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 4.1 QC X-ray 4.2 QC Fundamentals

 4.3 QC Step Method 4.4 Practice Problems
QC Fundamentals
Here’s a rundown of the QC fundamentals, the main things you need to know about this question type before you solidify your approach with the step method presented later in the chapter. Just like in Problem Solving, we’ll present you with multiple approaches, which you’ll choose from, depending on the specifics of each question you face. Here’s what we’ll cover:
• Disguised Problem Solving
• Math Logic
• Shortcuts for Number Problems
• Shortcuts for Variable Problems
• Working with Diagrams
Disguised Problem Solving
To drive home the point that comparison is the raison d’être of this question type, some books and courses suggest that you’ll never need to perform calculations on QCs. We believe that’s taking things too far. The truth is, sometimes calculating the values to compare them is the fastest, easiest, and surest way to go, especially on questions in the beginning of the section where the CAT program is simply getting a feel for whether you know how the QC question format works. Despite the relatively odd format, some QC questions are little more than Problem Solving questions in disguise and benefit from the kinds of standard applications of math concepts we discussed in the previous chapter. The only difference is that once you get your answer, you won’t search for it among the choices but will instead compare your answer to the quantity or expression presented in the other column. Let’s turn the page and look at a few examples.
Single-Column Calculations
A very common QC format consists of a typical Problem Solving type of question in Column A and a number to compare your answer to in Column B. Here’s an example (the same one you saw earlier in the X-ray):

A certain bread recipe calls for water, flour, and yeast to be mixed in a ratio of 4:5:2 ounces, respectively.

 Column A Column B The amount of flour needed to make 220 ounces of bread according to the recipe 100 ounces
(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.
There’s simply nothing to do here but use one of the Math 101 concepts, in this case, the mixture ratio formula: Set x equal to the factor by which each element is multiplied in the final mixture. In this case, that produces the equation 4x + 5x + 2x = 220. Adding the x terms gives 11x = 220, and dividing both sides by 11 gives x = 20. Flour corresponds to the 5 term in the ratio, so there are (5)(20) = 100 ounces of flour in the bread. Simply comparing that with the number in Column B tells us that the columns are equal, and C is correct.
Double-Column Calculations
Other QCs may require you to use your Problem Solving skills to figure out the values in both columns. Here’s a fairly simple example that you might see near the beginning of the Math section:

x + 3y = 15

y + 12 = 16

 Column A Column B x y
(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.
Now, we could knock ourselves out looking for clever, elegant shortcuts to solve this, but the truth is, it’s just not necessary. (Later, you’ll see examples for which such shortcuts are appropriate.) The math is far from difficult, so as the sneaker commercial implores, we should just do it. Subtracting 12 from both sides of the second equation allows us to determine that y = 4. Substituting 4 for y in the first equation gives us x + (3)(4) = 15, which simplifies to x + 12 = 15 and x = 3. Four is bigger than 3, so Column B is bigger than Column A, meaning choice B is correct.
This may seem fairly simple, but at least one QC question will most likely contain relatively basic math while really testing whether you understand the question format. Here’s another example of a QC that’s nothing more than a Problem Solving challenge without the standard choices:

20 < n < 22

 Column A Column B
(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.
Column B contains a straightforward fraction addition, and we can use the Magic X to determine that the sum equals 6 × 1, or 6, + 2 × 1, or 2, divided by the product of the denominators, 12. That comes out to , or . (If you’re still not up on the Magic X, review it in Math 101. If you prefer adding simple fractions the standard way, that’s fine too.) As for Column A, 21 is as good as any value to substitute for n, since n is limited to values between 20 and 22. It works out nicely too: simplifies to , so the quantities are equal, right?
Wrong. The quantities could be equal, but there are more numbers between 20 and 22 than just 21. Remember, fractions and decimals exist between any two integers. If the question doesn’t specify integers only, then fractions and decimals are in play as well. If we set n equal to 20.5, then the value of Column A would be larger than . Similarly, if we make n equal to 21.5, then Column A would be smaller than . So the correct answer is D—there’s not enough information to make a surefire determination. Note that if the test makers included the additional requirement that “n is an integer,” then the quantities would be equal and the answer would be C, since 21 is the only integer between 20 and 22.
The bottom line is that some QCs merely require you to apply your Math 101 concepts in standard ways to come up with values to use in your comparisons. When you have to “power through,” Problem Solving–style, do it. Some questions will naturally be more difficult than others, but the way you go about it should be familiar enough.
Many other QCs, however, call for a different approach. There are, in fact, numerous ways to cut QCs down to size so that you don’t have to power through full-fledged calculations. The first involves using a bit of cleverness to save you time and effort.
Math Logic
The GRE test makers point out that the Math section tests more than just your knowledge of math concepts; it also tests something they call quantitative reasoning. There are enough big words floating around in GRE lingo, so we’ll just call this math logic. It refers to the ability to reason logically using math concepts, rather than simply crunching the numbers. Let’s look at a couple of forms of math logic at work.
Basic Deductive Reasoning
Some questions hinge on a basic deduction. It’s not that such deductions are necessarily difficult or obscure—many test takers simply don’t know they’re supposed to look for them. That’s part of that old-school mindset that we discussed in chapter 1, the mindset that you need to change if you want to succeed on this section. While it’s true that to a certain extent deductive reasoning, or cleverness, can’t be taught, we can teach you to recognize situations in which these things may be required. Our practice questions will also help you hone the math logic skills you already possess. Consider, for example, the following QC:

j = 3.25

k = 4.25

 Column A Column B The average (arithmetic mean) of j, k – 1, j + 1, and 7 The average (arithmetic mean) of 7, k, k, and 3.25
(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.
A few bits of math logic can greatly simplify your task here. First, the numbers given for j and k are unwieldy, so the clever test taker will realize right off the bat that she’s probably not expected to just plug them into the arithmetic mean formula to solve for the quantities in both columns. Sure, you could try that, but it might be a long slog, fraught with the risk of careless mistakes. Having realized that there’s probably a shortcut, the clever test taker might then realize that 3.25 is exactly one less than 4.25; in other words, that j = k – 1. This is a particularly helpful deduction since k – 1 happens to be a term in Column A. Hey, that also means that k = j + 1, another term that shows up in Column A. 3.25 in Column B is merely the same as j, so recognizing that might help our cause too.
If you think the cleverness ends there, think again: The average formula states that average is equal to the sum of the terms divided by the number of the terms. It follows that adding the same number to two averages containing the same number of terms would be a wash, which means we can ignore the 7 altogether. If we were looking for the actual averages here, we’d have to use the 7, but since we’re only interested in comparing the two averages, we can ditch it. All of these realizations fall into the category of math logic; using reason in the context of Math 101 concepts to simplify the QC.
Now that we’ve taken it this far, we may as well solve it. Through the deductions above, we’ve essentially translated the columns into this:

j = 3.25

k = 4.25

j = k – 1

k = j + 1

 Column A Column B The average (arithmetic mean) of j, j, and k The average (arithmetic mean) of k, k, and j
We went ahead and changed the 3.25 in Column B into j to resemble the j’s in Column A—another example of “mirroring,” a concept we’ve already seen and will discuss more later. Since k is larger than j, the average of two ks and a j must be larger than the average of two js and a k, so B is correct. Notice again how math reasoning entered the picture based on an understanding of how averages work, even here in the final step.
Overall, a deduction, a couple of clever substitutions, and the strategic deletion of the 7s allowed us to compare the quantities without actually doing any math. Try to use reason in place of brute force whenever the quantities in the question seem unmanageable. But if the quantities look easy enough, go ahead and power through.
Missing Info
Another way to use math logic is to home in on the essentials of the concept under consideration to help you recognize right off the bat that you don’t have enough information to answer the question. It may be possible to select choice D without crunching any numbers if you notice that a piece is missing from a required formula. For example, see if you can spot what’s missing here:

Train x traveled 175 miles to Daling Farm at an average rate of 70 miles per hour. Train y traveled to Daling Farm at an average rate of 85 miles per hour.

