^{10}}
Quantity B22
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
The geometric sum is 2−12102-\tfrac1{2^{10}}: each term closes half the remaining gap to 2 without reaching it. B is greater, by 11024\tfrac1{1024}.
56 Medium ≈ 85th percentile
Compare the two quantities.

xx is an integer and x2<20x^2<20.

Quantity AThe greatest possible value of xx
Quantity BThe number of possible values of xx
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
x2<20x^2<20 allows x=−4,−3,…,4x=-4,-3,\dots,4. The greatest value is 4, and there are 9 values. B is greater.
57 Medium ≈ 75th percentile
A fair die is rolled twice. Compare the two quantities.
Quantity AThe probability that the sum is 8
Quantity BThe probability that the sum is 5
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2), five outcomes. Sum 5: (1,4), (2,3), (3,2), (4,1), four outcomes. A is greater, 536\tfrac5{36} against 436\tfrac4{36}.
58 Easy ≈ 60th percentile
A recipe uses flour and sugar in the ratio 5:25:2 by weight. How many grams of sugar go with 350 grams of flour?
  1. 100
  2. 120
  3. 140
  4. 160
  5. 175
Show answer
Answer: C. 140
Each part is 350/5=70350/5=70 grams, so sugar is 2×70=1402\times70=140 grams.
59 Medium ≈ 80th percentile
If 3<x<73<x<7 and 5<y<95<y<9, which of the following could be the value of x−yx-y?
  1. −8-8
  2. −6-6
  3. 11
  4. 22
  5. 33
Show answer
Answer: C. 11
x−yx-y is smallest when xx is small and yy large, and largest when xx is large and yy small: −6<x−y<2-6<x-y<2. The endpoints are excluded because the inequalities are strict, so of the choices only 1 is possible.
60 Medium ≈ 85th percentile
A quantity is normally distributed with mean 50 and standard deviation 10. Approximately what percent of values lie between 40 and 70?
  1. 68%
  2. 75%
  3. 82%
  4. 90%
  5. 95%
Show answer
Answer: C. 82%
About 34% lie between 40 and 50, 34% between 50 and 60, and 13.5% between 60 and 70 (half of the 95% − 68% band). Total about 82%.
61 Medium ≈ 85th percentile
What is the least positive integer kk for which 2k>1,000,0002^k>1{,}000{,}000?
  1. 17
  2. 18
  3. 19
  4. 20
  5. 23
Show answer
Answer: D. 20
210=10242^{10}=1024, just over a thousand, so 220=10242≈1,048,5762^{20}=1024^2\approx1{,}048{,}576, just over a million. 2192^{19} is half that, about 524,000. So k=20k=20.
62 Easy ≈ 65th percentile
What is the distance between the points (−2,3)(-2,3) and (4,−5)(4,-5) in the coordinate plane?
  1. 8
  2. 10
  3. 12
  4. 14
  5. 52\displaystyle \sqrt{52}
Show answer
Answer: B. 10
The horizontal gap is 6 and the vertical gap is 8, so the distance is 36+64=10\sqrt{36+64}=10, a 6-8-10 right triangle.
63 Medium ≈ 75th percentile
A circle has circumference 10π10\pi. A chord of the circle has length 8. How far is the chord from the centre?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 41\displaystyle \sqrt{41}
Show answer
Answer: B. 3
The radius is 5. The perpendicular from the centre bisects the chord, giving a right triangle with hypotenuse 5 and one leg 4, so the distance is 3.
64 Hard ≈ 90th percentile
How many four-digit positive integers are divisible by 5 and have only even digits?
  1. 64
  2. 80
  3. 96
  4. 100
  5. 125
Show answer
Answer: D. 100
Even digits are 0, 2, 4, 6, 8. Divisible by 5 with an even last digit forces a last digit of 0. The first digit is 2, 4, 6 or 8 (4 choices), the middle two digits 5 choices each: 4×5×5=1004\times5\times5=100.
65 Easy ≈ 65th percentile
The average (arithmetic mean) of five test scores is 82. What score on a sixth test would raise the average to 84?
Show answer
Answer: 9494
Six tests averaging 84 total 504; the first five total 410. The sixth must be 504−410=94504-410=94.
66 Medium ≈ 80th percentile
How many positive integers less than 500 are both perfect squares and perfect cubes?
Show answer
Answer: 22
A number that is both a square and a cube is a sixth power. 16=11^6=1 and 26=642^6=64 are below 500; 36=7293^6=729 is not. So 2.
67 Medium ≈ 80th percentile
An arithmetic sequence has 20 terms. Its first term is 6 and the sum of all 20 terms is 500. What is the common difference?
Show answer
Answer: 22
The sum is 202(2⋅6+19d)=10(12+19d)=500\tfrac{20}2\big(2\cdot6+19d\big)=10(12+19d)=500, so 12+19d=5012+19d=50 and d=2d=2.
68 Hard ≈ 90th percentile
A committee of 4 is chosen from 6 men and 5 women. How many different committees include at least 2 women?
Show answer
Answer: 215215
Count directly: 2 women (52)(62)=150\binom52\binom62=150, 3 women (53)(61)=60\binom53\binom61=60, 4 women (54)=5\binom54=5, total 215. Or subtract from all (114)=330\binom{11}4=330 committees the ones with no woman (15) or one woman (5×20=1005\times20=100): 330−115=215330-115=215.
69 Easy ≈ 65th percentile
What is the positive difference between the two solutions of x2−10x+21=0x^2-10x+21=0?
Show answer
Answer: 44
x2−10x+21=(x−3)(x−7)x^2-10x+21=(x-3)(x-7), so the solutions are 3 and 7, and they differ by 4.
70 Hard ≈ 95th percentile
What is the greatest prime factor of 38−13^8-1?
Show answer
Answer: 4141
Factor as a difference of squares repeatedly: 38−1=(34−1)(34+1)=80×82=24⋅5×2⋅413^8-1=(3^4-1)(3^4+1)=80\times82=2^4\cdot5\times2\cdot41. The greatest prime factor is 41. Computing 6560 and then trial-dividing is slower and error-prone.
71 Medium ≈ 75th percentile
A closed rectangular box measures 3 by 4 by 12. What is the length of the longest straight line that fits inside it, from one corner to the opposite corner?
Show answer
Answer: 1313
The space diagonal is 32+42+122=169=13\sqrt{3^2+4^2+12^2}=\sqrt{169}=13. It is two Pythagorean steps: the base diagonal is 5, and then 52+122=13\sqrt{5^2+12^2}=13.
72 Medium ≈ 75th percentile
Compare the two quantities.
Quantity AThe remainder when 2202^{20} is divided by 3
Quantity BThe remainder when 3203^{20} is divided by 4
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: C. The two quantities are equal.
2≡−1(mod3)2\equiv-1 \pmod 3, so 220≡(−1)20=12^{20}\equiv(-1)^{20}=1. 3≡−1(mod4)3\equiv-1 \pmod 4, so 320≡13^{20}\equiv1. Both remainders are 1: equal.
73 Medium ≈ 85th percentile
Compare the two quantities.

