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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 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 is an integer which of the following must be an integer?

1. 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

Column A Column B

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

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

## GRE Math Problems: Answers & Explanations

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

If we focus just on the , we can see that the result must be positive. Stepping back and looking at the entire equation we substitute u for , to get . Solving for absolute value, we get the following:

Thus, and . Because u must be positive, we discount the second result. Next, we have to find in the original , which we had substituted with u. Replacing u with 7 we get:

and

and .

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 is an integer which of the following must be an integer?

1. None of the above

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 , 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 and .

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

For D. if and equals an integer, but 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]

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. is greater than 100 so we can discount it.

Working in this fashion we can add the following instances:

.

Therefore, there are five instances.

### Explanation to Sample GRE Math Question #4

Question Type: Quantitative Comparison

Concept: Exponents and Fractions

Level: 155 – 160

Column A Column B

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

If x is less than 0 the answer is B. If x is , the answer is A. Therefore, the answer is D.

### Explanation to Sample GRE Math Question #5

Question Type: Multiple Choice

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)

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.

### 36 Responses to GRE Math Questions & Practice

1. Ash August 10, 2016 at 1:30 am #

Hi,

In the solution to the 1st Problem,

it says |x – 2| = -2 in the last but second step, which in actuality should have been

|x – 2| = -7. Kindly, make this change so that it doesn’t confuse ppl like me 😛

Thanks!

• Magoosh Test Prep Expert August 15, 2016 at 2:46 am #

Hi Ash 🙂

As we mentioned in another comment, at this point, we only want to consider the positive value for u, u = 7. So, when we solve for the value of x, we get

|x-2| = u
u = 7

|x – 2| = 7
x – 2 = 7 or x – 2 = -7

And we find that x = 9 or x = -5.

I hope this clears things up!

2. Taka August 9, 2016 at 8:24 am #

Hi Chris,
Tks for great practice question. As for #2, why did you choose ‘root’ mostly as picking number when you examine each choices. The condition in this question is ‘integer’ so I used fraction (eg.1/2). Actually I got wrong on ‘ab’ because I thought ab must be integer always. How I can improve math sense for picking up number?

• Magoosh Test Prep Expert August 13, 2016 at 9:53 pm #

If you’re dealing with a squared variable, it’s a good idea to immediately think about that variable in terms of square root, if you’re trying to figure out whether or not the variable is an integer. A square root of an integer, once squared, simply becomes the integer itself. But a square root of an integer that is not squared will very often NOT be an integer. That’s why the root is one of the first things you should look at when considering integers in an equation such as (a^2)(b).

Fractions are a bad first place to go for a very simple reason— operations with fractions are very difficult to do in your head. On the GRE, attempting to use fractions when you have other options will always slow you down and put you at risk of making mistakes.”

There’s no one trick to developing good number sense. But there are a few different kinds of things you can do to build up this skill over time while you prep for the GRE. The trick is to learn how to experiment with numbers and even play games with them on a regular basis. If you’re not a math person, this may sound strange. To wrap your mind around the idea of number sense and how to build it, I recommend checking out Magoosh’s GRE video lesson on developing number sense. This is a very important key concept to mastering GRE quants, so we offer this lesson as a free sample, even to non-subscribers.

• Taka August 14, 2016 at 2:08 am #

Tks a lot!! It was very helpful.

3. Daniel August 9, 2016 at 7:48 am #

Hello Chris,

Being extra picky, the path is not of uniform width. Being w the “general” width, at the corners the width is w√2. For the path to be of uniform width, the corners would need to be rounded. This way the answer would involve \pi somewhere.

But I believe this consideration would make the question harder than the usual GRE General question.

Thank you for the questions!

4. Maahi August 6, 2016 at 9:55 am #

In the first question,I think you meant |x-2|=-3 instead of |x-2|=-2. If so,then x will be 5,right?

• Magoosh Test Prep Expert August 15, 2016 at 2:37 am #

Hi Maahi,

This part of the explanation is a bit confusing. We have solved for u = 7, -3. And at this point, we discount u = -3, because an absolute value cannot equal a negative number. So, if we only consider u = 7, we get

|x – 2| = 7
x – 2 = 7 or x – 2 = -7

Solving for x, we see that

x – 2 = 7 –> x = 9
x – 2 = -7 –> x = -5

Our solutions are therefore x = 9 or x = -5.

