GRE Book Excerpt Exclusive! Quantitative Comparisons: Basics And Game Plan

 

Whether you’ve spent a lot of time with Quantitative Comparison (QC) problems, or whether you’re brand new to the problem format, it’s worth reviewing the basics of QC and developing a game plan for this problem type.

For QC problems, you need to compare Quantity A with Quantity B to decide whether:

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

The first three answer choices have an implied always before the words “greater” and “equal.” Think of the answer choices this way:

1. Quantity A is always greater.
2. Quantity B is always greater.
3. The two quantities are always equal.
4. The relationship cannot be determined from the information given, or no consistent relationship exists. For instance, it might be that most of the time, Quantity A is greater, but in just one case, either Quantity B is greater or the two quantities are equal.

For example:

First, you’re given some information up front: x is an integer greater than 0. You’ll need to restrict your analysis to cases where x = 1, 2, 3, etc. That’s still an infinite number of possibilities. But whenever you are given information up front, unpack that information. Take the time to understand it so that you can apply it correctly. 

Next, try a number that fits the given constraint to see what happens. For instance, if x is 4, then Quantity A is 4(6) = 24.

What does that mean for your answer choices? Think it through before you keep reading.

If Quantity A is 24 and Quantity B is 25, then in this one case, Quantity B is greater. Therefore, choices (A) and (C) are no longer possible, since Quantity A cannot always be greater, and the two quantities cannot always be equal.

But that doesn’t mean the answer is (B)! It’s true that (B) is greater in at least one case, but there’s not enough evidence yet to conclude that (B) is always greater. You’re down to either (B) or (D). What next?

The current hypothesis (based on one lonely data point) is that (B) is always greater. To break that hypothesis, you’d need to find a contradictory case—a value for which Quantity A is greater or for which the two quantities are equal. Can you think of such a value?

Try another case or two. Keep track of your work systematically so that you can spot any patterns.

Quantity B is greater in all three cases—so it must be greater all the time. Right?

Not so fast. Examine the pattern. When going from x = 1 to x = 4, the value of Quantity A increases. But when you increase x again, this time to 7, the value of Quantity A decreases.

What’s going on? Somewhere between x = 1 and x = 7, the value of Quantity A maxes out. But have you found that max yet? Since both the 4 and 7 cases result in values that are close but a bit too low to beat out the value of 25 for Quantity B, try another value for x between 4 and 7 to see whether you can break the “B is always greater” hypothesis.

When x = 5, Quantity A is 5(5) = 25. In that case, and only in that case, the values in the two quantities are equal. Therefore, Quantity B is not always greater.

Although Quantity B greater for literally an infinite number of values of x, just one counterexample is enough to make (B) the wrong answer. Quantity B is not always greater.

So the correct answer is (D) because sometimes Quantity B is greater and sometimes the two quantities are equal.

D is different from ABC

Answer choices (A), (B), and (C) all have an implied always. (D) is different. It’s the sometimes choice: Sometimes the comparison goes one way, sometimes it goes another way. You can even think of (D) as “not always.”

Answer (D) is different from the other answer choices. We’ll come back to this idea shortly.

The need for a game plan

QC can be more complicated than it might seem at first glance. The answer choices are fixed, but they don’t all work the same way. Answer (D) is different, so you’ll need to treat it differently somehow. And there are a lot of other potential complications that you haven’t even seen yet.

Before you start layering in all of those other complications, you need a QC Game Plan. 

That is, you need a solid, straightforward, universal approach that will work for any Quantitative Comparison problem.

That Game Plan is what we’ll develop in this chapter. We’ll start with the simplest version in Round 1. In Rounds 2, 3, and 4, we’ll add more nuances and techniques that will help you to tackle harder problems—while still using your straightforward Game Plan framework, every single time.

Game Plan, Round 1

Take a look at this example again:

How do you approach this, or any other, QC problem?

At its core, the QC Game Plan has just two steps:

1. Simplify
2. Pick Numbers

Every tactic and technique for QC will fit under one of these two steps.

Simplify

First, simplify what you’re given. 

Simplify means “make simpler,” of course: reduce the complexity, combine pieces or break them apart, and so on. 

It also means “make sense of.” Simplifying a piece of information means making sense of it: making it more concrete and understandable.

As you consider how to simplify, take extra care with the upfront information (“x is an integer greater than 0″). Namely, always unpack upfront information.

One great way to do that is to translate from math-y language (“x is an integer greater than 0″) to concrete examples (“x is 1, 2, 3, etc.”). This translation helps to minimize the chances that you accidentally use the wrong kind of numbers.

In the preceding problem, this upfront information wound up not mattering too much. But if that information were “x is not an integer” or “x is less than 0,” then the answer would have been different. You wouldn’t have been able to use the case x =  5, so the correct answer would have been (B), not (D).

Also try to simplify what you’re given in the columns. Quantity B is already as simple as it can get: 25. But Quantity A is a more complex expression. You could write it in a different way. But should you distribute and rewrite x(10 − x) as 10x − x2 ?

