About this GRE Math practice test

The ultimate challenge of GRE Quant lies in escaping the clever traps woven into Quantitative Comparisons. These practice drills target high-yield concepts across Algebra, Geometry, and Data Analysis, delivering detailed explanations that don't just solve the math but show you exactly why the distractors are statistically engineered to look correct.

GRE Quantitative Reasoning Practice Test sample questions

These starter questions help you launch a math mock test quickly. Swap them with your own worksheet, notebook, or textbook questions any time.

1. 1. What is the greatest prime factor of 2,310?

2. 2. Arrange the following from least to greatest: 5/8, 0.62, 3/5.

3. 3. A retailer raises a shirt's price by 20%, then during a clearance sale lowers the new price by 20%. The final price is what percent of the original price?

4. 4. In a class, the ratio of boys to girls is 5:7. If there are 12 more girls than boys, how many students are in the class in total?

5. 5. For how many integer values of x is |x - 4| < 3 true?

6. 6. Simplify: √50 + √18 - √8.

7. 7. If 4(x - 3) - 2 = 3(x + 1), what is the value of x?

8. 8. If 2x + y = 11 and x - y = 1, what is the value of x + y?

9. 9. If -3 < 2x + 1 ≤ 7, what is the range of possible values of x?

10. 10. If x² - 5x - 14 = 0 and x > 0, what is the value of x?

11. 11. The function g is defined by g(x) = x² - 4x + 7. What is the minimum value of g(x)?

12. 12. If 3^(2x+1) = 27^(x-1), what is the value of x?

13. 13. In triangle ABC, the measure of angle A is twice the measure of angle B, and the measure of angle C is 20° more than the measure of angle B. What is the measure of angle B?

14. 14. In parallelogram PQRS, angle P measures 65°. What is the measure of angle Q, the angle adjacent to angle P?

15. 15. A chord of a circle has length 16 and is located 6 units from the center. What is the radius of the circle?

16. 16. What is the distance between the points (-2, 3) and (4, -5) in the coordinate plane?

17. 17. A rectangular box has edge lengths 3, 4, and 12. What is the length of the box's longest interior diagonal?

18. 18. A data set consists of six numbers: 4, 7, 7, 9, 12, and x. If the mean of the data set is 8, what is the value of x?

19. 19. What is the median of the data set 3, 5, 5, 8, 9, 11, 14, 20?

20. 20. Set P = {10, 10, 10, 10} and Set Q = {5, 10, 10, 15}. Both sets have the same mean. Which set has the greater standard deviation?

21. 21. A bag contains 5 green, 3 yellow, and 2 orange marbles. If two marbles are drawn at random without replacement, what is the probability that both are yellow?

22. 22. A committee of 3 people is to be selected from a group of 4 men and 5 women. How many different committees can be formed that include at least 2 women?

23. 23. The table below shows the number of books a bookstore sold in four genres during one week: Fiction: 120; Mystery: 80; Biography: 45; Science: 55 (Total: 300 books). What percent of the books sold that week were mystery or science books?

24. 24. Using the same weekly sales data (Fiction: 120, Mystery: 80, Biography: 45, Science: 55), suppose the following week the ratio of fiction books sold to mystery books sold is the same as it was this week. If 150 mystery books are sold the following week, how many fiction books are sold?

25. 25. Quantitative Comparison — x is a negative integer. Column A: x². Column B: x³. (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given. Which is correct?

26. 26. Quantitative Comparison — Column A: the arithmetic mean of 3, 7, 11, and 15. Column B: the median of 3, 7, 11, and 15. (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given. Which is correct?

27. 27. Quantitative Comparison — x is a real number. Column A: (x + 3)². Column B: x² + 9. (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given. Which is correct?

28. 28. Quantitative Comparison — 0 < x < 1. Column A: x³. Column B: x². (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given. Which is correct?

29. 29. Quantitative Comparison — x and y are integers such that x > y > 0. Column A: x - y. Column B: 1. (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given. Which is correct?

30. 30. Quantitative Comparison — n is an integer greater than 2. Column A: n². Column B: 2n. (A) Column A is greater (B) Column B is greater (C) The two quantities are equal (D) The relationship cannot be determined from the information given. Which is correct?

Answer key & quick explanations

Short answers for the sample questions above. Use this to self-check before generating a fresh AI-built mock test.

1. 1. Greatest prime factor of 2,310

   11

   2,310 = 2 × 3 × 5 × 7 × 11, so its prime factorization contains five distinct primes. The largest of these is 11, so that is the greatest prime factor.

