QUANTITATIVE COMPARISON STRATEGIES
Just as with GRE Problem Solving, handling Quantitative Comparisons requires that
you know the fundamentals of arithmetic, algebra, and geometry—no doubt about
it—and that’s what the math review in the next part of this book is about. But the test
designers craft Quantitative Comparisons to gauge not only your math knowledge, but
also yourmental agility, flexibility, andcreativity inapplying it.Quantitative Compari-
sons are designed, for example, to measure your ability to do the following:
• See the dynamic relationships between numbers.
• Recognize the easiest, quickest, or most reliable way to compare quantities.
• Visualize geometric shapes and relationships between shapes.
In this section of the chapter, you’ll learn some strategies and techniques that dem-
onstrate these skills. These aren’t merely tricks and shortcuts. Your facility in
applying these techniques, along with your knowledge of substantive rules of math, is
exactly what the test designers are attempting to measure.
Know What Concept Is Being Tested
Each Quantitative Comparison question focuses on a particular math concept. The
first thing to do is look at the centered information and the expressions in the columns
to determine what’s being covered. This step can often help you get a head start in
making the comparison.
4. 2x
2
1 9x 5 5
Column A
Column B
x 25
The correct answer is (D). Notice that the centered equation is quadratic,
and that the two quantitative expressions essentially ask you to find the value
of x. You know that quadratic equations have two roots, and this is the concept
that’s probably being tested here. The two roots might be the same or they may
differ.
Now you know what you need to do and why. First, rewrite the centered
equation in standard form: 2x
2
1 9x 2 5 5 0. Now, factor the trinomial
expression into two binomials: (2x 2 1)(x 1 5) 5 0. It should now be clear that
there are two different roots of the equation:
1
2
and 25. Since
1
2
.25, but 25=
25, the correct answer is choice (D).
There May Be an Easier Way
You shouldn’t have to perform involved calculations to make a comparison. A few
simple calculations may be required, but if you’re doing a lot of number crunching or
setting up and solving of equations, you’ve probably missed the mathematical prin-
ciple that you’re being tested for. Put your pencil down and focus on the concept, not
the process.
Chapter 8: Quantitative Comparison 183
TIP
The examples here involve a
variety of math concepts and
all are at least moderately
difficult. If you have trouble
with a concept, focus on it
during the math review later in
this part of the book.
NOTE
Not every Quantitative
Comparison question you
encounter on theGRE is
suitable for one of these
strategies, but some are—so
it’s a good idea to become
familiar with them.
www.petersons.com
5. Column A Column B
~26!
11
~26!
3
~211!
8
~211!
3
The correct answer is (A). To make the comparison, you could multiply each
base number by itself multiple times, then figure out quotients. But that would
be a horrendous waste of time, and most GRE test-takers would at least realize
that they can simplify each fraction by canceling exponents:
()
()
()
−
−
=−
6
6
6
11
3
8
and
()
()
()
−
−
=−
11
11
11
8
3
5
Now, instead of multiplying 26 by itself 8 times, and then multiplying 211 by
itself 5 times to compare the two products, look at the signs and the exponents for
a moment. Quantity A is a negative number raised to an even power, so it must
have a positive value. Quantity B is a negative number raised to an odd power, so
it must have a negative value. There’s no need for all that calculating. Quantity
A must be greater.
Choose (D) Only If You Need More Information
Remember: Pick choice (D) only if you can’t make a comparison without having more
information. But if the comparison at hand involves numbers only, you’ll always be
able to calculate specific numerical values for both expressions (assuming you have
time to do the math). You certainly don’t need more information just to compare the
size of two specific numbers.
6. x M y 5 x(x 2 y)
Column A
Column B
(21 M 22) M (1 M 2) (1 M 22) M (21 M 2)
The correct answer is (C). The centered information contains variables, but
calculating the two quantities involves only numbers. Thus, the correct answer
cannot be (D). Just by recognizing this fact, you’ve improved your odds of
selecting the correct answer choice by 25 percent!
Let’s go ahead and compare the two quantities. First, Quantity A: Apply the
operation (defined by the symbol M) to each parenthesized pair, then apply it
again to those results:
(21 M 22) 521(21 2 [22]) 521(1) 521
(1 M 2) 5 1(1 2 2) 5 1(21) 521
Apply the defined operation again, substituting 21 for both x and y:
(21 M 21) 521(21 2 [21]) 521(0) 5 0
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Quantity A equals zero (0). Now let’s calculate Quantity B in the same manner:
(1 M 22) 5 1(1 2 [22]) 5 1(3) 5 3
(21 M 2) 521(21 2 2) 521(23) 5 3
Apply the defined operation again, substituting 3 for both x and y:
(3 M 3) 5 3(3 2 3) 5 3(0) 5 0
As you can see, Quantity (B) also equals zero (0).
Consider All Possibilities for Variables
When comparing expressions involving variables, pay careful attention to any cen-
tered information that restricts their value.Aside from any such restriction, be sure to
consider positive and negative values, as well as fractions and the numbers zero (0)
and 1. Comparisons often depend on which sort of number is used. In these cases, the
correct answer would be choice (D).
