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GRE Math Questions & Practice

According to ETS (the makers of the GRE), the GRE Quantitative Reasoning measure is a test that assesses your basic math skills, understanding of simple math concepts, ability to reason quantitatively, and aptitude for solving problems using quantitative methods.

This is really just a fancy way of saying that you better know your math concepts and be able to apply them in tricky situations. The Quantitative Reasoning test is as much a test of your ability to stay cool under pressure as it is a test of your knowledge of math. Whether you’re a math whiz or not, the best way to prepare for the GRE math problems that you’ll see on test day is to try some sample GRE math questions, check your answers, and then review the explanations.

Below are five GRE practice questions that will help you to sharpen your math skills and determine the approximate level at which you’re scoring. These questions cover the range of concepts you could expect to see on the exam. Remember that you’ll receive a GRE score on a scale of 130 to 170 for each section. If you struggle with the 165 – 170 range problems in this post, don’t despair. These questions get tricky. Perhaps you are only struggling with a certain concept area. No worries – you can click the link above each question to access even more GRE math practice problems dealing with a certain concept.

You’ll find answers and explanations at the end of the post. Good luck! (And no peeking!)

GRE Math Questions

sample gre math questions

Sample GRE Math Question #1

Question Type: Multiple Answer Questions (Choose all that apply)

Concept: Absolute Value/Algebra

Level: 145 – 150

What are all the possible solutions of | |x – 2| – 2| = 5?

  1. -5
  2. -3
  3. -1
  4. 7
  5. 9

Sample GRE Math Question #2

Question Type: Multiple Choice

Concept: Symbolic Reasoning/Exponents

Level: 165 – 170

If (a^2)(b) is an integer which of the following must be an integer?

  1. a
  2. b
  3. ab
  4. b^2
  5. None of the above

Sample GRE Math Question #3

Question Type: Numeric Entry

Concept: Prime Numbers/Factors

Level: 150 – 155

How many positive integers less than 100 are the product of three distinct primes? [ ]

Sample GRE Math Question #4

Question Type: Quantitative Comparison

Concept: Exponents/Fractions

Level: 155 – 160

-1<x<100

Column A Column B
x^3 x^6

 

  1. The quantity in Column A is greater
  2. The quantity in Column B is greater
  3. The two quantities are equal
  4. The relationship cannot be determined from the information given

Sample GRE Math Question #5

Question Type: Multiple Choice

Concept:: Geometry/Variables in Answer Choices

Level: 160 – 165

A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x? (160 – 165)

  1. x sqrt{2}
  2. 2 sqrt{x} - sqrt{2}
  3. {sqrt{2}/2} - {x/4}
  4. x sqrt{2} - {x/2}
  5. {sqrt{2x}/2} - {sqrt{x}/2}

Answers to GRE Example Questions

1. (A, E)
2. E
3. 5
4. D
5. E

GRE Math Problems: Answers & Explanations

gre example questions

The math practice questions you attempted above ranged from relatively easy to very difficult. By this time, I hope that you’ve attempted each problem and checked your answers. Now, unless you’re incredibly well prepared for the GRE Quant section, I’m guessing that you didn’t get 100% of the questions correct. Even if you did, I’d bet that you took more time per question than you’d have liked to.

So! With that in mind, I highly (HIGHLY) recommend that you review each and every explanation given below. Maybe you’ll learn something new that’ll help you on test day.

Explanation to Sample GRE Math Question #1

Question Type: Multiple Answer Questions (Choose all that apply)

Concept: Absolute Value/Algebra

Level: 145 – 150

What are all the possible solutions of | |x – 2| – 2| = 5?

  1. -5
  2. -3
  3. -1
  4. 7
  5. 9

Answers: A, E
If we focus just on the delim{|}{x-2}{|}, we can see that the result must be positive. Stepping back and looking at the entire equation we substitute u for delim{|}{x-2}{|}, to get delim{|}{u-2}{|}=5. Solving for absolute value, we get the following:

u-2=5
u-2=-5

Thus, u=7 and u=-3. Because u must be positive, we discount the second result. Next, we have to find x in the original delim{|}{x-2}{|}, which we had substituted with u. Replacing u with 7 we get:

delim{|}{x-2}{|}=u

delim{|}{x-2}{|}=7

x-2=7 and x-2=-7

x=9 and x = -5 .

