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# Revised GRE Math Practice Questions: Explanations!

A couple of weeks ago I wrote a Revised GRE Math practice questions post. The math practice questions ranged from relatively easy to very difficult. At the time I provided answers but not explanations. One of my hopes was that students would be able to reason on their own why a certain answer is correct. I was also curious to see if anyone would post explanations.

Understandably, a few of you wanted explanations, whether you were stumped or because you were curious if your reasoning was valid. As for the second point, some one did post an explanation (thanks, Ahmad!)

Below are my explanations.

## 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.

## 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.

## 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.

## 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.

## 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)

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).

Check out some more GRE math practice!

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### 24 Responses to Revised GRE Math Practice Questions: Explanations!

1. 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

2. 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 ๐

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

Sure,

If the area of the small square is x, then each side is โ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 โ2x. If we subtract the length of a side of the small square from a side of the large square, that leaves us with โ2x – โx. Remember that there are two parts of the path, so we have to divide by 2: โ2x/2 – โx/2, which is (E).

Hope that helps!

• SK August 24, 2015 at 4:05 pm #

I get the โ2x โ โ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?

• 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.

• radhe radhe June 5, 2016 at 12:05 am #

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!

3. 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 ๐
____________

Question Type: Quantitative Comparison

Concept: Exponents and Fractions
Level: 155 โ 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.

4. 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.’

5. 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.

6. 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..

7. 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 :).

8. 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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