7. p . 0
p Þ 1
Column A
Column B
p
23
p
4
The correct answer is (D). Here’s the general rule that applies to this
problem:
p
p
x
x
−
=
1
. Hence, if p . 1, then p
23
must be a fraction less than 1
while p
4
is greater than 1, and Quantity B is greater. On the other hand, if p ,
1, then the opposite is true. In short, which quantity is greater depends on
whether 0 is greater than 1 or is a fraction between 0 and 1.
Rewrite a Quantity to Resemble Another
If you have no idea how to analyze a particular problem, try manipulating one or both
of the expressions until they resemble each other more closely. You may be able to
combine numbers or other terms, do some factoring, or restate an expression in a
slightly different form.
8. Column A
Column B
4 2 x
2
(2 1 x)(2 2 x)
The correct answer is (C). Perhaps you recognized that (4 2 x
2
)isthe
difference of two squares (2
2
and x
2
) and that the following equation applies:
a
2
2 b
2
5 (a 1 b)(a 2 b)
If so, then you saw right away that the two quantities are the same. If you
didn’t recognize this, you could restate Quantity B by multiplying (2 2 x)by
(2 1 x) using the FOIL method:
(2 1 x)(2 2 x) 5 4 1 2x 2 2x 2 x
2
5 4 2 x
2
Rewriting Quantity B, then simplifying it, reveals the comparison. The two
quantities are equal.
Chapter 8: Quantitative Comparison 185
ALERT!
Remember: Just because a
Quantitative Comparison
question does involve one or
more variables, it doesn’t
necessarily mean that choice
(D) is the correct answer.
www.petersons.com
Add or Subtract Across Columns to Simplify the Comparison
If both expressions include the same term, you can safely “cancel” that term from each
one either by adding or subtracting it. This technique may help simplify one or both of
the expressions, thereby revealing the comparison to you. Remember: You don’t
change the comparative value of two expressions merely by adding or subtracting the
same terms from each one.
9. Column A
Column B
The sum of all integers
from 19 to 50, including 19
and 50
The sum of all integers from 21 to 51,
including 21 and 51
The correct answer is (B). The two number sequences have in common
integers 21 through 50. So you can subtract (or cancel) all of these integers
from both sides of the comparison. That leaves you to compare (19 1 20) in
Column A to 51 in Column B. You can now see clearly that Quantity B is
greater.
Be Wary of Multiplying or Dividing Across Columns
To help simplify the two expressions, you can also multiply or divide across columns,
but only if you know for sure that the quantity you’re using is positive. Multiplying or
dividing two unequal terms by a negative value reverses the inequality; the quantity
that was the greater one becomes the smaller one. So think twice before performing
either operation on both expressions.
10. x . y
Column A
Column B
x
2
yxy
2
The correct answer is (D). To simplify this comparison, you may be tempted
to divide both columns by x and by y. But that would be wrong! Although you
know that x is greater than y, you don’t know whether x and y are positive or
negative. You can’t multiply or divide by a quantity across columns unless
you’re certain that the quantity is positive. If you do this, here is what could
happen: Dividing both sides by xy would leave the comparison between x and y.
Given x . y, you’d probably select (A). But if you let x 5 1 and y 5 0, on this
assumption, both quantities 5 0 (they’re equal). Or let x 5 1 and y 521. On
this assumption, Quantity A 521 and Quantity B 5 1. Since these results
conflict with the previous ones, you’ve proven that the correct answer is
choice (D).
Solve Centered Equations for Values
If the centered information includes one or more algebraic equations and you need to
know the value of the variable(s) in those equations to make the comparison, then
there’s probably no shortcut to avoid doing the algebra.
PART IV: Quantitative Reasoning186
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11. x 1 y 5 6
x 2 2y 5 3
Column A
Column B
xy
The correct answer is (A). The centered information presents a system of
simultaneous linear equations in two variables, x and y. The quickest way to solve
for x is probably by subtracting the second equation from the first:
Substitute this value for y in either equation. Using the first one:
x
x
+=
=
16
5
Since both equations are linear, you know that each variable has one and only
one value. Since x 5 5 and y 5 1, Quantity A is greater than Quantity B.
Don’t Rely on Geometry Figure Proportions
For Quantitative Comparison questions involving geometry figures, never try to make
a comparison by visual estimation or by measuring a figure. Even if you see a note
stating that a figure is drawn to scale, you should always make your comparison
based on your knowledge of mathematics and whatever nongraphical data the
question provides, instead of by “eyeballing” the figure.
12.
Column A Column B
yz
The correct answer is (D). If you were to measure the two angles (y° and z°)
by eye, you might conclude that they are the same or that one is slightly
greater than the other, and then select choice (A), (B), or (C) accordingly. But
that would be a mistake. Your task is to compare y and z without resorting to
visual measurement. Given that any circle contains a total of 360°, you know
Chapter 8: Quantitative Comparison 187
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that y+z=360 2 220 = 140. Also, since x, y, and z are supplementary (their
sum is 180 because the three angles combine to form a straight line), you know
that x = 40. But the problem does not supply enough information for you to
determine the degree measure of either y or z. (All you know is that their sum
is 140.)