A faster way is to plug in the answer choices to see which ones work.

Explanation to Sample GRE Math Question #2

Question Type: Multiple Choice

Concept: Symbolic Reasoning/Exponents

Level: 165 – 170

If (a^2)(b) is an integer which of the following must be an integer?

  1. a
  2. b
  3. ab
  4. b^2
  5. None of the above

Answer: E
Let’s choose numbers to disprove each case. By the way, the word disprove is very important here – the question says ‘must’ so by picking numbers that prove the case, we are not necessarily proving that an answer choice must always be an integer.

For A. I can choose sqrt{3}=a, and b is any integer. Because a is not an integer, A. is not correct.

For B. it’s a bit tricky. However, if you keep in mind that there are no constraints in the problem stating that a cannot equal b, we can make a=3^{1/4} and b=sqrt{3}.

For C. we can choose the same numbers to show that ab is not an integer.

For D. if b=3^{1/4} and a=3^{3/8} (a^2)(b) equals an integer, but b^2 does not.

Explanation to Sample GRE Math Question #3

Question Type: Numeric Entry

Concept: Prime Numbers/Factors

Level: 150 – 155

How many positive integers less than 100 are the product of three distinct primes? [5]

Answer: 5
Let’s write out some primes: 2, 3, 5, 7, 11, 13, and 17.

I’m stopping at 17 because the smallest distinct primes, 2 and 3, when multiplied. by 17 give us 102. Therefore 13 is the greatest prime conforming to the question. Here is one instance. 2*5*13 is greater than 100 so we can discount it.

Working in this fashion we can add the following instances:

2*3*5

2*3*7

2*3*11

2*5*7

3*5*7 is too great

2*3*13=78.

Therefore, there are five instances.

Explanation to Sample GRE Math Question #4

Question Type: Quantitative Comparison

Concept: Exponents and Fractions

Level: 155 – 160

-1<x<100

Column A Column B
x^3 x^6

 

  1. The quantity in Column A is greater
  2. The quantity in Column B is greater
  3. The two quantities are equal
  4. The relationship cannot be determined from the information given

Answer: D
If x is less than 0 the answer is B. If x is 0 <x<1, the answer is A. Therefore, the answer is D.

Explanation to Sample GRE Math Question #5

Question Type: Multiple Choice

Concept: Geometry/Variables in Answer Choices

Level: 160 – 165

A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x? (160 – 165)

  1. x sqrt{2}
  2. 2 sqrt{x} - sqrt{2}
  3. {sqrt{2}/2} - {x/4}
  4. x sqrt{2} - {x/2}
  5. {sqrt{2x}/2} - {sqrt{x}/2}

Answer: E
If the area of the small square is x, then each side is √x. The area of the large square is 2x (you want to add the area of the small square to that of the path), leaving us with sides of √2x. If we subtract the length of a side of the small square from a side of the large square, that leaves us with √2x – √x. Remember that there are two parts of the path, so we have to divide by 2: √2x/2 – √x/2, which is (E).

GRE Practice Questions: An Important Takeaway

When practicing GRE math problems, the key is to figure out why you answered questions incorrectly. At first this process can be frustrating. But remember, by forcing yourself to figure out the answer instead of immediately turning to an explanation, you will understand the problem at a deeper level and be less likely to miss a similar problem in the future. This is the best (and really the only) way to improve your score on the GRE.

If you are still unsure about the answers to the problems above, let me know by leaving a comment, and I will provide an explanation. Also, don’t forget to try out Magoosh GRE. We offer hundreds of practice problems for all sections of the GRE, and they all come with text and video explanations.

Finally, if you’re looking for more free Magoosh practice questions, then check out our GRE Math Practice post.

 

Editor’s Note: This post was originally published in January 2012 and has been updated for freshness, accuracy, and comprehensiveness.

By the way, students who use Magoosh GRE improve their scores by an average of 8 points on the new scale (150 points on the old scale.) Click here to learn more.

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