 Column A Column B The time it took train x to travel to Daling Farm The time it took train y to travel to Daling Farm
(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.
The handy distance formula should pop into your head immediately: rate × time = distance. Now, we have two of those variables for train x but only one variable for train y. So while we can figure out the time it took train x to get to Daling Farm, we’re missing a key piece of information regarding train y’s journey: namely, how far it traveled. The fact that y traveled faster than x might lead some to conclude that y got there sooner. However, y may have come from much farther away, so D is correct.
The moral is this: As soon as you call up a formula to use in a QC question, first check to see if you have all the relevant facts you need to solve it. If not, don’t waste another second—choose D and move on.
Shortcuts for Number Problems
QCs come in two main varieties: problems that contain variables and problems that don’t. In this section, we provide you with strategies to simplify and shorten your work on QCs containing only numbers. Some of these strategies may overlap with math logic, but we think they’re important enough to present in their own category.
Chopping Choice D
We ended the previous section with a discussion of one good reason to select choice D: not enough information. Now we’re going to tell you a reason not to choose it: when the quantities in the two columns contain no variables.
Why can’t D be the answer if the columns contain only numbers? Because values are values. It doesn’t matter if you can’t figure out the size of one or both; the fact remains that one must be bigger than the other, or they must be equal. There are no other alternatives. If the columns contain only numbers, never choose D. Even if you find yourself totally baffled, if there are no variables involved, guess among A, B, and C. If nothing else, you’ll increase your odds to one out of three.
Approximating
We discussed the value of approximating in chapter 1’s General Math Strategies. Approximating comes in particularly handy in some QCs since, as you’ve already seen, you don’t always need to calculate precise answers to effectively compare the quantities. As a quick refresher, here’s the approximation example we presented in that earlier chapter, already in QC form:
 Column A Column B 48% of 54 11% of 273
(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.
We demonstrated how one could quickly eyeball these quantities to approximate their values: Less than half of 54 is less than 27, and more than 10% of 273 is more than 27. A quick dose of solid math logic helps us to choose B without having to perform any tedious calculations. You’ll get lots of practice with approximating when we get to Data Interpretation in the following chapter.
Positive vs. Negative Numbers
The test makers get a lot of mileage out of positive and negative numbers. Sometimes they’ll test your understanding in a straightforward Problem Solving question. You know the kind: “If such and such is true, which of the following must be positive,” or something along those lines. But a solid understanding of what makes numbers positive or negative can really come in handy in QCs. You won’t need to perform difficult or time-consuming calculations if you can instead compare the two quantities on the basis of where they fall in relation to the zero point. As always, the concept makes the most sense in the context of an example, so let’s do one. See how this QC strikes you:

p < 0

 Column A Column B
(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.
Many test takers would assume that because we don’t know the exact value of p here, the answer must be “not enough information,” choice D. Those test takers would be wrong. The additional information tells us that p is less than 0, which means p is a negative number (which is why we include this one in this section on number problems). The numerator and denominator of the fraction in Column A must both be negative since the product of a negative and a positive number is negative. Since the numerator and denominator in Column A are both negative, the fraction itself must be positive. The fraction in Column B, however, is negative, since two negatives in the numerator multiply out to a positive, which is then divided by a negative denominator. So without even knowing the value of p, we can use our knowledge of number properties to determine that Column A is positive and Column B is negative, which means choice A is correct.
Also keep in mind that anything multiplied by 0 is 0, and that 0 divided by any number is also 0. If 0 turns up as one of the quantities, determining whether the quantity in the other column is positive or negative (or also 0) will give you the answer without having to figure out its exact value.
Greater than 1 vs. Less than 1
They say 1 is the loneliest number. We don’t know about that, but we do know that for the sake of GRE QCs, 1 is one of the most helpful numbers around. Some QCs that appear to contain complex calculations can be solved by noting whether each quantity is greater or less than 1. Consider the following:
 Column A Column B
(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.
Who wants to add the unwieldy numbers in Column A, or face the nightmare in Column B? Not you, that’s who. Luckily there’s a quick way around this. Did you notice anything interesting about the fractions in Column A? would be exactly equal to , so must be more than . Same for the second fraction: is equal to , so must be more than . We now have two numbers, each greater than , added together, which means Column A is more than 1.
The quantity in Column B, on the other hand, must be less than 1. That’s because a fraction between 0 and 1 raised to a power results in a smaller fraction between 0 and 1, and multiplying two fractions between 0 and 1 results in another fraction between 0 and 1. Without having to crunch through the difficult math, we can still determine that Column A must be greater than Column B by checking each quantity’s relationship to the number 1.
Mirroring
Mirroring is a technique whereby you make the values given in one column look like the values in the other column. Notice we’re not saying you should solve for either quantity—just get them into the same basic form so you’ll be able to figure out which, if either, is bigger. We’ve seen this technique in action twice already: in the introduction to the chapter, where we expressed 1446 as 1212 to compare it to 1112 more easily, and in the averages questions, where we used some cleverness to delete an identical number and express both columns in simple terms of j and k. Here we’ll tackle a mirroring example with numbers; later we’ll see more examples of mirroring in problems containing variables. (You may note that the question below does contain a variable, but it’s defined numerically in the additional information, so we’ll consider this a numbers problem.) Try to work some mirroring magic on the following:

