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Practice · GMAT · Quantitative Reasoning

GMAT Quant practice questions

92 questions with worked solutions. Problem solving across arithmetic, algebra, word problems, number properties and probability, as in the Focus Edition.

A 90 in Quantitative Reasoning is the 100th percentile and an 89 only the 96.6th, so at the top the total is usually won in Verbal and Data Insights. GMAC's August 2026 tables are in what score is the 99th percentile on the GMAT.

12 easy · 46 medium · 34 hard · Percentile levels 65th to 99th · what the levels mean

Practise timed: 21 questions · 45 min All GMAT topics

1 Easy ≈ 70th percentile
A shop raises the price of a jacket by 20%, then offers 25% off the new price. The final price is what percent of the original price?
  1. 80%
  2. 85%
  3. 90%
  4. 95%
  5. 100%
Show answer
Answer: C. 90%
Multiply the factors: 1.2×0.75=0.91.2\times0.75=0.9. Successive percentage changes multiply; they do not add, which is why "+20% then −25%" is not "−5%".
2 Medium ≈ 80th percentile
How many positive integers less than 1,000 are divisible by 6 but not by 9?
  1. 111
  2. 116
  3. 122
  4. 133
  5. 166
Show answer
Answer: A. 111
Multiples of 6 below 1,000: ⌊999/6⌋=166\lfloor 999/6\rfloor=166. A number divisible by both 6 and 9 is divisible by lcm⁡(6,9)=18\operatorname{lcm}(6,9)=18: ⌊999/18⌋=55\lfloor 999/18\rfloor=55. So 166−55=111166-55=111.
3 Hard ≈ 90th percentile
If nn is a positive integer and n2n^2 is divisible by 72, what is the largest positive integer that must divide nn?
  1. 6
  2. 12
  3. 18
  4. 24
  5. 36
Show answer
Answer: B. 12
72=23⋅3272=2^3\cdot3^2. For 23∣n22^3\mid n^2 the exponent of 2 in nn must be at least 2; for 32∣n23^2\mid n^2 the exponent of 3 must be at least 1. So 22⋅3=122^2\cdot3=12 must divide nn, and n=12n=12 shows nothing larger is forced (144=2⋅72144=2\cdot72).
4 Medium ≈ 85th percentile
Machine A can complete an order in 6 hours and machine B in 9 hours. A works alone for 2 hours, then B joins it until the order is finished. How many hours does the order take in total?
  1. 3.6
  2. 4.0
  3. 4.2
  4. 4.4
  5. 5.0
Show answer
Answer: D. 4.4
In 2 hours A does 26=13\tfrac26=\tfrac13 of the order. Together they work at 16+19=518\tfrac16+\tfrac19=\tfrac5{18} per hour, so the remaining 23\tfrac23 takes 23÷518=125=2.4\tfrac23\div\tfrac5{18}=\tfrac{12}5=2.4 hours. Total 2+2.4=4.42+2.4=4.4.
5 Hard ≈ 95th percentile
Five different positive integers have an average (arithmetic mean) of 12. What is the greatest possible value of their median?
  1. 14
  2. 15
  3. 16
  4. 17
  5. 18
Show answer
Answer: E. 18
The sum is 60. To push the median mm up, make everything else as small as it can be: the two smallest are 1 and 2, the two largest m+1m+1 and m+2m+2. Then 1+2+m+(m+1)+(m+2)=601+2+m+(m+1)+(m+2)=60, so 3m=543m=54 and m=18m=18 (the list 1, 2, 18, 19, 20).
6 Easy ≈ 70th percentile
If 2x⋅4x+1=12822^{x}\cdot4^{x+1}=128^{2}, what is the value of xx?
  1. 4
  2. 5
  3. 6
  4. 7
  5. 8
Show answer
Answer: A. 4
Write everything as a power of 2: 2x⋅22x+2=2142^x\cdot2^{2x+2}=2^{14}, so 3x+2=143x+2=14 and x=4x=4.
7 Hard ≈ 90th percentile
A box holds 4 red and 6 blue marbles. Three marbles are drawn at random without replacement. What is the probability that at least two are red?
  1. 110\displaystyle \frac{1}{10}
  2. 16\displaystyle \frac{1}{6}
  3. 15\displaystyle \frac{1}{5}
  4. 310\displaystyle \frac{3}{10}
  5. 13\displaystyle \frac{1}{3}
Show answer
Answer: E. 13\displaystyle \frac{1}{3}
There are (103)=120\binom{10}{3}=120 equally likely draws. Exactly two red: (42)(61)=36\binom42\binom61=36. Three red: (43)=4\binom43=4. So 40120=13\tfrac{40}{120}=\tfrac13.
8 Medium ≈ 85th percentile
A committee of 3 is to be chosen from 5 men and 4 women. How many different committees contain at least one woman?
  1. 60
  2. 64
  3. 70
  4. 74
  5. 80
Show answer
Answer: D. 74
Count the complement. All committees: (93)=84\binom93=84. All-male committees: (53)=10\binom53=10. So 84−10=7484-10=74. Adding the cases "one woman, two women, three women" gives 40+30+4=7440+30+4=74 as a check.
9 Hard ≈ 95th percentile
If xx and yy are positive integers with x2−y2=45x^2-y^2=45, how many ordered pairs (x,y)(x,y) are possible?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 6
Show answer
Answer: C. 3
Factor: (x−y)(x+y)=45(x-y)(x+y)=45 with 0<x−y<x+y0<x-y<x+y. The factor pairs are (1,45),(3,15),(5,9)(1,45),(3,15),(5,9), all odd, so each gives integers: (23,22),(9,6),(7,2)(23,22),(9,6),(7,2). Three pairs.
10 Medium ≈ 75th percentile
A car covers the first half of a journey at 40 miles per hour and the second half, of equal distance, at 60 miles per hour. What is its average speed for the whole journey, in miles per hour?