aa and bb are positive and a+b=1a+b=1.

Quantity Aabab
Quantity B14\tfrac14
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: D. The relationship cannot be determined from the information given.
ab=a(1−a)ab=a(1-a) is largest at a=12a=\tfrac12, where it equals 14\tfrac14. So ab≤14ab\le\tfrac14: equal when a=b=12a=b=\tfrac12, smaller otherwise. Both outcomes are possible, so the relationship cannot be determined. "Never bigger" is not the same as "always smaller".
74 Medium ≈ 85th percentile
Compare the two quantities.

x2=4xx^2=4x

Quantity Axx
Quantity B22
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: D. The relationship cannot be determined from the information given.
x2−4x=0x^2-4x=0 gives x=0x=0 or x=4x=4. Dividing both sides by xx loses the root 0. One value is below 2 and the other above: cannot be determined.
75 Easy ≈ 70th percentile
Compare the two quantities.
Quantity AThe number of minutes in a week
Quantity B10,00010{,}000
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: A. Quantity A is greater.
7×24×60=10,0807\times24\times60=10{,}080, just above 10,000.
76 Medium ≈ 75th percentile
Compare the two quantities.

An integer kk is chosen at random from 1 to 100 inclusive.

Quantity AThe probability that kk is a perfect square
Quantity BThe probability that kk is a prime number
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
There are 10 perfect squares from 1 to 100 (121^2 to 10210^2) and 25 primes, so the probabilities are 0.10 and 0.25.
77 Hard ≈ 95th percentile
Compare the two quantities.
Quantity A0.9100.9^{10}
Quantity B0.350.35
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.
Show answer
Answer: B. Quantity B is greater.
0.92=0.810.9^2=0.81, 0.94≈0.6560.9^4\approx0.656, 0.98≈0.4300.9^8\approx0.430, and 0.910=0.98×0.81≈0.3490.9^{10}=0.9^8\times0.81\approx0.349. Slightly below 0.35. The usual shortcut, 0.910≈e−1≈0.3680.9^{10}\approx e^{-1}\approx0.368, overestimates here, which is why the question is worth working exactly.
78 Medium ≈ 75th percentile
The median of seven consecutive odd integers is 21. What is the product of the smallest and the largest of them?
  1. 375
  2. 391
  3. 405
  4. 425
  5. 441
Show answer
Answer: C. 405
The integers run from 21 − 6 = 15 to 21 + 6 = 27, stepping by 2. 15×27=40515\times27=405.
79 Hard ≈ 90th percentile
For how many positive integers nn is 2n<n32^n<n^3?
  1. 6
  2. 7
  3. 8
  4. 9
  5. 10
Show answer
Answer: C. 8
n=1n=1 fails (2>12>1). From n=2n=2 to n=9n=9 it holds: 29=512<7292^9=512<729. At n=10n=10, 1,024>1,0001{,}024>1{,}000, and after that the exponential pulls away for good. That is n=2,…,9n=2,\dots,9: 8 values.
80 Medium ≈ 80th percentile
A class has 12 girls and 8 boys. Two students are picked at random. What is the probability that both are boys?
  1. 425\displaystyle \tfrac4{25}
  2. 1495\displaystyle \tfrac{14}{95}
  3. 219\displaystyle \tfrac2{19}