n > 2

 Column A Column B
(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.
The key is to recognize the repeated terms and cancel them out. Since the terms appear in both quantities, we can disregard them since they add the same amount to both and are thus a wash. That reduces the problem to a comparison between and, with the condition that n is greater than 2. As n gets larger and larger, the fraction gets smaller and smaller. is less than, and even substituting fractions between 2 and 3 for n doesn’t change the relationship; Column B will still be less than if n is greater than 2, so A is correct.
Notice that if instead of the condition n > 2 the test makers specified n > 1, then D would be correct, since then the quantities could be equal (if n is exactly 2) or Column B could even be greater (if n is between 1 and 2). So you have to pay very careful attention to all aspects of QC questions, since the slightest change in the quantities, or the additional information provided, could make a big difference.
This is a fairly simple example of mirroring. You saw some more difficult examples earlier, and you’ll see a particularly tough mirroring example involving exponents in the practice set at the end of the chapter. But right now, more mirroring is on hand as we move to our discussion of how to handle QCs containing variables.
Shortcuts for Variable Problems
The strategies discussed in the previous section should give you a great sense of the options you have when dealing with QCs made up only of numbers. However, many QCs contain variables as well, so let’s take a look at a couple of powerful approaches to handling these, beginning with a continuation of the mirroring technique.
Mirroring
The same mirroring technique that applies to pure numbers applies to quantities containing variables: If you can manipulate the expressions in Columns A and B to resemble each other, you may not need to actually solve anything to get the answer. One way to mirror quantities with variables is to reduce or multiply them out, and again we saw an example of this back in chapter 1. We’ll repeat it here for your convenience to remind you of the technique, but this time in QC form:

x = 33.87

 Column A Column B 32
(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.
While we might consider this one basically a Problem Solving question in disguise, the mess in Column A needs some serious work since we really don’t want to plug 33.87 into it. A root canal may be more fun. But we can get around that by mirroring the numerator and denominator, in this case by multiplying terms in the numerator, or by factoring the denominator. As we saw back in chapter 1, the denominator factors into (3x + 15)(x + 1), and we can cancel these terms out of both the numerator and denominator, leaving the whole expression as (x – 2). If we then simply substitute 33.87 for x, the value in Column A becomes 31.87, which is less than 32. So Column B is bigger, and choice B is correct.
See how you do with this one:

18n = 90k

12m = 72k

k is positive

 Column A Column B n m
(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.
Note first that your task is to compare n and m, and we can mirror them by manipulating the equations in the additional information to express both of these variables in terms of k. Dividing both sides of the first equation by 18 yields n = 5k, and dividing both sides of the second equation by 12 gives us m = 6k. Now we have quantities that are much easier to compare since they look a lot alike:
n = 5k
m = 6k
Given that k is positive, m must be bigger than n since 6 times a positive number is bigger than 5 times that same positive number. B is therefore correct.
Note, however, that if they said that k was negative, then the answer would switch to A, since 6 times a negative number is more negative (i.e., smaller) than 5 times the same negative number. And we hope you anticipated the final possibility: They might leave out the additional information entirely, in which case we wouldn’t know which was bigger, since it changes depending on whether k is positive or negative. For that matter, the quantities could even be equal if we make k zero. With no information on k, we couldn’t tell for sure which is bigger, and D would be correct.
The mirroring technique puts us in the ballpark, but we still have to be mighty careful when considering just what values our QC variables might take. If only there was a strategy to help us test out various numbers in such a situation . . .
Wait—there is! And we call it . . .
The FONZ
Sure, the rest of the world knows the Fonz as the leather-jacketed epitome of cool from the 1970s TV show Happy Days. Let them have their sitcom silliness—to you, it’s an acronym for values to test when substituting numbers into QC questions. In fact, it’s a specialized version of the “making up numbers” technique you learned about in the previous chapter. Here’s what it stands for:
F = fractions O = one N = negative numbers Z = zero
When the quantities provided include variables, you can plug in these values and see what you get. These are particularly good values to try, since they each have special properties that often reveal if more than one relationship between the quantities is possible. As you know, if the relationship between the quantities changes as you try different values, then D is correct. If not, the correct answer will be A, B, or C, depending on the exact relationship that has emerged. Let’s consider a simple example to see how this works:
 Column A Column B x 2 x 3
(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.
Your first reaction might be that the quantity in Column B is greater; a totally understandable reaction, since x3 is greater than x2 for most whole numbers: 43 is greater than 42, 103 is greater than 102, and so on. So it’s looking like B may be correct.
The FONZ, however, begs to differ. The number 1, the O in FONZ, is the simplest test. If we substitute 1 for x, the quantities are equal, and we can stop right there: The answer must be D since we’ve now seen that the quantity in Column B might be bigger, but the two quantities might also be the same. Perhaps you tried a fraction instead—the F in FONZ. If x = , then x2 = , and x3 = . Under these circumstances, Column A is bigger. Whenever the relationship changes depending on the value of the variables, the fourth oval, what we call choice D, is correct.
Let’s try one more before we move on:
 Column A Column B
(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.
The easiest FONZ number types to try are usually 0 and 1, so it makes sense to try these first. If z is 0, then Column A = 0 and Column B = 0 + 2, or 2. If z is 1, then Column A = –5 and Column B = 2.5. Column B contains the larger quantity in both cases, but let’s not jump the gun; we’ve only done a half FONZ maneuver so far. (If that sounds like a wrestling move, that’s not inappropriate considering how we’re wrestling with GRE math.) Let’s try a negative number next, and an easy one at that: say, –1. If z = –1, then Column A = 5 (remember that multiplying two negatives yields a positive), and Column B =. Ah, so the tables have turned, and now A is larger than B. No need to even try a fraction, as we know by now that the answer must be D.
As mentioned earlier, the FONZ technique is part of the general “making up numbers” strategy. If there are no restrictions on the variables provided by the additional information above the columns, then by all means FONZ away, beginning with 0 and 1 and moving on to fractions and negatives if necessary. However, if there are restrictions provided, such as “x < 0” or “x is a positive even integer,” then be sure to test out only permitted values when attempting to pinpoint the relationship between the columns.
You’ll get more FONZing practice in the practice set at the end of the chapter. Before we finish up this section on QC fundamentals and move on to the step method, let’s first consider one important point about QCs containing diagrams.
Working with Diagrams
When you see a diagram in a Problem Solving question, you can generally assume that it’s drawn to scale. However, the opposite is true for QCs: If the test makers intend the diagram to be to scale, they’ll tell you. If they don’t say a word about the picture, then there may be a trap afoot, since the actual values of angles or areas or other aspects of the figure may be different from how they appear. For example, an angle that looks like a right angle may in fact have a value of 60° when you work out the math. Answering QCs based on how the pictures attached to them look can be risky business. Take the following, for example:

The perimeter of rectangle ABCD is 12.

 Column A Column B The area of rectangle ABCD 7
(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.
The figure is described as a rectangle but actually looks like a square. That’s certainly permissible, as a square is a special form of rectangle with all its sides equal. But just because ABCD looks like a square doesn’t mean it actually is, since there’s no indication that the figure is drawn to scale. No doubt, the test makers would be checking to see whether you simply assumed the figure is a square just because it looks like the sides are equal. If you did, then you’d simply divide the perimeter, 12, by 4 to come up with four equal sides of length 3, since the length of each side of a square is the perimeter divided by 4:
In this case, the area would be 3 × 3 = 9, and you’d choose A because 9 is larger than 7. However, with no indication that the sides must in fact be equal, there’s no reason that our rectangle couldn’t look like this:
This rectangle also has a perimeter of 12, so we haven’t broken any rules. But note that its area has dwindled to 5 × 1 = 5, less than the 7 in Column B. If we go simply by the original picture provided, Column A seems bigger. If we remember that the picture is not necessarily drawn to scale, we find that another relationship is possible. D is therefore the choice to select.
 Jump to a New ChapterIntroductionMeet GRE MathMath 101Problem SolvingQuantitative ComparisonsData InterpretationMeet GRE VerbalSentence CompletionsReading ComprehensionAntonymsAnalogiesMeet GRE EssaysThe Issue EssayThe Argument EssayPracticing with Practice TestsThe Future of the GRETop 15 GRE Test Day TipsFinal Thought
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