  1. 45
  2. 48
  3. 50
  4. 52
  5. 55
Show answer
Answer: B. 48
Take each half as 120 miles. Time: 3+2=53+2=5 hours for 240 miles, so 48 mph. Equal distances give the harmonic mean, 2⋅40⋅6040+60=48\tfrac{2\cdot40\cdot60}{40+60}=48, not the arithmetic mean 50.
11 Medium ≈ 80th percentile
The ratio of boys to girls in a class is 3 : 5. After 6 more boys join, the ratio becomes 3 : 4. How many girls are in the class?
  1. 24
  2. 30
  3. 36
  4. 40
  5. 45
Show answer
Answer: D. 40
Let the class be 3k3k boys and 5k5k girls. Then 3k+65k=34\tfrac{3k+6}{5k}=\tfrac34, so 12k+24=15k12k+24=15k and k=8k=8. Girls: 5k=405k=40.
12 Hard ≈ 90th percentile
What is the sum of all values of xx for which ∣x−3∣+∣x+2∣=7|x-3|+|x+2|=7?
  1. −3
  2. −1
  3. 0
  4. 1
  5. 4
Show answer
Answer: D. 1
For x≥3x\ge3: 2x−1=72x-1=7, so x=4x=4. For x≤−2x\le-2: 1−2x=71-2x=7, so x=−3x=-3. For −2<x<3-2<x<3 the left side is the constant 5. The solutions are 4 and −3, with sum 1.
13 Hard ≈ 90th percentile
What is the units digit of 347⋅7223^{47}\cdot7^{22}?
  1. 0
  2. 1
  3. 3
  4. 7
  5. 9
Show answer
Answer: C. 3
Units digits of powers of 3 cycle 3,9,7,13,9,7,1; 47=4⋅11+347=4\cdot11+3, so 3473^{47} ends in 7. Powers of 7 cycle 7,9,3,17,9,3,1; 22=4⋅5+222=4\cdot5+2, so 7227^{22} ends in 9. 7⋅9=637\cdot9=63: units digit 3.
14 Medium ≈ 75th percentile
$10,000 is invested at 10% annual interest, compounded every six months. How much interest is earned in the first year?
  1. $1,000
  2. $1,025
  3. $1,050
  4. $1,100
  5. $1,102.50
Show answer
Answer: B. $1,025
Each half-year pays 5%: 10,000×1.052=11,02510{,}000\times1.05^2=11{,}025. Interest is $1,025. The extra $25 over simple interest is interest earned on the first half-year's interest.
15 Hard ≈ 95th percentile
How many positive divisors of 720 are perfect squares?
  1. 3
  2. 4
  3. 5
  4. 6
  5. 9
Show answer
Answer: D. 6
720=24⋅32⋅5720=2^4\cdot3^2\cdot5. A square divisor needs even exponents: 2 to the power 0, 2 or 4 (three choices), 3 to the power 0 or 2 (two), 5 to the power 0 (one). 3⋅2⋅1=63\cdot2\cdot1=6: they are 1, 4, 9, 16, 36, 144.
16 Easy ≈ 65th percentile
After a 30% discount a shirt costs xx dollars. What was its original price, in dollars?
  1. 7x10\displaystyle \frac{7x}{10}
  2. 13x10\displaystyle \frac{13x}{10}
  3. 10x7\displaystyle \frac{10x}{7}
  4. 17x10\displaystyle \frac{17x}{10}
  5. 10x3\displaystyle \frac{10x}{3}
Show answer
Answer: C. 10x7\displaystyle \frac{10x}{7}
The sale price is 70% of the original: 0.7p=x0.7p=x, so p=10x7p=\tfrac{10x}{7}. Adding 30% back (1.3x1.3x) is the classic trap, because the 30% was taken off a larger number.
17 Medium ≈ 80th percentile
The sum of the first nn positive even integers is 2,450. What is nn?
  1. 49
  2. 50
  3. 70
  4. 98
  5. 1225
Show answer
Answer: A. 49
2+4+⋯+2n=n(n+1)2+4+\dots+2n=n(n+1). Solve n(n+1)=2450=49⋅50n(n+1)=2450=49\cdot50: n=49n=49.
18 Medium ≈ 85th percentile
Six workers, working at the same constant rate, build 4 walls in 10 days. How many days would 9 such workers need to build 12 walls?
  1. 12
  2. 15
  3. 18
  4. 20
  5. 24
Show answer
Answer: D. 20
One wall takes 6⋅104=15\tfrac{6\cdot10}{4}=15 worker-days. Twelve walls take 180 worker-days; with 9 workers that is 20 days.
19 Medium ≈ 80th percentile
If 1x+1y=14\frac1x+\frac1y=\frac14 and xy=48xy=48, what is x+yx+y?
  1. 12
  2. 14
  3. 16
  4. 24
  5. 48
Show answer
Answer: A. 12
Combine the fractions: 1x+1y=x+yxy\tfrac1x+\tfrac1y=\tfrac{x+y}{xy}. So x+y48=14\tfrac{x+y}{48}=\tfrac14 and x+y=12x+y=12. There is no need to find xx and yy.
20 Hard ≈ 90th percentile
What is the remainder when 21002^{100} is divided by 7?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 6
Show answer
Answer: B. 2
23=8≡1(mod7)2^3=8\equiv1\pmod 7. 100=3⋅33+1100=3\cdot33+1, so 2100=(23)33⋅2≡22^{100}=(2^3)^{33}\cdot2\equiv2.
21 Medium ≈ 80th percentile
Which of the following changes to the list 4, 8, 12, 16, 20 leaves its standard deviation unchanged?
  1. Adding 3 to every number
  2. Multiplying every number by 2
  3. Adding the number 12 to the list
  4. Removing the number 12
  5. Replacing 20 with 24
Show answer
Answer: A. Adding 3 to every number
Standard deviation measures spread about the mean. Shifting every value by 3 moves the mean by 3 and leaves every deviation the same. Doubling doubles the spread. Adding or removing the mean (12) changes the average squared deviation, and replacing 20 with 24 widens the list.