There’s no hard and fast rule. The factored form x(10 − x) is at least as simple as the distributed form 10x − x2 . In fact, many will probably find it easier to plug numbers into the original x(10 − x) form. It’s your call—but do pause to consider what you think will be best for you. That few seconds of investment will save you time and energy as you work through the rest of the problem.

Pick Numbers

There are QC problems for which you stop at step 1. Why? Step 1 is sometimes enough to finish the problem.

As you simplify, you pull on a thread, you keep pulling…and the whole problem unravels. You have the answer, and you’re done. Be ready to stop there, enter your answer, and go on to the next problem.

But for other QC problems, you simplify as far as you can, and then you can’t simplify any further. What next?

Now you pick numbers, as shown in the example problem.

First, pick a number, any number that you think is easy, that follows the constraints given in the problem. Don’t even think too much about what number to pick; as long as it follows the given facts, you can pick it.

In this problem, if x = 4, then Quantity A is 4(6) = 24. Because Quantity B is 25, you can rule out answer choices (A) and (C). With just one test number, you can eliminate half of the answer choices. That’s always the case, in fact! You can always eliminate two of the four answer choices by picking just one test number per variable and working out what Quantities A and B are. (In this problem, there’s just one variable, so it was only necessary to pick one test number.)

The two answer choices you can eliminate on your first try are either (A) and (B), (A) and (C), or (B) and (C). You can never eliminate (D) after you test your first case. That’s because (D) is different: It’s the sometimes answer. 

In this example, the first case shows that (B) is greater. But you can’t know yet whether Quantity B is always going to be greater; you only know that it’s greater in this one case. Maybe a different test case will have a different result. Or maybe it won’t. 

At this point, don’t look for another case that also makes Quantity B greater. Instead, look for a case that gives you a different result.

In other words, try to prove (D). Be a skeptic about (A), (B), and (C), the “always” answer choices. Just one single counterexample can disprove any of those three choices and leave you with (D). So be on the hunt for those counterexamples. For this problem, that means finding a case in which Quantity A is greater or a case in which the two are equal.

In the search for numbers that will produce different results, consider numbers that have different characteristics. Make your first case relatively easy to test, as x = 4 was, but make your second number weird, if possible. 

One memory device that captures both easy and weird cases is ZONEF (“Zone-F”) = Zero One Negatives Extremes Fractions. (You could also switch the word Decimals in for Fractions to get ZONED as an alternative mnemonic.)

Here’s what ZONEF looks like in number line form:

 

The easiest numbers to test (−1, 0, and 1) are on the bottom of the number line. The top shows groupings of numbers that have different characteristics. Depending on the problem, a negative number could return quite a different result compared to a positive number.

The example QC problem says that x is an integer greater than 0, though, so that rules out zero, negatives, and fractions on this one. You’ve already tried x = 4. Maybe you should try the smallest possible value of x, the value 1? And 1 is an odd number, while 4 is even, so that’s probably a good value to try next.

It gives the same result, though—Quantity B is still greater. Try one more—something greater than 4 this time. But x = 7 also results in Quantity B being greater. 

Generally speaking, you’re not going to try more than 3 or, at most, 4 cases. Once you’ve tried 3, if you keep getting the same result, try one more thing before concluding that you have found an always answer: Look at the pattern (if any) created by the three values that you tested. (This is also why it’s important to try three values that have different characteristics.)

In this problem, there was an interesting pattern: The value of Quantity A first increased but then decreased again, providing the hint to try another value somewhere around the number 4. On a different problem, the pattern might actually prove to you that it’s impossible to find a counterexample, in which case you can conclude that this is in fact the always answer. (Sometimes, there’s no pattern or you can’t see it; in that case, go with the likelihood that you have found the always answer.)

As you’re looking for weird numbers, remember these four ranges of the number line: positive numbers, negative numbers, fractions between 0 and 1, fractions between 0 and −1. The boundary numbers −1, 0, and 1 are also worth considering separately, because they each have special properties. 

It may seem like a lot of work to test all of these ranges for every Quantitative Comparison problem that involves variables. It would be! But you won’t need to test 7+ cases.

First, many problems include constraints that will knock out some categories of numbers. Second, one single counterexample is enough to make (D) the correct answer. Third, as you learn the math and practice QC, you’re going to get better at identifying which kinds of numbers will produce different results in various math scenarios, so you will be able to test fewer options.

Let’s add some sub-steps to the two core steps:

1. Simplify
   1. Unpack upfront
   2. Might be done here!
2. Pick Numbers
   1. Try to prove (D)
   2. Easy, then weird (ZONEF)
   3. After 3 cases, look for the pattern, then make a call

{That’s it for our excerpt! We hope you learned something. If you like what you’ve read, you can find the rest of this chapter—and a lot more!—in Manhattan Prep’s GRE Math Strategies book. If you’re looking for other free lessons and practice problems, sign up for a free Starter Kit syllabus on our site.}

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Stacey Koprince is a Manhattan Prep instructor based in Montreal, Canada and Los Angeles, California. Stacey has been teaching the GMAT, EA, GRE, and LSAT  for more than 15 years and is one of the most well-known instructors in the industry. Stacey loves to teach and is absolutely fascinated by standardized tests.