2. 2. Ordering 5/8, 0.62, 3/5

   3/5 < 0.62 < 5/8

   Converting to decimals, 3/5 = 0.6 and 5/8 = 0.625, while 0.62 is given directly. Comparing 0.6, 0.62, and 0.625 shows the correct increasing order is 3/5, then 0.62, then 5/8.

3. 3. 20% increase then 20% decrease

   96%

   Increasing by 20% multiplies the price by 1.2, and decreasing that result by 20% multiplies it by 0.8. Since 1.2 × 0.8 = 0.96, the final price is 96% of the original — a classic trap where students expect exactly 100%.

4. 4. Boys:girls 5:7 ratio, 12 more girls

   72

   Let boys = 5x and girls = 7x. Since girls exceed boys by 12, 7x − 5x = 2x = 12, so x = 6. Total students = 5x + 7x = 12x = 72.

5. 5. Integers satisfying |x-4|<3

   5

   The inequality |x - 4| < 3 means -3 < x - 4 < 3, or 1 < x < 7. The integers strictly between 1 and 7 are 2, 3, 4, 5, and 6 — five values.

6. 6. Simplify √50+√18-√8

   6√2

   Each radical simplifies to a multiple of √2: √50 = 5√2, √18 = 3√2, and √8 = 2√2. Combining like terms gives 5√2 + 3√2 - 2√2 = 6√2.

7. 7. Solve 4(x-3)-2=3(x+1)

   x = 17

   Distributing gives 4x - 12 - 2 = 3x + 3, or 4x - 14 = 3x + 3. Subtracting 3x from both sides and adding 14 gives x = 17.

8. 8. System: 2x+y=11, x-y=1

   7

   From the second equation, x = y + 1. Substituting into the first equation gives 2(y+1) + y = 11, so 3y = 9 and y = 3, making x = 4. Therefore x + y = 7.

9. 9. Solve -3<2x+1≤7

   -2 < x ≤ 3

   Subtracting 1 from all three parts gives -4 < 2x ≤ 6. Dividing everything by 2 (a positive number, so the inequality direction is unchanged) gives -2 < x ≤ 3.

10. 10. Solve x²-5x-14=0, x>0

    x = 7

    The quadratic factors as (x - 7)(x + 2) = 0, giving x = 7 or x = -2. Since the problem specifies x > 0, the only valid solution is x = 7.

11. 11. Minimum of g(x)=x²-4x+7

    3

    For a parabola ax² + bx + c with a > 0, the minimum occurs at x = -b/(2a) = 4/2 = 2. Evaluating g(2) = 4 - 8 + 7 = 3 gives the minimum value.

12. 12. Solve 3^(2x+1)=27^(x-1)

    x = 4

    Rewriting 27 as 3³ turns the right side into 3^(3x-3). Setting exponents equal (since the bases match) gives 2x + 1 = 3x - 3, so x = 4.

13. 13. Triangle angle B given relations

    40°

    Let angle B = b; then angle A = 2b and angle C = b + 20. Since the angles sum to 180°, 2b + b + (b + 20) = 180, so 4b = 160 and b = 40°.

14. 14. Parallelogram adjacent angle

    115°

    In any parallelogram, consecutive (adjacent) angles are supplementary because the two sides between them are cut by a transversal of parallel lines. Since 180° - 65° = 115°, angle Q measures 115°.

15. 15. Chord length 16, 6 from center

    10

    A radius drawn to the midpoint of a chord is perpendicular to the chord and bisects it, forming a right triangle with legs 8 (half the chord) and 6 (distance to center). By the Pythagorean theorem, the radius = √(8² + 6²) = √100 = 10.

16. 16. Distance between (-2,3) and (4,-5)

    10

    Using the distance formula, the horizontal change is 4-(-2)=6 and the vertical change is -5-3=-8. The distance is √(6² + 8²) = √100 = 10, a 6-8-10 right triangle.

17. 17. Diagonal of box 3x4x12

    13

    The space diagonal of a rectangular box with edges a, b, c is √(a²+b²+c²). Here that is √(9+16+144) = √169 = 13.

18. 18. Six numbers, mean 8, find x

    9

    A mean of 8 over six numbers means the total sum is 48. The five known numbers sum to 4+7+7+9+12 = 39, so x = 48 - 39 = 9.

19. 19. Median of 8 numbers

    8.5

    With eight ordered values, the median is the average of the 4th and 5th terms. Here those are 8 and 9, so the median is (8+9)/2 = 8.5.

