• How complex is this question? (How many steps are involved in solving it? Does it
require setting up equations, or does it require merely a few quick calculations?)
• Do I have a clue, off the top of my head, how I would begin solving this problem?
Determine how much time you’re willing to spend on the problem, if any. Recognizing
a “toughie” when you see it may save you valuable time; if you don’t have a clue, take
a guess and move on.
Step 2: Size Up the Answer Choices
Before youattempt tosolve theproblem athand, examinethe answerchoices. Theycan
provide helpful clues about how to proceed in solving the problem and what sort of
solution you should be aiming for. Pay particular attention to the following:
• Form: Are the answer choices expressed as percentages, fractions, or decimals?
Ounces or pounds? Minutes or hours? If the answer choices are expressed as
equations, are all variables together on one side of the equation? As you work
through the problem, convert numbers and expressions to the same form as the
answer choices.
• Size: Are the answer choices numbers with extremely small values? Greater
numbers? Negative or positive numbers? Do the answer choices vary widely in
value? If they’re tightly clustered in value, you can probably disregard decimal
points and extraneous zeros when performing calculations, but be careful about
rounding off your figures. Wide variation in value suggests that you can easily
eliminate answer choices that don’t correspond to the general size of number
suggested by the question.
• Other distinctive properties and characteristics: Are the answer choices
integers? Do they all include a variable? Do they contain radical signs (roots) or
exponents? Is there a particular term, expression, or number that they have in
common?
Step 3: Look for a Shortcut to the Answer
Before plunging headlong into a problem, ask yourself whether there’s a quick way to
determine the correct answer. If the solution is a numerical value, perhaps only one
answer choice is in the ballpark. Or you might be able to identify the correct answer
intuitively, without resorting to equations and calculations.
Step 4: Set Up the Problem and Solve It
If your intuition fails you, grab your pencil and do whatever computations, algebra, or
other procedures are needed to solve the problem. Simple problems may require just a
few quick calculations. However, complex algebra and geometry questions may require
setting up and solving one or more equations.
Step 5: Verify Your Selection Before Moving On
After solving the problem, if your solution does not appear among the answer choices,
check your work—youobviouslymade atleastone mistake.Ifyour solutiondoes appear
amongthechoices,don’tcelebratequiteyet.Althoughthere’sagoodchanceyouranswer
iscorrect,it’spossibleyouransweriswrong and thatthetestdesignersanticipatedyour
Chapter 7: Problem Solving 153
NOTE
Remember: The computerized
GRE testing system adjusts the
difficulty level of your questions
according to previous
responses. If you respond
incorrectly to tough questions,
fewer of them will come up
later in that section.
www.petersons.com
error by including that incorrect solution as an answer choice. So check the question to
verifythatyourresponsecorresponds to what thequestioncallsforinvalue,expression,
units of measure, and so forth. If it does, and if you’re confident that your work was
careful and accurate, don’t spend any more time checking your work. Confirm your
response and move on to the next question.
Applying the 5-Step Plan
Let’s apply thesefive stepsto twoGRE-style ProblemSolving questions.Question 1is a
story problem involving changes in percent. (Story problems might account for as many
as one half of your Problem Solving questions.) This question is relatively easy—
approximately 80 percent of test takers respond correctly to questions like this one:
1. If Susan drinks 10% of the juice from a 16-ounce bottle immediately before
lunch and 20% of the remaining amount with lunch, approximately how
many ounces of juice are left to drink after lunch?
(A) 4.8
(B) 5.5
(C) 11.2
(D) 11.5
(E) 13.0
Step 1: This problem involves the concept of percent—more specifically, percent
decrease. The question is asking you to perform two computations in sequence. (You’ll
use the result of the first computation to perform the second one.) Percent questions
tend to be relatively simple. All that is involved here is a two-step computation.
Step 2: The five answer choices in this question provide two useful clues:
Notice that they range in value from 4.8 to 13.0—a broad spectrum. But what
general size should we be looking for in a correct answer to this question? Without
crunching any numbers, it’s clear that most of the juice will still remain in the
bottle, even after lunch. So you’re looking for a value much closer to 13 than to 4.
You can eliminate choices (A) and (B).
Notice that each answer choice is carried to exactly one decimal place, and that
the question asks for an approximate value. These two features are clues that you
can probably round off your calculations to the nearest “tenth” as you go.
Step 3: You already eliminated choices (A) and (B) in step 1. But if you’re on your toes,
you can eliminate all but the correct answer without resorting to precise calculations.
Look at the question from a broader perspective. If you subtract 10% from a number,
then 20% from the result, that adds up to a bit less than a 30% decrease from the
original number. 30% of 16 ounces is 4.8 ounces. So the solution must be a number
that is a bit larger than 11.2 (16 2 4.8). Answer choice (D), 11.5, is the only one that
works.
PART IV: Quantitative Reasoning154
ALERT!
