Practice Quantitative Reasoning questions for the Graduate Record Examinations. Every question includes a full explanation of why the correct answer is right and why the tempting distractors are wrong.
10 questions available · medium difficulty · GRE · Free, no registration required
10 example questions with full explanations. Use the interactive practice above to work through the complete set.
Question 1medium
Column A: (x + 2)² when x = −5
Column B: (x − 2)² when x = −5
Compare the two quantities.
- A.Column B is greater✓
- B.Column A is greater
- C.The two quantities are equal
- D.The relationship cannot be determined from the information given
Correct answer: A
Substitute x = −5 into each expression. Column A: (−5 + 2)² = (−3)² = 9. Column B: (−5 − 2)² = (−7)² = 49. Since 49 > 9, Column A is greater.Bcommon error is to forget that squaring a negative number yields a positive result, or to distribute before substituting and make a sign error. Because x is given as a fixed value (−5), no ambiguity exists and the answer is definitively B.
Question 2medium
The table below shows the quarterly revenue (in thousands of dollars) for a company.
Q1: $120k | Q2: $150k | Q3: $180k | Q4: $90k
Revenue in Q3 accounts for approximately what percent of the total annual revenue?
Correct answer: B
Total annual revenue = 120 + 150 + 180 + 90 = 540. Q3 share = 180 / 540 = 1/3 ≈ 33.3%, making B the closest answer. Choice A (25%) reflects the mistaken assumption that each quarter is an equal share. Choice B(30%) may arise from dividing 180 by 600, an incorrect total. Choice D (40%) could result from dividing Q3 by only the other three quarters (180/450 = 40%), a common ratio-vs.-percent error.
Question 3medium
A train travels 240 miles at a constant speed. If the train had traveled 20 miles per hour faster, it would have completed the trip exactly 1 hour sooner. What is the train's original speed, in miles per hour?
Correct answer: C
Let r = original speed. Then 240/r − 240/(r + 20) = 1. Multiplying through: 240(r + 20) − 240r = r(r + 20), giving 4800 = r² + 20r, or r² + 20r − 4800 = 0. Factoring: (r − 60)(r + 80) = 0, so r = 60 (rejecting the negative root). Verification: at 60 mph the trip takes 4 hours; at 80 mph it takes 3 hours — exactly 1 hour less. Choice A and B are plausible guesses that do not satisfy the equation; choice D (80) is the faster speed, a common mix-up between original and adjusted values.
Question 4medium
Without using a calculator, which of the following values is closest to √147?
- A.11.8
- B.12.5
- C.13.2
- D.12.1✓
Correct answer: D
Note that 12² = 144 and 13² = 169, so √147 lies between 12 and 13, but much closer to 12. To refine: 12.1² = 146.41 and 12.2² = 148.84, so √147 is between 12.1 and 12.2, nearest to 12.1. Choice A (11.8) falls below 12, contradicting 12² = 144 < 147. Choice B (12.5) equals √156.25, too large. Choice C (13.2) is near √174, far too large. Anchoring to perfect squares (144 and 169) and then narrowing with one iteration of estimation is the intended strategy.
Question 5medium
When positive integer n is divided by 7, the remainder is 4. What is the remainder when 3n is divided by 7?
Correct answer: A
If n divided by 7 leaves remainder 4, then n = 7q + 4 for some non-negative integer q. Multiplying both sides by 3 gives 3n = 21q + 12. Since 21q is divisible by 7, the remainder depends only on 12 ÷ 7, which gives 7 × 1 + 5, so the remainder is 5. Choice D (12) is a common error from simply multiplying the remainder by 3 without reducing modulo 7. Choices C and A do not follow from any standard misapplication and serve as additional distractors.
Question 6medium
A circle is inscribed in a square with side length 8. What is the area of the region inside the square but outside the circle?