20. 20. Compare sd of Set P and Set Q

    Set Q

    Set P has no variation (all values equal 10), so its standard deviation is 0. Set Q has deviations of -5, 0, 0, and 5 from its mean of 10, giving a positive standard deviation, so Set Q's is greater even though both means are identical — a reminder that sd measures spread, not center.

21. 21. P(both yellow) from bag of marbles

    1/15

    There are 3 yellow marbles out of 10 total. Drawing without replacement, P(both yellow) = (3/10)×(2/9) = 6/90 = 1/15.

22. 22. Committees with at least 2 women

    50

    Count committees with exactly 2 women and 1 man: C(5,2)×C(4,1) = 10×4 = 40. Add committees with 3 women: C(5,3) = 10. The total is 40 + 10 = 50.

23. 23. DI: percent mystery+science

    45%

    Mystery and science books together total 80+55 = 135 out of 300 total books sold. That is 135/300 = 0.45, or 45%.

24. 24. DI: fiction books next week

    225

    This week's ratio of fiction to mystery books is 120:80, which simplifies to 3:2. If mystery books next week are 150, fiction books = 150 × (3/2) = 225, preserving the same ratio.

25. 25. QC: x² vs x³, x negative integer

    Column A is greater

    Since x is negative, x² is always positive, while x³ (a negative number times a positive square) is always negative. A positive number is always greater than a negative one, so Column A is greater for every negative integer x — try x=-1 (1 vs -1) or x=-5 (25 vs -125) to confirm. Column B is never greater, the columns are never equal, and the relationship never varies, ruling out the other three choices.

26. 26. QC: mean vs median of 3,7,11,15

    The two quantities are equal

    The mean is (3+7+11+15)/4 = 36/4 = 9. The median, the average of the two middle terms 7 and 11, is also (7+11)/2 = 9. Because both values compute to exactly 9, neither column is greater, and the relationship doesn't depend on any unknown, so 'cannot be determined' is also wrong.

27. 27. QC: (x+3)² vs x²+9, x real

    The relationship cannot be determined from the information given

    Expanding gives (x+3)² = x² + 6x + 9, so the difference between the columns is exactly 6x. If x > 0, Column A is greater (e.g., x=1: 16 vs 10); if x < 0, Column B is greater (e.g., x=-1: 4 vs 10); if x = 0, they're equal (9 vs 9). Since all three outcomes are possible depending on x, the relationship cannot be determined.

28. 28. QC: x³ vs x² for 0<x<1

    Column B is greater

    For any x strictly between 0 and 1, multiplying x² by x (a positive number less than 1) always shrinks it, so x³ < x² throughout the entire interval — check x=0.5 (0.125 vs 0.25) or x=0.9 (0.729 vs 0.81). Because this holds for every value in the given range with no exception, Column B is greater, and 'equal' or 'cannot be determined' don't apply.

29. 29. QC: x-y vs 1, integers x>y>0

    The relationship cannot be determined from the information given

    Since x and y are integers with x > y > 0, the smallest possible difference is 1 (e.g., x=2, y=1), which makes the columns equal, but larger gaps are equally valid (e.g., x=10, y=1 gives a difference of 9, making Column A greater). Because both 'equal' and 'Column A is greater' are genuinely possible depending on the integers chosen, and Column B can never be greater, the relationship cannot be pinned down.

30. 30. QC: n² vs 2n for integer n>2

    Column A is greater

    The difference n² - 2n factors as n(n-2), which is positive whenever n > 2, so Column A exceeds Column B for every allowed integer — check n=3 (9 vs 6) or n=10 (100 vs 20). Because n=2 (where they'd be equal) is explicitly excluded and the difference only grows for larger n, there's no case producing equality or a Column B advantage.

Curriculum Mapping & Learning Guide

Use this breakdown to identify which skills each question tests and guide post-test review.

GRE Math units covered

1. Chapter 1: Arithmetic: integers, divisibility, primes, fractions, decimals, percentages
2. Chapter 2: Arithmetic: ratio, proportions, absolute value, powers and roots
3. Chapter 3: Algebra: linear equations and inequalities, systems of equations
4. Chapter 4: Algebra: quadratic equations, functions and graphs, exponents
5. Chapter 5: Geometry: lines, angles, triangles, quadrilaterals, circles
6. Chapter 6: Geometry: coordinate geometry, 3D figures
7. Chapter 7: Data Analysis: mean, median, mode, standard deviation, percentiles
8. Chapter 8: Data Analysis: probability, counting, data interpretation (charts/tables)
9. Chapter 9: Quantitative Comparison: strategy and column comparison

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