Many Problem Solving
questions are designed to
reward you for recognizing
easier, more intuitive ways to
find the correct answer—so
don’t skip step 3. It’s worth
your time to look for a
shortcut.
www.petersons.com
Step 4: If your intuition fails you, work out the problem. First, determine 10% of 16,
then subtract that number from 16:
16 3 0.1=1.6
16 2 1.6 = 14.4
Susan now has 14.4 ounces of juice. Now perform the second step. Determine 20% of
14.4, then subtract that number from 14.4:
14.4 3 0.2=2.88
Round off 2.88 to the nearest tenth (2.9), then subtract:
14.4 2 2.9 = 11.5
Step 5: The decimal number 11.5 is indeed among the answer choices. Before moving
on, however, ask yourself whether your solution makes sense—in this case, whether
the size of our number (11.5) “fits” what the question asks for. If you performed step 2,
you should already realize that 11.5 is in the ballpark. If you’re confident that your
calculations were careful and accurate, confirm your response choice (D), and move on
to the next question. The correct answer is (D).
Question 2 involves the concept of arithmetic mean (simple average). This question is
moderately difficult. Approximately 60 percent of test takers respond correctly to
questions like it.
2. The average of 6 numbers is 19. When one of those numbers is taken
away, the average of the remaining 5 numbers is 21. What number was
taken away?
(A) 2
(B) 8
(C) 9
(D) 11
(E) 20
Step 1: This problem involves the concept of arithmetic mean (simple average). To
handle this question, you need to be familiar with the formula for calculating the
average of a series of numbers. Notice, however, that the question does not ask for the
average but for one of the numbers in the series. This curveball makes the question a
bit tougher than most arithmetic-mean problems.
Step 2: Scan the answer choices for clues. Notice that the middle three are clustered
close together in value. So take a closer look at the two outliers: choices (A) and (E).
Choice (A) would be the correct answer to the question: “What is the difference
between 19 and 21?” But this question is asking something entirely different, so you
can probably rule out choice (A) as a “red herring” choice. Choice (E) might also be a
red herring, since 20 is simply 19 + 21 divided by 2. If this solution strikes you as too
simple, you’ve got good instincts! The correct answer is probably choice (B), (C), or (D).
If you’re pressed for time, guess one of these, and move on to the next question.
Otherwise, go to step 3.
Chapter 7: Problem Solving 155
ALERT!
In complex questions, don’t
look for easy solutions.
Problems involving algebraic
formulas generally aren’t
solved simply by adding (or
subtracting) a few numbers.
Your instinct should tell you to
reject easy answers to these
kinds of problems.
www.petersons.com
Step 3: If you’re on your toes, you might recognize a shortcut here. You can solve this
problem quickly by simply comparing the two sums. Before the sixth number is taken
away, the sum of the numbers is 114 (6 3 19). After removing the sixth number, the
sum of the remaining numbers is 105 (5 3 21). The difference between the two sums
is 9, which must be the value of the number removed.
Step 4: If you don’t see a shortcut, you can solve the problem conventionally. The
formula for arithmetic mean (simple average) can be expressed this way:
AM =
sum of terms in the set
number of terms in the set
In the question, you started with six terms. Let a through f equal those six terms:
19
6
=
+++++abcdef
114 =+++++abcdef
fabcde=−++++
(
)
114
Letting f = the number that is removed, here’s the arithmetic mean formula, applied
to the remaining five numbers:
21
5
=
++++abcde
105 = a + b + c + d + e
Substitute 105 for (a + b + c + d + e) in the first equation:
f =1142 105
f =9
Step 5: If you have time, check to make sure you got the formula right, and check
your calculations. Also make sure you didn’t inadvertently switch the numbers 19 and
21 in your equations. (It’s remarkably easy to commit this careless error under time
pressure!) If you’re satisfied that your analysis is accurate, confirm your answer and
move on to the next question. The correct answer is (C).
Question 3 involves the concept of proportion. This question is moderately difficult.
Approximately 50 percent of test takers respond correctly to questions like it.
PART IV: Quantitative Reasoning156
ALERT!
On the GRE, committing a
careless error, such as
switching two numbers in a
problem, is by far the leading
cause of incorrect responses.
www.petersons.com
3. If p pencils cost 2q dollars, how many pencils can you buy for c cents?
[Note: 1 dollar = 100 cents]
(A)
pc
q2
(B)
pc
q200
(C)
50pc
q
(D)
2pq
c
(E) 200pcq
Step 1: The first step is to recognize that instead of performing a numerical compu-
tation, you’re task in Question 3 is to express a computational process in terms of
letters. Expressions such as these are known as literal expressions, and they can be
perplexing if you’re not ready for them. Although it probably won’t be too time-
consuming, it may be a bit confusing. You should also recognize that the key to this
question is the concept of proportion. It might be appropriate to set up an equation to
solve for c. Along the way, expect to convert dollars into cents.
Step 2: The five answer choices provide two useful clues:
Notice that each answer choice includes all three letters (p, q, and c)—therefore so
should your solution to the problem.
Notice that every answer but choice (E) is a fraction. So anticipate building a
fraction to solve the problem algebraically.