22 Hard ≈ 95th percentile
What is the greatest integer kk such that x3−xx^3-x is divisible by kk for every positive integer xx?
  1. 2
  2. 3
  3. 6
  4. 12
  5. 24
Show answer
Answer: C. 6
x3−x=(x−1)x(x+1)x^3-x=(x-1)x(x+1) is a product of three consecutive integers, so it always contains a multiple of 2 and a multiple of 3: it is divisible by 6. At x=2x=2 it equals exactly 6, so no larger kk works for every xx.
23 Medium ≈ 80th percentile
A 20-litre solution is 30% acid. How many litres of pure acid must be added to make a solution that is 50% acid?
  1. 5
  2. 6
  3. 7
  4. 8
  5. 10
Show answer
Answer: D. 8
There are 6 litres of acid now. After adding aa litres: 6+a20+a=12\tfrac{6+a}{20+a}=\tfrac12, so 12+2a=20+a12+2a=20+a and a=8a=8.
24 Medium ≈ 75th percentile
If aa and bb are positive, a2+b2=20a^2+b^2=20 and ab=8ab=8, what is a+ba+b?
  1. 4
  2. 25\displaystyle 2\sqrt5
  3. 6
  4. 210\displaystyle 2\sqrt{10}
  5. 8
Show answer
Answer: C. 6
(a+b)2=a2+b2+2ab=20+16=36(a+b)^2=a^2+b^2+2ab=20+16=36, and a+b>0a+b>0, so a+b=6a+b=6. (Indeed a,ba,b are 2 and 4.)
25 Medium ≈ 80th percentile
In a sequence, a1=2a_1=2 and an+1=3an−1a_{n+1}=3a_n-1 for every n≥1n\ge1. What is a5a_5?
  1. 41
  2. 122
  3. 125
  4. 365
  5. 366
Show answer
Answer: B. 122
a2=5a_2=5, a3=14a_3=14, a4=41a_4=41, a5=122a_5=122.
26 Hard ≈ 99th percentile
How many even four-digit numbers have four different digits?
  1. 1792
  2. 2016
  3. 2296
  4. 2520
  5. 4536
Show answer
Answer: C. 2296
Split on the last digit. If it is 0: 9⋅8⋅7=5049\cdot8\cdot7=504. If it is 2, 4, 6 or 8 (4 ways), the first digit cannot be 0 or the last digit (8 ways), and the middle two are chosen from the remaining 8 digits in order (8⋅78\cdot7): 4⋅8⋅56=17924\cdot8\cdot56=1792. Total 504+1792=2296504+1792=2296.
27 Medium ≈ 75th percentile
A trader buys 100 items at $8 each. She sells 70% of them at $12 each and the rest at $6 each. What is her profit as a percentage of cost?
  1. 20%
  2. 22.5%
  3. 25%
  4. 27.5%
  5. 30%
Show answer
Answer: D. 27.5%
Cost $800. Revenue 70⋅12+30⋅6=840+180=102070\cdot12+30\cdot6=840+180=1020. Profit $220, which is 220800=27.5%\tfrac{220}{800}=27.5\%.
28 Easy ≈ 70th percentile
If x<y<0x<y<0, which of the following must be positive?
  1. x+yx+y
  2. x−yx-y
  3. x2y\displaystyle x^2y
  4. yx−1\displaystyle \frac{y}{x}-1
  5. xyxy
Show answer
Answer: E. xyxy
The product of two negatives is positive, so xy>0xy>0. The others: x+y<0x+y<0; x−y<0x-y<0 since x<yx<y; x2y<0x^2y<0; and since ∣y∣<∣x∣|y|<|x|, 0<yx<10<\tfrac yx<1, so yx−1<0\tfrac yx-1<0.
29 Medium ≈ 80th percentile
The average (arithmetic mean) of 11 consecutive integers is 23. What is the product of the smallest and the largest of them?
  1. 475
  2. 504
  3. 520
  4. 529
  5. 575
Show answer
Answer: B. 504
For consecutive integers the mean is the middle term, 23, so they run from 18 to 28. 18⋅28=50418\cdot28=504.
30 Hard ≈ 95th percentile
An integer is chosen at random from 1 to 100 inclusive. What is the probability that it is divisible by 3 or by 5, but not by both?
  1. 0.29
  2. 0.35
  3. 0.41
  4. 0.47
  5. 0.53
Show answer
Answer: C. 0.41
Divisible by 3: 33. By 5: 20. By both, i.e. by 15: 6. "Exactly one" removes the overlap from both counts: 33+20−2⋅6=4133+20-2\cdot6=41, so 0.410.41.
31 Easy ≈ 70th percentile
A quantity is increased by 40% and the result is then decreased by 25%. Overall, the quantity
  1. decreases by 5%
  2. does not change
  3. increases by 5%
  4. increases by 15%
  5. increases by 65%
Show answer
Answer: C. increases by 5%
1.4×0.75=1.051.4\times0.75=1.05. Percentage changes multiply, so the two moves do not cancel and do not add.
32 Medium ≈ 85th percentile
The product of three consecutive positive integers is 210. What is their sum?
  1. 15
  2. 18
  3. 21
  4. 24
  5. 30
Show answer
Answer: B. 18
210=5⋅6⋅7210=5\cdot6\cdot7 (the cube root of 210 is close to 6, so start there). The sum is 18.
33 Hard ≈ 90th percentile
What is the remainder when 7357^{35} is divided by 5?
  1. 0
  2. 1
  3. 2
  4. 3
  5. 4
Show answer
Answer: D. 3
7≡2(mod5)7\equiv2\pmod 5, and powers of 2 cycle 2,4,3,12,4,3,1 modulo 5. Since 35=4⋅8+335=4\cdot8+3, the remainder is the third in the cycle: 3.
