Official GRE Quantitative Comparison Practice Questions

Correct Answer is (B).

This particular GRE question is one that many past high-scoring test takers have struggled with greatly. At Menlo Coaching, we constantly emphasize to all our students how important it is not to get trapped by an all-too-tempting “con” that the exam is setting you up to fall for.

For example, the big “con” in this question that students frequently fall for is trying to rewrite the initial equation by expanding the perfect squares (which unintentionally ends up making the problem more complicated than it really is).

The resulting equation would look as follows:
`(x + 5)*(x + 5) – (y – 3)*(y – 3) = 0`


At this point, the equation can be further expanded using FOIL rule (First, Outside, Inside, Last) to obtain:

`(x^2 + 5x + 5x + 25) – (y^2 – 3y – 3y + 9) = 0`

Which further simplifies to:
`(x^2 + 10x + 25) – (y^2 – 6y + 9) = 0`

Unfortunately, this is the trap the GRE is hoping you will fall into. The question is now a much more complicated double quadratic problem containing two different trinomials and two distinct variables, which will ultimately prove significantly more difficult to solve.

A best practice principle that all Menlo GRE tutoring students are taught is to first take a moment to critically assess what each problem is testing you on and then determine the best course of action to solve it efficiently, rather than creating overly-complex algebraic expressions. This requires students to develop more of a “soft” skill that can be trained with the proper guidance and expert coaching.
Here’s a much more efficient approach to this question.

Since the GRE is a standardized exam that tests a set body of mathematical knowledge again and again, it is often helpful to be familiar with some of the most commonly tested mathematical concepts on the exam. For example, when it comes to exponents, the single most commonly tested quadratic on the GRE is a ‘difference of squares’.

The general form of this equation is as follows:
`a^2 – b^2 = 0` which can immediately be factored down to `(a + b)*(a – b) = 0`.

Consequently, the key to solving this problem efficiently is being able to see that the initial equation given is essentially just a ‘difference of squares’, albeit in a slightly more complicated form, where `a = x + 5` and `b = y – 3`.

Since `a^2 – b^2 = 0` simplifies to `(a + b)*(a – b) = 0`, it follows that `(x + 5)^2 – (y – 3)^2 = 0` can be simplified to `((x + 5) + (y – 3))*((x + 5) – (y – 3)) = 0`.

This equation can further be simplified to:
`(x + 5 + y – 3)*(x + 5 – y + 3) = 0`

After you combine the like integer terms, you get:
`(x + y + 2)*(x – y + 8) = 0`

Since you know `x+y=2`, by substituting this into the first bracket, you obtain:
`(2 + 2)*(x – y + 8) = 0` which equals `(4)*(x – y + 8) = 0`

Given that the right side of the equation equals 0, this means one of the two brackets on the left side must also equal zero. Since the first bracket equals 4, this means the second bracket must equal 0.
If `x – y + 8 = 0` then `x – y = -8 `

Given that Quantity A equals x – y, this means Quantity A equals -8. Lastly, given that Quantity B equals 0 and the value of 0 is greater than -8, the correct answer is (B).

What are our final key takeaways here?
Firstly, almost all medium-to-hard GRE questions follow the same rule. There is an initial “con” that the test makers are hoping you will fall for.

Tip: Don’t fall into their trap! Good test takers will always look to spot the “con” in the problem before jumping in and just attempting to do the math without critically assessing the situation first.

Secondly, it is invaluable to develop the “soft” skill of more intuitively being able to see what math concept the question is really testing you on.

Tip: Top test takers will always be on the lookout for commonly tested patterns that will inevitably appear on all “standardized” exams. Here, the key is noticing how similar the initial equation is to a traditional difference of squares. Spot this up front and it makes all the difference in the world in being able to solve this problem quickly, efficiently, and (most importantly) correctly.

Correct Answer is (D).

This next problem is equally challenging but in a different way. At first glance, it seems to be a fairly straightforward algebra question containing two basic expressions, one in Quantity A and one in Quantity B.

