Official GRE Quantitative Comparison Practice Questions

The correct answer is (A).

A common error on GRE quantitative comparison questions is to default to a middle-school math class mindset—going through all the calculations step by step as if you were going to have to “show your work.” The GRE doesn’t want to see how you arrived at the right answer; it only rewards you for getting there quickly.

It’s important to understand that questions on this section of the exam are designed to look more technical than they really are—tempting you to solve them in that systematic, time-intensive way when you really don’t have to.

This one, for example, is a simple function question: `p^∇` could just as well be written `f(p)`, and it would mean the same thing. The function simply tells us to add the input value, `p`, to its reciprocal, `1/p`.

The fact that the two quantities are not presented in the same way—we have a fraction in A, and a decimal in B—is also designed to make straightforward numbers and operations look unfamiliar.

To cut through this, seeing that the function you’ve been given involves fractions, you should immediately convert Quantity B into a fraction. 3.5 becomes `7/2`. Right away, you’ll see that this is the reciprocal of `2/7`, which is of course not a coincidence.

Now, Quantity A asks us to start from `2/7` and apply the function defined above, twice. But remember, as a quantitative comparison question, this is not asking us to find the exact value of Quantity A—only whether it’s bigger than Quantity B. Let’s see whether we actually need to go through the calculations to figure that out.

Plugging our fractions into the function above, we get `2/7+7/2`. And we can already stop. We don’t even need to work out the result of this operation.

We are adding a positive value to `7/2`, which is Quantity B. By definition, we can only end up with something greater than Quantity B. So the correct answer is (A).

We could continue the calculation, applying the same function again to the result of this first function, but it would be a complete waste of our precious time. Applying the same function again can only result, again, in a bigger number. We already know that Quantity A is greater than Quantity B. It doesn’t matter how much bigger.

As long as you can understand the function you’re being shown, you will pretty much inevitably arrive at the right answer to this question. But what separates a good performance from a bad one is how little work you do to get there. On the GRE, you need to spend your time wisely, saving the big time investments for the questions where they’re really needed.

The correct answer is (D).

Test takers who handle quantitative comparison questions well have two common characteristics: they are flexible, and they are intentional in the steps they take to find the answer. These questions reward those qualities because they make it tempting to pour a ton of low- or even no-value-added time into the question. Avoid doing so, and you’re ahead of the pack.

Take this example. The way people tend to waste time here is with a lot of random number-picking. They eventually get to the answer, but how quickly it happens is down to luck, because they have no clear strategy for which numbers to pick. Let’s work out a more systematic approach to this question.

First, pay attention to that `x!=1` condition. Why is it there? Well, if you try plugging in 1, you’ll realize that for Quantity A, it results in a denominator of `1-1=0`. Dividing by zero of course results in an undefined expression. Clearly, the question writer wanted to rule out the possibility of “breaking the question” in this way.

From here, the vast majority of test takers find that their first instinct is to plug in 2. It can’t be 1, so 2 is the logical next choice. That gives us:

Quantity A Quantity B `(2^2-2)/(2-1)=2` `2+2=4`

4 is obviously greater than 2. But we can’t simply conclude that Quantity B is always greater based on this example. We need to determine whether that’s always the case.

Note that we’ve narrowed down our options: Either Quantity B is always greater (B), or it is sometimes greater and sometimes not, so the relationship cannot be determined (D). In other words, we can answer (D) as soon as we can find any situation where Quantity A is greater or the two are equal.

But this is where the question can become very time-consuming if we resort to random number-picking. We might even fail to land on a number that gives a different outcome and end up throwing in the towel, assuming that (B) is probably correct.

Instead, we want to home in on numbers that are likely to be the exception to the rule. But how?

In this case, it involves looking carefully at the two expressions representing our two quantities. In both, we have at least one instance of the variable `x`, for which we don’t have a defined value. So let’s cut `x` out entirely, in both expressions. What are we left with?

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

If there’s no `x`, then Quantity A and Quantity B are equal. In what situation is there no `x`? When `x=0`, of course!

Nothing in the question ruled out the idea that `x` could be equal to 0, so this is enough to get us to the correct answer, option (D).

Now, the idea that the two quantities would be equal if we got rid of the variables might not have jumped out at you right away. That’s OK. Let’s look at another way to get there.

What makes it tricky to compare the two quantities at first glance is simply the fact that they are presented in different ways. First we have a complicated-looking expression involving a square and a fraction. Then we have a much more straightforward expression that involves only simple addition.

To make them play nicely together, we want to get both expressions into the same format by removing that distracting denominator in Quantity A. This involves a bit of basic algebra.

To get rid of the denominator `x-1`, we can of course multiply Quantity A by `x-1`, canceling it out:

`(x^2-2)/(x-1)*(x-1)=x^2-2`

Now we apply the same function to Quantity B so that they remain comparable:

`(x+2)*(x-1)`

In this case, you’ll have to dig out the FOIL method, multiplying `x` by `x`, then `x` by `-1`, then `2` by `x`, then `2` by `-1`, resulting in:

`x^2+2x-x-2`

This can be simplified further to:

`x^2+x-2`

So we have our two quantities:

Quantity A Quantity B `x^2-2` `x^2+x-2`

In this format, it is much easier to spot that the only difference between the two expressions is the middle part, adding `x`. From this we can draw the conclusion that if `x=0`, the two quantities will be equal, whereas if it is any other value, they will not be equal—and pick option (D).

Working intentionally, putting the two expressions into a format that gives us the greatest possible visibility on how they differ, allows us to reliably discern the right answer here. We’d probably have gotten there with random number-picking too—but with no guarantee of doing so in a timely fashion. And on the GRE, you can’t afford to be gambling with your time.