34 Easy ≈ 70th percentile
A 12-litre mixture is 25% juice. How many litres of water must be added to make it 20% juice?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
Show answer
Answer: C. 3
The juice stays at 3 litres. We need 312+w=15\tfrac{3}{12+w}=\tfrac15, so 12+w=1512+w=15 and w=3w=3.
35 Hard ≈ 95th percentile
How many integers from 1 to 200 inclusive are divisible by 3 but not by 4?
  1. 34
  2. 50
  3. 58
  4. 66
  5. 84
Show answer
Answer: B. 50
Multiples of 3: ⌊200/3⌋=66\lfloor200/3\rfloor=66. Those also divisible by 4 are the multiples of 12: ⌊200/12⌋=16\lfloor200/12\rfloor=16. So 66−16=5066-16=50.
36 Medium ≈ 75th percentile
If xx is 25% of yy, and yy is 40% of zz, then xx is what percent of zz?
  1. 6.5%
  2. 10%
  3. 15%
  4. 62.5%
  5. 65%
Show answer
Answer: B. 10%
x=0.25⋅0.4z=0.1zx=0.25\cdot0.4z=0.1z. Chain the fractions rather than adding the percentages.
37 Hard ≈ 95th percentile
A bag holds 3 red, 4 green and 5 blue counters. Two are drawn at random without replacement. What is the probability that they are the same colour?
  1. 19144\displaystyle \frac{19}{144}
  2. 16\displaystyle \frac16
  3. 14\displaystyle \frac14
  4. 522\displaystyle \frac{5}{22}
  5. 1966\displaystyle \frac{19}{66}
Show answer
Answer: E. 1966\displaystyle \frac{19}{66}
Same colour: (32)+(42)+(52)=3+6+10=19\binom32+\binom42+\binom52=3+6+10=19 out of (122)=66\binom{12}2=66.
38 Medium ≈ 85th percentile
The sum of five consecutive even integers is 130. What is the largest of them?
  1. 26
  2. 28
  3. 30
  4. 32
  5. 34
Show answer
Answer: C. 30
The middle term is 130/5=26130/5=26, so the integers are 22, 24, 26, 28, 30.
39 Hard ≈ 99th percentile
In how many ways can 8 identical sweets be shared among 3 children so that each child gets at least one?
  1. 15
  2. 21
  3. 28
  4. 36
  5. 56
Show answer
Answer: B. 21
Give each child one sweet first, then split the remaining 5 freely: (5+22)=(72)=21\binom{5+2}{2}=\binom72=21 (stars and bars).
40 Hard ≈ 90th percentile
If 3x+1=81 x−13^{x+1}=81^{\,x-1}, what is xx?
  1. 1
  2. 53\displaystyle \frac53
  3. 2
  4. 52\displaystyle \frac52
  5. 3
Show answer
Answer: B. 53\displaystyle \frac53
81=3481=3^4, so x+1=4(x−1)x+1=4(x-1), giving 3x=53x=5 and x=53x=\tfrac53.
41 Medium ≈ 80th percentile
$5,000 is invested at 8% a year, compounded annually. What is the interest earned over three years, to the nearest penny?
  1. $1,200.00
  2. $1,248.64
  3. $1,298.56
  4. $1,331.00
  5. $1,400.00
Show answer
Answer: C. $1,298.56
5000(1.083−1)=5000(1.259712−1)=1298.565000(1.08^3-1)=5000(1.259712-1)=1298.56. Simple interest would give $1,200, so the answer must be a little above it.
42 Hard ≈ 95th percentile
For how many positive integers nn does n!n! end in exactly six zeros?
  1. 0
  2. 1
  3. 4
  4. 5
  5. 6
Show answer
Answer: D. 5
The number of trailing zeros is the number of factors of 5. From n=25n=25 to n=29n=29 it is ⌊n/5⌋+⌊n/25⌋=5+1=6\lfloor n/5\rfloor+\lfloor n/25\rfloor=5+1=6; at n=30n=30 it jumps to 7. Five values.
43 Medium ≈ 80th percentile
A car travels for 2 hours at 60 miles per hour and then for 3 hours at 40 miles per hour. What is its average speed, in miles per hour?
  1. 44
  2. 46
  3. 48
  4. 50
  5. 52
Show answer
Answer: C. 48
Total distance 120+120=240120+120=240 miles in 5 hours: 48 mph. The plain average of 60 and 40 is wrong because more time is spent at the lower speed.
44 Medium ≈ 85th percentile
How many integers satisfy ∣2x−5∣≤3|2x-5|\le3?
  1. 2
  2. 3
  3. 4
  4. 5
  5. Infinitely many
Show answer
Answer: C. 4
−3≤2x−5≤3-3\le2x-5\le3 gives 1≤x≤41\le x\le4: the integers 1, 2, 3, 4.
45 Hard ≈ 90th percentile
How many positive integers less than 100 have an odd number of positive divisors?
  1. 7
  2. 9
  3. 10
  4. 49
  5. 50
Show answer
Answer: B. 9
Divisors pair up except when a number is a perfect square, so only squares have an odd count: 1,4,9,…,811,4,9,\dots,81, nine of them.
46 Medium ≈ 80th percentile
A machine loses 20% of its value each year. After three years it is worth $12,800. What was it worth when new?
  1. $16,000
  2. $20,000
  3. $24,000
  4. $25,000
  5. $32,000
Show answer
Answer: D. $25,000
0.83=0.5120.8^3=0.512, so the original value is 12,800/0.512=25,00012{,}800/0.512=25{,}000.
47 Hard ≈ 95th percentile
Of 60 people, 35 like tea, 40 like coffee and 10 like neither. How many like both?
  1. 15
  2. 20
  3. 25
  4. 30
  5. 35
Show answer
Answer: C. 25
60−10=5060-10=50 like at least one, so both =35+40−50=25=35+40-50=25.
48 Hard ≈ 90th percentile
If aa and bb are positive integers and ab=0.325\dfrac ab=0.325 exactly, what is the least possible value of bb?