However, the key to answering this question correctly (and doing so in a timely manner) is far from simple.

There are several ways a student can go wrong here. First off, all algebra questions on the GRE can be solved one of two different ways:
1) algebraically; or
2) plugging in numbers.

A truly novice test taker might resort to approaching all algebra questions on the GRE exam algebraically. This, however, would be a major strategic mistake.

Luckily though, those who use official GRE study resources to prepare for the exam would already know to “plug in numbers” as ETS explicitly advises students to do so on all GRE quantitative comparison questions (containing algebra).

For example, in this question, since `x != 1`, a likely starting point for many test takers would likely be `x = 2` (the smallest integer larger than 1). Using this value, we get the following:

Quantity A: `(2^2-2)/(2-1)`

Quantity B: 2+2

This simplifies to:

Quantity A: 4 – 2 = 2

Quantity B: 4

Since Quantity B is greater than Quantity A (4 is greater than 2), the unwitting student may simply select answer choice (B) here and move on to the next question. However, doing so would mean committing another major strategic error (that of only plugging in once). The general rule of thumb on GRE quantitative comparison questions is that you need to plug in at least twice (and try to obtain the opposite result the second time you plug in).

Unfortunately, many novice test takers unfamiliar with the quantitative comparison question format may simply be unaware of this and end up getting this question wrong without even realizing they have been trapped. It’s at this point where quantitative comparisons begin to get infinitely more complex.

Let’s take things up another level though…

Even for students who know to plug in twice, the “softer” skill of knowing how to correctly identify the right numbers to select to successfully obtain the opposite result is oftentimes easier said than done.

For example, let’s say we use x = 3 instead of x = 2.

Using this new value for x, we get the following:

Quantity A: `(3^2-2)/(3-1)`

Quantity B: 3 + 2

This simplifies to:

Quantity A: `7/2 = 3.5`

Quantity B: 5

Once again, answer choice (B) seems to be correct. It is at this point where even a relatively well-trained test taker who knows to plug in twice but fails to “see” the right numbers to choose, may still end up getting this question wrong.

Why is that? Well, simply put, there are a plethora of different number property criteria, other than using a bigger (or a smaller) number, any one of which could potentially provide us with the opposite result we are looking for.

Furthermore, given that all numbers on the GRE are assumed to be real (unless otherwise noted), a common trap many students fall into is failing to consider other unusual (but still possible) values that x could be and not plugging those numbers in accordingly (i.e. positive and negative numbers, integers and fractions, etc.).

Keeping this in mind, let’s remember to consider a negative number and use `x = -1`.

Using this value for x, we get the following:

Quantity A: `((-1)^2 – 2)/(-1 – 1)`

Quantity B: -1 + 2

This simplifies to:

Quantity A: `(-1)/-2`

Quantity B: 1

Unfortunately, once again, answer choice (B) still seems to be correct.

Let’s also not forget to consider a fractional or decimal value here and try `x = 1/2`.

Using this value for x, we get the following:

Quantity A: `((1/2)^2-2)/(-(1/2)-1)`

Quantity B: `((-1)/2)+2`

This simplifies to:

Quantity A: `(-1.75)/(-1.5)`

Quantity B: 1.5

Which further simplifies to:

Quantity A: 1.16

Quantity B: 1.5

Even here, answer choice (B) still seems correct.

So, does this mean (B) is right? Well, maybe. But maybe not.

This is where high-level GRE quantitative comparison questions can become incredibly tricky (and time-consuming). One of the primary ways ETS makes these questions more challenging is by structuring the parameters of the question in such a way that there are very few ways to truly get the opposite result. In fact, on the highest-level of such questions, there is sometimes only one number (or one way) to reach the correct final answer.

How then do we know if we’ve missed out on locating that key unique number?

Furthermore, how do we know when this number doesn’t exist and recognize that it’s better to just stop plugging in now and move on?