We can also see that the solutions are x = -5 or x = 9 and not x = 5 by plugging these values in for x:

||x – 2| – 2| = 5

x = -5
||-5 – 2| – 2| = 5
||-7| – 2| = 5
7 – 2 = 5 (YES!)

x = 9
||9 – 2| – 2| = 5
||7| – 2| = 5
7 – 2 = 5 (YES!)

x = 5
||5 – 2| – 2| = 5
||3| – 2| = 5
3 – 2 = 1 not 5 (NO!)

I hope this clears up your doubts! 🙂

5. Peter August 5, 2016 at 2:06 pm #

Hello Chris,

Do you have any tip on how to avoid careless mistakes? What I am noticing during my practice is that I don’t really have much problem in answering questions. However, I do end up making caress less mistakes that cost me points. For example, if a question asks what is the probability a certain event will not happen, I end up answering the probability that certain event will happen. I have repeatedly told myself to read the questions carefully and still end up making mistakes. I noticed that on average, I make one silly mistake every six problem. Sometimes, I don’t read questions carefully and other times I make silly calculation mistakes. I really don’t know how to deal with this. Any tip will be appreciated.

• Magoosh Test Prep Expert August 6, 2016 at 8:23 am #

Hi Peter 🙂

Happy to help! First, it’s great that you’ve already thought about some of the circumstances under which you make these types of mistakes. With that in mind, overall, I’d recommend that you take your time with each question. Reading each question carefully is key to answering what’s actually being asked. If not, it’s very unlikely that you’ll get the right answer.

After that, keep in mind that we often make mistakes towards the end of a question. This happens to the best of us in different situations: we drop our guard when we see the finish line. We relax. We lose focus. We rush. All of that doesn’t help us to do well. So, when you see the finish line, when you are nearing the end of the problem, focus even more. Don’t rush.

Also, you should ask yourself if you make these types of mistakes more often when you’re tired. If so, taking a short break to stretch, move around, and drink water might be all that you need to refocus. Obviously you won’t be able to take a stretch break in the middle of the test, but you should absolutely do so during your study periods. For every hour of study, let your mind drift off to somewhere else for 5 minutes. Then return to answering problems.

As for test day, closing your eyes momentarily can offer a nice break. Keep them closed and count to 10 or 20. Try to push all thoughts out of your mind and just focus on taking long, purposeful breaths. This will help to fight off any exhaustion.

I would also recommend reading the following blog post about how to stay focused during the test. Some of these tips can be applied to your study time as well 🙂

How to Stay Focused on the GRE

I hope this helps! Happy studying 🙂

6. Salwan June 29, 2014 at 2:40 pm #

Chris, thank you very much for your help for us as gre student, but because maybe we have no time to see the explanation we got confused about the answer. Please make sure that you mention the right one. have a great day

7. Ahmed October 20, 2013 at 3:21 am #

Whats the answer for the last (Geometry) question? (the path around the square garden)

• Chris Lele October 21, 2013 at 12:04 pm #

Hi Ahmed,

The answer is (E). Let me know if you need an explanation :).

• Nikki October 23, 2013 at 6:34 pm #

I’d like an explanation please 🙂

• Chris Lele October 24, 2013 at 11:30 am #

Sure,

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).

Hope that helps!

• SK August 24, 2015 at 4:05 pm #

I get the √2x – √x part, but I’m still a little confused about why you divide by 2…what do you mean there are two parts of the path?

• rihan September 10, 2016 at 7:59 pm #

i was wondering when the question said that the square and path both have an area of x i understood it that “area square”+”area path”=x

• Magoosh Test Prep Expert September 11, 2016 at 6:59 am #

Hi Rihan,

Happy to clarify 🙂 That part of the prompt reads: “If the path and the garden both have an area of x”. This means that the path has an area of x and the garden has an area of x, not that the sum of the two areas is equal to x. It expresses the idea that both spaces have areas equal to x, not that together the spaces have an area of x. Some examples of wording that would indicate that [area square] + [area path] = x are:

* “Together, the area of the path and of the garden is equal to x”
* “The sum of the area of the path and the garden is equal to x”

Do you see how these wordings indicate that the sum of the two areas is x, while the original prompt is saying that each space (the square and the path) have an individual area of x?