  1. 8
  2. 13
  3. 25
  4. 40
  5. 1000
Show answer
Answer: D. 40
0.325=3251000=13400.325=\tfrac{325}{1000}=\tfrac{13}{40} in lowest terms, so the smallest denominator is 40.
49 Hard ≈ 95th percentile
Two fair dice are rolled. What is the probability that the product of the two numbers is a multiple of 6?
  1. 16\displaystyle \frac16
  2. 14\displaystyle \frac14
  3. 13\displaystyle \frac13
  4. 512\displaystyle \frac5{12}
  5. 12\displaystyle \frac12
Show answer
Answer: D. 512\displaystyle \frac5{12}
Count the favourable rolls by the first die: 1 gives 1 way, 2 gives 2, 3 gives 3, 4 gives 2, 5 gives 1, 6 gives 6. That is 15 of 36, or 512\tfrac5{12}.
50 Hard ≈ 95th percentile
Working alone, A finishes a job in 10 days; working together, A and B finish it in 6 days. B works alone for 5 days and then leaves. How many more days does A need to finish the job?
  1. 5
  2. 6
  3. 623\displaystyle 6\frac23
  4. 712\displaystyle 7\frac12
  5. 10
Show answer
Answer: C. 623\displaystyle 6\frac23
B's rate is 16−110=115\tfrac16-\tfrac1{10}=\tfrac1{15}, so in 5 days B does 13\tfrac13. The remaining 23\tfrac23 takes A 23÷110=203=623\tfrac23\div\tfrac1{10}=\tfrac{20}3=6\tfrac23 days.
51 Easy ≈ 65th percentile
The price of a lamp is increased by 25%. By what percent must the new price be reduced to bring it back to the original price?
  1. 20%
  2. 22%
  3. 25%
  4. 30%
  5. 3313%\displaystyle 33\tfrac13\%
Show answer
Answer: A. 20%
The new price is 1.25P1.25P. To return to PP you multiply by 11.25=0.8\tfrac1{1.25}=0.8, a 20% cut. The reduction is smaller than the increase because it is taken from a larger base.
52 Easy ≈ 70th percentile
The average (arithmetic mean) of six numbers is 15. When one of the numbers is removed, the average of the remaining five is 14. What number was removed?
  1. 16
  2. 18
  3. 20
  4. 21
  5. 24
Show answer
Answer: C. 20
The six numbers total 6×15=906\times15=90 and the five total 5×14=705\times14=70. The removed number is 90−70=2090-70=20. Work with totals, not averages.
53 Easy ≈ 65th percentile
If x2−y2=48x^2-y^2=48 and x+y=12x+y=12, what is the value of xx?
  1. 4
  2. 8
  3. 10
  4. 12
  5. 16
Show answer
Answer: B. 8
x2−y2=(x+y)(x−y)x^2-y^2=(x+y)(x-y), so x−y=48/12=4x-y=48/12=4. Adding x+y=12x+y=12 gives 2x=162x=16, x=8x=8.
54 Medium ≈ 80th percentile
For how many positive integers nn is 1<20n<51<\dfrac{20}{n}<5?
  1. 11
  2. 12
  3. 13
  4. 14
  5. 15
Show answer
Answer: E. 15
20n>1\tfrac{20}n>1 gives n<20n<20 and 20n<5\tfrac{20}n<5 gives n>4n>4. So n=5,6,…,19n=5,6,\dots,19: 15 integers.
55 Hard ≈ 95th percentile
The sum 1+2+3+⋯+n1+2+3+\dots+n is a three-digit number whose three digits are all the same. What is nn?
  1. 36
  2. 37
  3. 40
  4. 44
  5. 45
Show answer
Answer: A. 36
The sum is n(n+1)2=111d\tfrac{n(n+1)}2=111d for a digit dd, so n(n+1)=222d=2⋅3⋅37⋅dn(n+1)=222d=2\cdot3\cdot37\cdot d. One of nn, n+1n+1 must be a multiple of 37, and both are below 45, so the pair is (36,37)(36,37) or (37,38)(37,38). 36⋅37=1332=222⋅636\cdot37=1332=222\cdot6 works, giving 666. 37⋅38=140637\cdot38=1406 is not a multiple of 222. So n=36n=36. Spotting the prime 37 in 111=3×37111=3\times37 is the whole problem.
56 Easy ≈ 70th percentile
A train 200 metres long passes a signal post in 10 seconds. At the same speed, how many seconds does it take to cross a bridge 400 metres long?
  1. 24
  2. 30
  3. 36
  4. 40
  5. 60
Show answer
Answer: B. 30
Speed is 200/10=20200/10=20 metres a second. To clear the bridge the front must travel the bridge plus the train's own length, 400+200=600400+200=600 metres, which takes 30 seconds.
57 Medium ≈ 80th percentile
$8,000 is invested for two years at 5% a year. How much more interest is earned if the interest is compounded annually rather than simple?
  1. $5
  2. $10
  3. $15
  4. $20
  5. $40
Show answer
Answer: D. $20
Simple: 2×400=$8002\times400=\$800. Compound: 8000(1.052−1)=8000×0.1025=$8208000(1.05^2-1)=8000\times0.1025=\$820. The $20 difference is 5% interest on the first year's $400 of interest.
58 Medium ≈ 85th percentile
What is the greatest integer kk for which 3k3^k is a factor of 20!20! (the product of the integers from 1 to 20)?
  1. 5
  2. 6
  3. 7
  4. 8
  5. 9
Show answer
Answer: D. 8
Multiples of 3 up to 20 contribute one factor each (⌊20/3⌋=6\lfloor20/3\rfloor=6), and multiples of 9 contribute one more (⌊20/9⌋=2\lfloor20/9\rfloor=2). Total 8.
59 Hard ≈ 95th percentile
How many three-digit positive integers contain exactly one digit 7?