At Menlo Coaching, we train our GRE tutoring students to follow a variety of key best practice principles that will allow them to spot subtle clues hiding in the questions that, once identified, will “unlock” the question and allow them to get to the right answer quickly and consistently.

Tip: Great test takers go one step further beyond simply considering real numbers and always use the context of the question to help identify what key or unique numbers should be considered.

How would that concept be applied to this question though?

Did you notice there was an initial condition listed in the question stem that x ≠ 1?

Why would they have said that?

Perhaps this is the missing clue that will be the key to “unlocking” this question.

One founding principle we teach our students is to recognize that there are two key numbers on the real number line that have very unusual number properties (0 and 1). And this question has, for some unknown reason, told us that we cannot use the number 1. Interestingly enough though, it never said anything about not using 0. As it turns out, this number will ultimately be the key to unlocking this question.

Let’s try x = 0 right now…

Using this as our final value for x, we get the following:

Quantity A: `((0)^2-2)/(0-1)`

Quantity B: 0 + 2

This simplifies to:

Quantity A: `(-2)/(-1)=2`

Quantity B: 2

As you can clearly see, in this case, the two quantities are the same, which is answer choice (C). However, since the previous numbers gave us (B) and this number gives us (C), we don’t actually know which quantity is bigger. Therefore, the right answer is (D).

GRE quantitative comparison questions that provide seemingly innocuous conditions (i.e. `x !=1`) in the question stem may in fact be hiding the key to solving the problem in plain sight.

To make sure you don’t miss out on testing counterintuitive numbers like 0 and 1, you should always aim to use the context of the GRE quantitative comparison question to guide you in your number picking.

Ultimately, this problem shows you how spotting this key “clue” up front would not only have allowed us to solve this problem easily but also allowed us to do so in just two rounds of plugging in as opposed to four or five, thus saving us significant exam time that can now be used more productively later on.

Correct Answer is (D).

This next problem is equally challenging but in a different way. At first glance, it seems to be a fairly straightforward algebra question containing two basic expressions, one in Quantity A and one in Quantity B.

However, the key to answering this question correctly (and doing so in a timely manner) is far from simple.

There are several ways a student can go wrong here. First off, all algebra questions on the GRE can be solved one of two different ways:
1) algebraically; or
2) plugging in numbers.

A truly novice test taker might resort to approaching all algebra questions on the GRE exam algebraically. This, however, would be a major strategic mistake.

Luckily though, those who use official GRE study resources to prepare for the exam would already know to “plug in numbers” as ETS explicitly advises students to do so on all GRE quantitative comparison questions (containing algebra).

For example, in this question, since x ≠ 1, a likely starting point for many test takers would likely be x = 2 (the smallest integer larger than 1). Using this value, we get the following:

Quantity A: ^{2² – 2}⁄_{2 – 1}

Quantity B: 2 + 2

This simplifies to:

Quantity A: 4 – 2 = 2

Quantity B: 4

Since Quantity B is greater than Quantity A (4 is greater than 2), the unwitting student may simply select answer choice (B) here and move on to the next question. However, doing so would mean committing another major strategic error (that of only plugging in once). The general rule of thumb on GRE quantitative comparison questions is that you need to plug in at least twice (and try to obtain the opposite result the second time you plug in).

Unfortunately, many novice test takers unfamiliar with the quantitative comparison question format may simply be unaware of this and end up getting this question wrong without even realizing they have been trapped. It’s at this point where quantitative comparisons begin to get infinitely more complex.

Let’s take things up another level though…

Even for students who know to plug in twice, the “softer” skill of knowing how to correctly identify the right numbers to select to successfully obtain the opposite result is oftentimes easier said than done.

For example, let’s say we use x = 3 instead of x = 2.