PART IV: Quantitative Reasoning188
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SUMMING IT UP
• The 45-minute Quantitative Reasoning section tests your proficiency in per-
forming arithmetical operations and solving algebraic equations and inequalities.
It also tests your ability to convert verbal information into mathematical terms;
visualize geometric shapes and numerical relationships; interpret data presented
in charts, graphs, tables, and other graphical displays; and devise intuitive and
unconventional solutions to conventional mathematical problems.
• Quantitative Comparison is one of two basic formats for questions in the Quan-
titative Reasoning section of theGRE (the other is Problem Solving).
• All Quantitative Comparison questions require you to compare two quantities,
one in Column A and one in Column B. You choose whether the quantity in
Column A is greater, the quantity in Column B is greater, the quantities are
equal, or the relationship cannot be determined from the information given.
These four are the only answer choices, and they’re the same for all Quantitative
Comparison questions. Information concerning one or both of the quantities to be
compared is centered above the two columns.
• You’ll make fewer calculations and solve fewer equations for Quantitative Com-
parison questions than for Problem Solving questions. You’re being tested mainly
on your ability to recognize and understand principles, not your ability to work
step-by-step toward a solution.
• Follow and review the six basic steps for handling GRE Quantitative Comparison
questions outlined in this chapter and apply them to this book’s Practice Tests.
Then review them again just before exam day.
Chapter 8: Quantitative Comparison 189
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Math Review: Number
Forms, Relationships,
and Sets
OVERVIEW
• Percents, fractions, and decimals
• Simplifying and combining fractions
• Decimal place values and operations
• Simple percent problems
• Percent increase and decrease
• Ratios and proportion
• Arithmetic mean, median, mode, and range
• Standard deviation
• Arithmetic sequences
• Geometric sequences
• Permutations
• Combinations
• Probability
• Summing it up
In this chapter, you’ll focus first on various forms of numbers and relation-
ships between numbers. Specifically, you’ll review the following basic tasks
involving percents, decimal numbers, and fractions (including ratios):
• combining fractions using the four basic operations
• combining decimal numbers by multiplication and division
• comparing numbers in percentage terms
• comparing percent changes with number changes
• rewriting percents, fractions, and decimal numbers from one form
to another
• determining ratios between quantities and determining quantities
from ratios
• setting up equivalent ratios (proportions)
• handling fractions and ratios
• handling ratios involving more than two quantities
• solving proportion problems with variables
chapter 9
191
Later in this chapter, you’ll explore the following topics, all of which involve
descriptive statistics and sets (defined groups) of numbers or other objects:
• Arithmetic mean—of simple average—and median (two ways that a set of
numbers can be described as a whole)
• Standard deviation (a quantitative expression of the dispersion of a set of mea-
surements)
• Arithmetic sequences (where there is a constant difference between successive
numbers)
• Geometric sequences (where each term is a constant multiple of the preceeding
one)
• Permutations (the possibilities for arranging a set of objects)
• Combinations (the possibilities for selecting groups of objects from a set)
• Probability (the statistical chances of a certain event, permutation, or combi-
nation occurring)
PERCENTS, FRACTIONS, AND DECIMALS
Any real number can be expressed as a fraction, a percent, or a decimal number. For
instance,
2
10
, 20%, and 0.2 are all different forms of the same quantity or value. GRE
math questions often require you to rewrite one form as another as part of solving the
problem at hand. You should know how to write any equivalent quickly and confi-
dently.
To rewrite a percent as a decimal, move the decimal point two places to the left (and
drop the percent sign). To rewrite a decimal as a percent, move the decimal point two
places to the right (and add the percent sign).
95% 5 0.95
0.004 5 0.4%
To rewrite a percent as a fraction, divide by 100 (and drop the percent sign). To
rewrite a fraction as a percent, multiply by 100 (and add the percent sign). Percents
greater than 100 are equivalent to numbers greater than 1.
810% 5
810
100
5
81
10
5 8
1
10
3
8
5
300
8
% 5
75
2
% 5 37
1
2
%
Percents greater than 100 or less than 1 (such as 457% and .067%) can be confusing,
because it’s a bit harder to grasp their magnitude. Here’s an illustrative example in
the Quantitative Comparison format. (If you need to review this format, see
Chapter 8.)
PART IV: Quantitative Reasoning192
ALERT!
Although the initial sections of
this chapter cover the most
basic of all the math review
topics in this book, don’t skip
them. The skills covered here
are basic building blocks for
other, more difficult types of
questions covered in later
sections of this math review.
www.petersons.com
. to each parenthesized pair, then apply it
again to those results:
(21 M 22) 521( 21 2 [22]) 521( 1) 521
(1 M 2) 5 1(1 2 2) 5 1 (21) 521
Apply the defined operation. 521
Apply the defined operation again, substituting 21 for both x and y:
(21 M 21) 521( 21 2 [21] ) 521( 0) 5 0
PART IV: Quantitative Reasoning184
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Quantity