  1. 225
  2. 234
  3. 243
  4. 252
  5. 261
Show answer
Answer: A. 225
7 in the hundreds place: the other two digits are any of 9 non-7 digits each, 9×9=819\times9=81. 7 in the tens place: the hundreds digit can't be 0 or 7 (8 choices) and the units digit has 9, so 72. 7 in the units place: likewise 72. Total 81+72+72=22581+72+72=225. The trap is to forget that the hundreds digit cannot be 0.
60 Easy ≈ 70th percentile
The ratio of aa to bb is 2:32:3 and the ratio of bb to cc is 4:54:5. What is the ratio of aa to cc?
  1. 2 : 5
  2. 8 : 15
  3. 3 : 5
  4. 4 : 5
  5. 5 : 6
Show answer
Answer: B. 8 : 15
ac=ab⋅bc=23⋅45=815\tfrac ac=\tfrac ab\cdot\tfrac bc=\tfrac23\cdot\tfrac45=\tfrac8{15}. Equivalently, scale to a common bb: a:b=8:12a:b=8:12 and b:c=12:15b:c=12:15.
61 Medium ≈ 75th percentile
A tank is 35\tfrac35 full. After 12 litres are drawn off, it is 13\tfrac13 full. What is the capacity of the tank, in litres?
  1. 30
  2. 36
  3. 40
  4. 42
  5. 45
Show answer
Answer: E. 45
The 12 litres are 35−13=415\tfrac35-\tfrac13=\tfrac4{15} of the tank, so the capacity is 12÷415=4512\div\tfrac4{15}=45.
62 Medium ≈ 80th percentile
If xx is an integer and 2<∣x−1∣<52<|x-1|<5, how many values can xx take?
  1. 1
  2. 2
  3. 3
  4. 4
  5. 8
Show answer
Answer: D. 4
∣x−1∣|x-1| must be 3 or 4, so x−1∈{−4,−3,3,4}x-1\in\{-4,-3,3,4\} and x∈{−3,−2,4,5}x\in\{-3,-2,4,5\}: four values. An absolute-value inequality gives two separate intervals, not one.
63 Hard ≈ 95th percentile
Seven different positive integers have an average (arithmetic mean) of 10 and a median of 12. What is the greatest possible value of the largest of them?
  1. 25
  2. 27
  3. 29
  4. 31
  5. 34
Show answer
Answer: A. 25
The total is 70. To make the largest as big as possible, make the others as small as the conditions allow: the three below the median are 1, 2, 3; the median is 12; the next two must exceed 12 and be different, so 13 and 14. That uses 1+2+3+12+13+14=451+2+3+12+13+14=45, leaving 70−45=2570-45=25.
64 Medium ≈ 85th percentile
Two cyclists start 60 km apart and ride towards each other along the same road at 12 km/h and 18 km/h. After how many minutes are they 15 km apart for the second time?
  1. 90
  2. 120
  3. 130
  4. 150
  5. 180
Show answer
Answer: D. 150
They close at 12+18=3012+18=30 km/h. They are 15 km apart first when they have closed 45 km (90 minutes), then pass each other, and are 15 km apart again when they have covered 60+15=7560+15=75 km together: 75/30=2.575/30=2.5 hours, or 150 minutes.
65 Medium ≈ 85th percentile
When the positive integer nn is divided by 5 the remainder is 3, and when it is divided by 7 the remainder is 4. What is the smallest such nn greater than 100?
  1. 103
  2. 113
  3. 118
  4. 123
  5. 158
Show answer
Answer: D. 123
List numbers that leave 4 on division by 7 (4, 11, 18, …) until one leaves 3 on division by 5: 18. The pattern repeats every 5×7=355\times7=35, so the solutions are 18+35k18+35k. The first above 100 is 18+105=12318+105=123.
66 Medium ≈ 85th percentile
What is the sum of the odd positive divisors of 72?
  1. 9
  2. 12
  3. 13
  4. 15
  5. 24
Show answer
Answer: C. 13
72=23⋅3272=2^3\cdot3^2. An odd divisor can use no factor of 2, so the odd divisors are the divisors of 9: 1, 3, 9. Their sum is 13.
67 Medium ≈ 75th percentile
Coffee A costs $6 a pound and coffee B costs $10 a pound. How many pounds of A must be mixed with 12 pounds of B to make a blend worth $7 a pound?
  1. 24
  2. 28
  3. 30
  4. 32
  5. 36
Show answer
Answer: E. 36
Each pound of A is $1 below the target price, and each pound of B is $3 above it. The deficits must balance: 1⋅x=3⋅121\cdot x=3\cdot12, so x=36x=36. Checking: (6⋅36+10⋅12)/48=336/48=7(6\cdot36+10\cdot12)/48=336/48=7.
68 Hard ≈ 95th percentile
In how many ways can 10 be written as the sum of three positive integers, if the order of the terms does not matter?
  1. 5
  2. 6
  3. 7
  4. 8
  5. 10
Show answer
Answer: D. 8
List them with a≤b≤ca\le b\le c: 1+1+8, 1+2+7, 1+3+6, 1+4+5, 2+2+6, 2+3+5, 2+4+4, 3+3+4. That is 8. Counting ordered triples instead would give (92)=36\binom92=36; listing in increasing order is what avoids repeats.
69 Medium ≈ 75th percentile
If 5a⋅25b=5125^{a}\cdot25^{b}=5^{12} and a−b=3a-b=3, what is the value of aa?
  1. 4
  2. 5
  3. 6
  4. 7
  5. 8
Show answer
Answer: C. 6
25b=52b25^b=5^{2b}, so a+2b=12a+2b=12. With a−b=3a-b=3, subtracting gives 3b=93b=9, b=3b=3, a=6a=6.
70 Medium ≈ 75th percentile
Each number in a data set is multiplied by 3, and then 4 is added to each result. If the standard deviation of the original set is dd, what is the standard deviation of the new set?