  4. 15\displaystyle \tfrac15
  5. 738\displaystyle \tfrac7{38}
Show answer
Answer: B. 1495\displaystyle \tfrac{14}{95}
820×719=56380=1495\tfrac{8}{20}\times\tfrac{7}{19}=\tfrac{56}{380}=\tfrac{14}{95}. Using 820\tfrac{8}{20} twice, as if the first student went back into the class, gives the trap answer 425\tfrac{4}{25}.
81 Medium ≈ 80th percentile
Line kk passes through the point (1,2)(1,2) and is perpendicular to the line y=2x+5y=2x+5. What is the xx-intercept of line kk?
  1. 3
  2. 4
  3. 5
  4. 6
  5. 7
Show answer
Answer: C. 5
Perpendicular slopes multiply to −1-1, so kk has slope −12-\tfrac12: y−2=−12(x−1)y-2=-\tfrac12(x-1). Setting y=0y=0: x−1=4x-1=4, so x=5x=5.
82 Medium ≈ 85th percentile
If xx is an integer and x2<9x^2<9, which of the following could be the value of x3−xx^3-x? Indicate all such values.

Select all that apply.

  1. −6
  2. −2
  3. 0
  4. 2
  5. 6
  6. 24
Show answer
Answer: A. −6; C. 0; E. 6
xx is one of −2,−1,0,1,2-2,-1,0,1,2. Then x3−x=(x−1)x(x+1)x^3-x=(x-1)x(x+1) takes the values −6,0,0,0,6-6,0,0,0,6. So −6, 0 and 6.
83 Medium ≈ 85th percentile
Which of the following are factors of 212−12^{12}-1? Indicate all such numbers.

Select all that apply.

  1. 3
  2. 5
  3. 7
  4. 11
  5. 13
  6. 17
Show answer
Answer: A. 3; B. 5; C. 7; E. 13
Factor as a difference of squares and cubes: 212−1=(26−1)(26+1)=63×65=32⋅7⋅5⋅132^{12}-1=(2^6-1)(2^6+1)=63\times65=3^2\cdot7\cdot5\cdot13. So 3, 5, 7 and 13; not 11 or 17.
84 Medium ≈ 80th percentile
In a geometric sequence the second term is 6 and the fifth term is 162. Which of the following are terms of the sequence? Indicate all such numbers.

Select all that apply.

  1. 2
  2. 12
  3. 18
  4. 54
  5. 108
  6. 486
Show answer
Answer: A. 2; C. 18; D. 54; undefined. 486
Three steps multiply by r3=1626=27r^3=\tfrac{162}6=27, so r=3r=3 and the first term is 2. The sequence is 2, 6, 18, 54, 162, 486, … 12 and 108 are not in it.
85 Medium ≈ 80th percentile
What is the sum of all the positive divisors of 48?
Show answer
Answer: 124124
48=24⋅348=2^4\cdot3. The divisor sum multiplies over prime powers: (1+2+4+8+16)(1+3)=31×4=124(1+2+4+8+16)(1+3)=31\times4=124.
86 Medium ≈ 80th percentile
A fair coin is tossed 5 times. What is the probability of getting more heads than tails? Give your answer as a decimal.
Show answer
Answer: 0.50.5
With an odd number of tosses there can be no tie, so either heads or tails is in the majority, and by symmetry each is equally likely: 0.5. Adding (53)+(54)+(55)=16\binom53+\binom54+\binom55=16 out of 32 confirms it.
87 Hard ≈ 90th percentile
If x>0x>0 and 1x−1−1x+1=112\dfrac1{x-1}-\dfrac1{x+1}=\dfrac1{12}, what is the value of xx?
Show answer
Answer: 55
The left side is (x+1)−(x−1)x2−1=2x2−1\tfrac{(x+1)-(x-1)}{x^2-1}=\tfrac{2}{x^2-1}. So x2−1=24x^2-1=24 and x=5x=5, taking the positive root.

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