- A.64 − 8π
- B.64 − 16π✓
- C.64 − 32π
- D.32 − 16π
Correct answer: B
A circle inscribed in a square with side length 8 has a diameter equal to 8, so its radius is 4. The area of the square is 8² = 64, and the area of the circle is π(4²) = 16π. The region inside the square but outside the circle is 64 − 16π. Choice A results from using radius 4 incorrectly as r² = 8. Choice C uses diameter 8 directly as r², doubling the circle's area. Choice D halves the square's area, a common error when confusing the inscribed figure's relationship to the square.
Question 7medium
If x² − 5x + 6 = 0 and x > 2, what is the value of x² + (1/x)?
Correct answer: C
Factoring x² − 5x + 6 = 0 gives (x − 2)(x − 3) = 0, so x = 2 or x = 3. Since x > 2, we must have x = 3. Then x² + 1/x = 9 + 1/3 = 28/3. Choice D (9) is a common error from computing x² but forgetting to add 1/x. Choice A (10) likely comes from computing 9 + 1 = 10, using 1 instead of 1/3. Choice B (31/3) may result from using x = 2 and computing 4 + 1/2, then incorrectly converting to thirds.
Question 8medium
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles drawn are red?
- A.25/64
- B.5/16
- C.2/7
- D.5/14✓
Correct answer: D
The probability that the first marble is red is 5/8. Given the first was red, 4 red marbles remain out of 7 total, so the probability the second is also red is 4/7. Multiplying gives (5/8) × (4/7) = 20/56 = 5/14. Choice A (25/64) is the result of treating the draws as independent with replacement: (5/8)². Choice B (5/16) is a common arithmetic error when simplifying. Choice C (2/7) results from computing C(5,2)/C(8,2) incorrectly as 10/35 = 2/7 — note C(5,2) = 10 and C(8,2) = 28, so the correct combinatorial answer is 10/28 = 5/14, confirming D.
Question 9medium
Set S = {2, 4, 6, 8, 10}. Every element of S is multiplied by 3 to form a new set T. Which of the following correctly describes the effect on the mean and standard deviation?
- A.Both the mean and the standard deviation are multiplied by 3.✓
- B.The mean is multiplied by 3; the standard deviation remains unchanged.
- C.The mean increases by 18; the standard deviation is multiplied by 9.
- D.The mean is multiplied by 3; the standard deviation increases by 3.
Correct answer: A
The mean of S is (2+4+6+8+10)/5 = 6. Multiplying every element by 3 scales the mean by 3 to 18. Standard deviation measures spread relative to the mean; scaling all values by a constant k multiplies the standard deviation by |k|, so the new standard deviation is 3 times the original. Choice C is wrong because spread also scales with multiplication. Choice A incorrectly states the mean increases additively by 18 (it is multiplied) and compounds the error by squaring the scale factor for SD. Choice D correctly handles the mean but wrongly treats the SD as shifting additively rather than scaling.
Question 10hard
How many 5-digit positive integers have the property that the product of all five digits equals 12? (Note: digits include 0–9, and a leading digit of 0 is not permitted, but 0 as any digit would make the product 0, so all digits must be from 1–9.)
Correct answer: B
We need ordered 5-tuples of digits from {1–9} whose product is 12. First, find all unordered multisets of five digits (each 1–9) with product 12. The valid factorizations of 12 into exactly five factors from 1–9 are: (1) {1,1,1,2,6}: product = 12 ✓; (2) {1,1,1,3,4}: product = 12 ✓; (3) {1,1,2,2,3}: product = 12 ✓. No other combinations work — for example {1,1,1,1,12} fails since 12 is not a single digit, and {2,2,1,3,1} is already case (3). Now count arrangements: Case (1): 5!/3! = 20 ways (three 1s repeated); Case (2): 5!/3! = 20 ways (three 1s repeated); Case (3): 5!/(2!·2!) = 30 ways (two 1s and two 2s repeated). Total = 20 + 20 + 30 = 70. Common errors include missing one of the three factorization families (giving 50 or 60) or double-counting arrangements (giving 80).