  1. dd
  2. 3d3d
  3. 3d+43d+4
  4. 9d9d
  5. 9d+49d+4
Show answer
Answer: B. 3d3d
Adding a constant moves every value and the mean together, so it leaves spread unchanged. Multiplying by 3 triples every distance from the mean, so it triples the standard deviation. The variance would be multiplied by 9.
71 Hard ≈ 95th percentile
Three fair six-sided dice are rolled. What is the probability that the sum is 10?
  1. 112\displaystyle \tfrac1{12}
  2. 19\displaystyle \tfrac19
  3. 18\displaystyle \tfrac18
  4. 536\displaystyle \tfrac5{36}
  5. 16\displaystyle \tfrac16
Show answer
Answer: C. 18\displaystyle \tfrac18
Count the unordered triples: 1+3+6, 1+4+5, 2+2+6, 2+3+5, 2+4+4, 3+3+4. Those with three different numbers can be ordered 6 ways and those with a repeat 3 ways: 6+6+3+6+3+3=276+6+3+6+3+3=27. So 27216=18\tfrac{27}{216}=\tfrac18.
72 Easy ≈ 70th percentile
If 0.1% of xx equals 5% of 20, what is xx?
  1. 100
  2. 200
  3. 1,000
  4. 2,000
  5. 10,000
Show answer
Answer: C. 1,000
5% of 20 is 1, and 0.1% is 11000\tfrac1{1000}, so x1000=1\tfrac x{1000}=1 and x=1000x=1000.
73 Medium ≈ 75th percentile
The first term of an arithmetic sequence is 7 and the 10th term is 61. What is the sum of the first 20 terms?
  1. 1,180
  2. 1,220
  3. 1,280
  4. 1,340
  5. 1,400
Show answer
Answer: C. 1,280
Nine steps take 7 to 61, so the common difference is 54/9=654/9=6. The 20th term is 7+19×6=1217+19\times6=121, and the sum is 20×7+1212=1,28020\times\tfrac{7+121}{2}=1{,}280.
74 Medium ≈ 85th percentile
If xx and yy are positive integers and 3x+5y=413x+5y=41, how many ordered pairs (x,y)(x,y) are possible?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6
Show answer
Answer: B. 3
41−5y41-5y must be a positive multiple of 3. Since 41≡241\equiv2 and 5y≡2y(mod3)5y\equiv2y \pmod 3, we need y≡1(mod3)y\equiv1\pmod 3: y=1,4,7y=1,4,7 give x=12,7,2x=12,7,2. The next, y=10y=10, makes xx negative. Three pairs.
75 Hard ≈ 90th percentile
What is the greatest common divisor of 220−12^{20}-1 and 212−12^{12}-1?
  1. 3
  2. 7
  3. 15
  4. 31
  5. 63
Show answer
Answer: C. 15
gcd⁡(2a−1, 2b−1)=2gcd⁡(a,b)−1\gcd(2^a-1,\,2^b-1)=2^{\gcd(a,b)}-1, the same Euclidean steps run on the exponents. gcd⁡(20,12)=4\gcd(20,12)=4, so the answer is 24−1=152^4-1=15.
76 Medium ≈ 80th percentile
A runner averages 8 minutes per mile for the first 3 miles and 10 minutes per mile for the next 2 miles. What is the runner's average speed over the 5 miles, to the nearest tenth of a mile per hour?
  1. 6.5
  2. 6.8
  3. 7.0
  4. 7.2
  5. 7.5
Show answer
Answer: B. 6.8
Total time is 3×8+2×10=443\times8+2\times10=44 minutes for 5 miles. Speed is 5÷4460=30044≈6.85\div\tfrac{44}{60}=\tfrac{300}{44}\approx6.8 miles per hour. Averaging the two paces first gives the wrong answer, because the runner spends different distances at each.
77 Medium ≈ 85th percentile
Of 100 students, 45 study French, 40 Spanish and 30 German. 15 study French and Spanish, 10 French and German, 8 Spanish and German, and 5 study all three. How many study none of the three?
  1. 8
  2. 13
  3. 18
  4. 23
  5. 28
Show answer
Answer: B. 13
Inclusion–exclusion: 45+40+30−15−10−8+5=8745+40+30-15-10-8+5=87 study at least one, so 100−87=13100-87=13 study none. The students in all three were subtracted three times in the pairs, so they are added back once.
78 Hard ≈ 95th percentile
How many different 4-letter arrangements can be made from the letters of the word LEVEL?
  1. 24
  2. 30
  3. 36
  4. 48
  5. 60
Show answer
Answer: B. 30
LEVEL has two Ls, two Es and one V; using four letters means leaving one out. Leave out V: LLEE, 4!2! 2!=6\tfrac{4!}{2!\,2!}=6. Leave out an L: LEEV, 4!2!=12\tfrac{4!}{2!}=12. Leave out an E: LLEV, 12. Total 30.
79 Medium ≈ 80th percentile
If f(x)=2x−3f(x)=2x-3 and g(x)=x2+1g(x)=x^2+1, what is f(g(2))−g(f(2))f(g(2))-g(f(2))?
  1. −3
  2. 0
  3. 3
  4. 5
  5. 7
Show answer
Answer: D. 5
g(2)=5g(2)=5, so f(g(2))=7f(g(2))=7. f(2)=1f(2)=1, so g(f(2))=2g(f(2))=2. The difference is 5. Composition is not commutative, which is the point of the question.
80 Medium ≈ 75th percentile
The list 2, 5, 9, 11, xx has a mean equal to its median. If x>11x>11, what is xx?
  1. 12
  2. 15
  3. 18
  4. 21
  5. 24
Show answer
Answer: C. 18
With x>11x>11 the median is 9. The mean is 27+x5=9\tfrac{27+x}{5}=9, so x=18x=18.
81 Hard ≈ 90th percentile
What is the sum of all positive integers nn for which n2−11n+24n^2-11n+24 is negative?