I hope this helps!

• Ragith Radhakrishnan April 19, 2016 at 3:45 am #

The outer figure is a square, so u have path in both side. You need to divide with 2 to get one side width.

i didnt get square problem..please give me the process by giving diagrams..thank you

• Magoosh Test Prep Expert July 6, 2016 at 12:42 pm #

Here’s a link to a figure to help explain this practice problem: .

As you can see in the figure, since the area of the green square is x, the side length of that square is sqrt(x). Similarly, the area of the larger square is 2x, so the side length of that square is sqrt(2x). We can also see that the side length of the larger square is equal to one side length of the green square plus two times the width of the path. Using the values we currently know, that is:

sqrt(2x) = sqrt(x) + 2?

where ? is the width of the path. When we solve for ? following the steps to the right of the figure, we find that ? = sqrt(2x)/2 – sqrt(x)/2

I hope this explanation helps!

8. Nicholle August 23, 2013 at 9:04 am #

On this question (see below), “C” is bolded as if it’s the answer, but it’s actually D (which is listed as the right answer but not highlighted). Just a friendly heads up 😉
____________

Question Type: Quantitative Comparison

Concept: Exponents and Fractions
Level: 155 – 160

-1<x<100

Column A

Column B

x^3 x^6

A.The quantity in Column A is greater
B.The quantity in Column B is greater
C.The two quantities are equal
D.The relationship cannot be determined from the information given

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.
_____________

• Chris Lele August 23, 2013 at 1:33 pm #

Thanks Nicole for pointing that out :). We’ll fix it right away.

9. kishor September 1, 2012 at 5:57 am #

in the prime number question, the answer is written as 6 but the explanation shows the answer as 5, so which may be the correct answer?

• Chris September 5, 2012 at 4:14 pm #

Sorry for the confusion, Kishor. That was a typo :). The answer should be ‘5.’

10. suratha rajan August 23, 2012 at 8:10 am #

(2w+vX) is the side of the big square and vX is the side of the small square then subtracting the area of big and small square we get the area of the path which is also X
can we write it as

(2W+vX)^2 – X = X

then take x on the other side making it 2X
then squaring it on both sides
we get

(2W+vX) =v2X

then W=vX/2

• Chris August 24, 2012 at 12:03 pm #

Hi Suratha,

I kind of lost you there :). At the very end you’ve solved interms of W, but I am not sure how that helps you get the answer. Also by adding extra variables–even if doing so is logically valid–I think it complicates things a little.

• suratha rajan August 24, 2012 at 12:33 pm #

Precisely my point Chris , the answer choice E is not simplified and if it is’nt then its kinda hard to understand or choose ! my final answer was “vX/2” which also the answer E being not simplified , w is width , and x is the area , cheers !!! good selection of problems looking forward to a lot of problems

• Chris August 27, 2012 at 11:52 am #

I see – got it!

We definitely have more such problems on the way :).

• Deepthi September 4, 2013 at 4:22 am #

Hi Suratha,

A small correction…
Option E is correct & cant be simplified.
u have considered (squareroot(2x) – squareroot(x)) = squareroot(x)
which is wrong.

11. Dhawal July 28, 2012 at 10:30 pm #

Hey Chris..!!! i just have 23 days for my GRe…give me some tips go score..help me out in quants specially..

12. nashaat June 28, 2012 at 1:42 pm #

Hello I believe there was a mistake with one of the problems about the prime numbers, there is a sixth one which is 2*3*13= 78, which is less than 100.

• Chris June 29, 2012 at 12:14 pm #

Hmmm, it seems as if 2 x 3 x 13 is written as part of the explanation but is inserted in the wrong place. I’ve put it in the right spot. Thanks for pointing this out :).

13. tabi January 20, 2012 at 4:36 am #

Thank you so much for the explanation!

• Chris January 20, 2012 at 12:12 pm #

Great! I’m happy it helped. Stay tuned for more math questions!

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