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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):
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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:
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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:
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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?
, 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.
. 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:
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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
|
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:
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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:
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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:
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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:
|
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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.
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:
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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.
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:
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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:
|
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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:
|
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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.
, 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:
|
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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.
. 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:
|
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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.
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