Using this new value for x, we get the following:

Quantity A: ^{3² – 2} ⁄_{3 – 1}

Quantity B: 3 + 2

This simplifies to:

Quantity A: ⁷⁄₂ = 3.5

Quantity B: 5

Once again, answer choice (B) seems to be correct. It is at this point where even a relatively well-trained test taker who knows to plug in twice but fails to “see” the right numbers to choose, may still end up getting this question wrong.

Why is that? Well, simply put, there are a plethora of different number property criteria, other than using a bigger (or a smaller) number, any one of which could potentially provide us with the opposite result we are looking for.

Furthermore, given that all numbers on the GRE are assumed to be real (unless otherwise noted), a common trap many students fall into is failing to consider other unusual (but still possible) values that x could be and not plugging those numbers in accordingly (i.e. positive and negative numbers, integers and fractions, etc.).

Keeping this in mind, let’s remember to consider a negative number and use x = -1.

Using this value for x, we get the following:

Quantity A: ^{(-1)² – 2} ⁄_{-1 – 1}

Quantity B: -1 + 2

This simplifies to:

Quantity A: ^-1⁄_-2

Quantity B: 1

Unfortunately, once again, answer choice (B) still seems to be correct.

Let’s also not forget to consider a fractional or decimal value here and try x = ½.

Using this value for x, we get the following:

Quantity A: ^{1/2² – 2}⁄_{-1/2 – 1}

Quantity B: ^-1⁄₂ + 2

This simplifies to:

Quantity A: ^-1.75⁄_-1.5

Quantity B: 1.5

Which further simplifies to:

Quantity A: 1.16

Quantity B: 1.5

Even here, answer choice (B) still seems correct.

So, does this mean (B) is right? Well, maybe. But maybe not.

This is where high-level GRE quantitative comparison questions can become incredibly tricky (and time-consuming). One of the primary ways ETS makes these questions more challenging is by structuring the parameters of the question in such a way that there are very few ways to truly get the opposite result. In fact, on the highest-level of such questions, there is sometimes only one number (or one way) to reach the correct final answer.

How then do we know if we’ve missed out on locating that key unique number?

Furthermore, how do we know when this number doesn’t exist and recognize that it’s better to just stop plugging in now and move on?

At Menlo Coaching, we train our GRE tutoring students to follow a variety of key best practice principles that will allow them to spot subtle clues hiding in the questions that, once identified, will “unlock” the question and allow them to get to the right answer quickly and consistently.

Tip: Great test takers go one step further beyond simply considering real numbers and always use the context of the question to help identify what key or unique numbers should be considered.

How would that concept be applied to this question though?

Did you notice there was an initial condition listed in the question stem that x ≠ 1?

Why would they have said that?

Perhaps this is the missing clue that will be the key to “unlocking” this question.

One founding principle we teach our students is to recognize that there are two key numbers on the real number line that have very unusual number properties (0 and 1). And this question has, for some unknown reason, told us that we cannot use the number 1. Interestingly enough though, it never said anything about not using 0. As it turns out, this number will ultimately be the key to unlocking this question.

Let’s try x = 0 right now…

Using this as our final value for x, we get the following:

Quantity A: ^{(0)² – 2}⁄_{0 – 1}

Quantity B: 0 + 2

This simplifies to:

Quantity A: ^-2⁄_-1 = 2

Quantity B: 2

As you can clearly see, in this case, the two quantities are the same, which is answer choice (C). However, since the previous numbers gave us (B) and this number gives us (C), we don’t actually know which quantity is bigger. Therefore, the right answer is (D).

GRE quantitative comparison questions that provide seemingly innocuous conditions (i.e. x≠ 1) in the question stem may in fact be hiding the key to solving the problem in plain sight.

To make sure you don’t miss out on testing counterintuitive numbers like 0 and 1, you should always aim to use the context of the GRE quantitative comparison question to guide you in your number picking.

Ultimately, this problem shows you how spotting this key “clue” up front would not only have allowed us to solve this problem easily but also allowed us to do so in just two rounds of plugging in as opposed to four or five, thus saving us significant exam time that can now be used more productively later on.