  1. 15
  2. 18
  3. 22
  4. 26
  5. 30
Show answer
Answer: C. 22
n2−11n+24=(n−3)(n−8)n^2-11n+24=(n-3)(n-8), negative strictly between the roots: n=4,5,6,7n=4,5,6,7, summing to 22.
82 Medium ≈ 85th percentile
A price rose by 10% in one year and by xx% the next. Over the two years it rose by 32%. What is xx?
  1. 18
  2. 20
  3. 21
  4. 22
  5. 24
Show answer
Answer: B. 20
1.1(1+x100)=1.321.1(1+\tfrac{x}{100})=1.32, so 1+x100=1.21+\tfrac{x}{100}=1.2 and x=20x=20. Subtracting, 32−10=2232-10=22, ignores compounding.
83 Hard ≈ 90th percentile
The product of the digits of a three-digit number is 24, and the number is odd. How many such numbers are there?
  1. 6
  2. 8
  3. 10
  4. 12
  5. 16
Show answer
Answer: B. 8
No digit can be 0, and the last digit must be odd and divide 24: 1 or 3. Last digit 1: the first two multiply to 24, as 38, 83, 46 or 64. Last digit 3: they multiply to 8, as 18, 81, 24 or 42. That is 8 numbers.
84 Medium ≈ 85th percentile
If x=6+6+6+⋯x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots}}}, what is xx?
  1. 2
  2. 6\displaystyle \sqrt6
  3. 3
  4. 6
  5. 9
Show answer
Answer: C. 3
The expression under the first root contains xx itself: x=6+xx=\sqrt{6+x}, so x2−x−6=0x^2-x-6=0, (x−3)(x+2)=0(x-3)(x+2)=0. xx is positive, so x=3x=3.
85 Medium ≈ 80th percentile
A bag holds red and blue marbles in the ratio 2 : 3. After 8 red marbles are added, the ratio is 6 : 5. How many blue marbles are in the bag?
  1. 10
  2. 12
  3. 15
  4. 18
  5. 20
Show answer
Answer: C. 15
Write the counts as 2k2k and 3k3k. Then 2k+83k=65\tfrac{2k+8}{3k}=\tfrac65, so 10k+40=18k10k+40=18k and k=5k=5. The blue marbles never changed: 15.
86 Hard ≈ 95th percentile
How many positive integers less than 1,000 have digits that sum to 6?
  1. 21
  2. 24
  3. 28
  4. 36
  5. 45
Show answer
Answer: C. 28
Write each number with three digits, padding with leading zeros: a+b+c=6a+b+c=6 with digits 0 to 9. Stars and bars gives (82)=28\binom{8}{2}=28, and no digit can exceed 6, so all count. 000 is excluded automatically because its digits sum to 0.
87 Medium ≈ 85th percentile
If 5a=25^a=2 and 2b=1252^b=125, what is abab?
  1. 2
  2. 3
  3. 4
  4. 5
  5. 6
Show answer
Answer: B. 3
Substitute: 125=2b=(5a)b=5ab125=2^b=(5^a)^b=5^{ab}, and 125=53125=5^3, so ab=3ab=3. No logarithms needed.
88 Medium ≈ 75th percentile
A shop sells pens at 3 for $2 and pencils at 4 for $1. Maria spends exactly $9 and buys 21 items in all. How many pens does she buy?
  1. 6
  2. 9
  3. 12
  4. 15
  5. 18
Show answer
Answer: B. 9
A pen costs 23\tfrac23 and a pencil 14\tfrac14 of a dollar. With pp pens, 23p+14(21−p)=9\tfrac23p+\tfrac14(21-p)=9; multiplying by 12, 8p+63−3p=1088p+63-3p=108, so p=9p=9. Check: 3 sets of pens for $6 and 3 sets of pencils for $3.
89 Hard ≈ 90th percentile
What is the remainder when 1!+2!+3!+⋯+50!1!+2!+3!+\cdots+50! is divided by 12?
  1. 0
  2. 3
  3. 6
  4. 9
  5. 11
Show answer
Answer: D. 9
From 4!=244!=24 on, every factorial is a multiple of 12, so only 1+2+6=91+2+6=9 matters. The remainder is 9.
90 Medium ≈ 85th percentile
An inlet fills a cistern in 10 hours and a drain empties it in 15 hours. If both are open from the moment the cistern is empty, how many hours does it take to fill?
  1. 20
  2. 25
  3. 30
  4. 35
  5. 40
Show answer
Answer: C. 30
Net rate: 110−115=3−230=130\tfrac1{10}-\tfrac1{15}=\tfrac{3-2}{30}=\tfrac1{30} of the cistern per hour, so 30 hours.
91 Hard ≈ 95th percentile
In how many ways can 6 people be split into 3 pairs, if the pairs are not labelled?
  1. 15
  2. 30
  3. 45
  4. 90
  5. 720
Show answer
Answer: A. 15
Take the first person: they can be paired with any of 5 others. Of the 4 left, take the first: 3 choices of partner. The last two pair themselves. 5×3×1=155\times3\times1=15. Counting (62)(42)(22)=90\binom62\binom42\binom22=90 and forgetting to divide by 3!=63!=6 for the order of the pairs gives the trap answer 90.
92 Medium ≈ 85th percentile
An integer xx is chosen at random from 1 to 30 inclusive. What is the probability that x2−1x^2-1 is divisible by 3?
  1. 13\displaystyle \tfrac13
  2. 12\displaystyle \tfrac12
  3. 35\displaystyle \tfrac35
  4. 23\displaystyle \tfrac23
  5. 34\displaystyle \tfrac34
Show answer
Answer: D. 23\displaystyle \tfrac23
x2−1=(x−1)(x+1)x^2-1=(x-1)(x+1). Of three consecutive integers x−1,x,x+1x-1,x,x+1 one is a multiple of 3; the product is divisible by 3 unless that one is xx. 20 of the 30 numbers are not multiples of 3, so the probability is 2030=23\tfrac{20}{30}=\tfrac23.

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