Step 3: Is there any way to answer this question besides setting up an algebraic
equation? Yes. In fact, there are two ways. One is to use easy numbers for the three
variables; for example, p =2,q =1,andc = 100. These simple numbers make the
question easy to work with:
“If 2 pencils cost 2 dollars, how many pencils can you buy for 100 cents?”
Obviously, the answer to this question is 1. So plug in the numbers into each answer
choice to see which choice provides an expression that equals 1. Only choice (B) works:
2 100
200 1
1
(
)
(
)
(
)
(
)
=
Another way to shortcut the algebra is to apply some intuition to this question. If you
strip away the pencils, p’s, q’s and c’s, in a very general sense the question is asking:
“If you can by an item for a dollar, how many can you buy for one cent?”
Since one cent (a penny) is
1
100
of a dollar, you can buy
1
100
of one item for a cent.
So you’re probably looking for a fractional answer with a large number such as 100 in
the denominator (as opposed to a number such as 2, 3, or 6). Choice (B) is the only that
appears to be in the right ballpark. And choice (B) is indeed the correct answer.
Chapter 7: Problem Solving 157
NOTE
On the GRE, expect to
encounter two or three “story”
problems involving literal
expressions (where the solution
includes not just numbers but
variables as well).
www.petersons.com
Step 4: You can also answer the question in a conventional manner, using algebra.
(This is easier said than done.) Here’s how to approach it:
• Express 2q dollars as 200q cents (1 dollar = 100 cents).
• Let x equal the number of pencils you can buy for c cents.
• Think about the problem “verbally,” then set up an equation and solve for x:
“p pencils is to 200q cents as x pencils is to c cents.”
“The ratio of p to 200q is the same as the ratio of x to c” (in other words, the two
ratios are proportionate). Therefore:
p
q
x
c200
=
pc
q
x
200
=
Step 5: Our solution,
pc
q200
, is indeed among the answer choices. If you arrived at this
solution using the conventional algebraic approach (step 4), you can verify your
solution by substituting simple numbers for the three variables (as we did in step 3).
Or if you arrived at your solution by plugging in numbers, you can check you work by
plugging in a different set of numbers, or by thinking about the problem conceptually
(as in step 3). Once you’re confident you’ve chosen the correct expression among the
five choices, confirm your choice, and then move on to the next question. The correct
answer is (B).
PROBLEM-SOLVING STRATEGIES
To handle GRE Quantitative questions (Problem Solving and Quantitative Compari-
sonsalike),you’llneed to be well-versedin thefundamental rulesof arithmetic, algebra,
and geometry:Your knowledge of these basics is, to a large extent, what’s being tested.
(That’s what the math review later in this part of the book is all about.) But when it
comes to Problem Solving questions, theGRE test designers are also interested in
gauging your mental agility, flexibility, creativity, and efficiency in problem solving.
Morespecifically,theydesignthesequestionstodiscoveryourabilitytodothefollowing:
• Manipulate numbers with a certain end result already in mind.
• See the dynamic relationships between numbers as you apply operations to them.
• Visualize geometric shapes and relationships between shapes.
• Devise unconventional solutions to conventional quantitative problems.
• Solve problems efficiently by recognizing the easiest, quickest, or most reliable
route to a solution.
This section of the chapter will help you develop these skills. The techniques you’ll
learn here are intrinsic to the GRE. Along with your knowledge of substantive rules of
math, they’re precisely what GRE Problem Solving questions are designed to
measure.
PART IV: Quantitative Reasoning158
NOTE
Don’t worry if you didn’t fully
understand the way we set up
and solved this problem. You’ll
learn more about how to
handle GRE proportion
questions in this book’s
math review.
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.
www.petersons.com
Read the Question Stem Very Carefully
Careless reading isby far theleading cause of wrong answers in GREProblem Solving,
so bedoubly sureyou answerthe precise question that’s being asked, andconsider your
responses carefully. Ask yourself, for example, whether the correct answer to the
question at hand is one of the following:
• an arithmetic mean or median
• a circumference or an area
• a sum or a difference
• a perimeter or a length of one side only
• an aggregate rate or a single rate
• a total time or average time
Also check to make sure of the following:
• you used the same numbers provided in the question.
• you didn’t inadvertently switch any numbers or other expressions.
• you didn’t use raw numbers where percentages were provided, or vice versa.
Always Check Your Work
Here are three suggestions for checking your work on a Problem Solving question:
Do a reality check. Ask yourself whether your solution makes sense for what the
question asks. (This check is especially appropriate for story problems.)
For questions in which you solve algebraic equations, plug your solution into the
equation(s) to make sure it works.
Confirm all of your calculations. It’s amazingly easy to commit errors in even the
simplest calculations, especially under GRE exam pressure.
Scan Answer Choices for Clues
For multiple-choice questions, scan the answer choices to see what all or most of them
have in common—such as radical signs, exponents, factorable expressions, or frac-
tions. Then try to formulate a solution that looks like the answer choices.
Chapter 7: Problem Solving 159
ALERT!
No calculators are provided or
allowed during the test, which
makes calculating on your
scratch paper and checking
those calculations that much
more crucial.
www.petersons.com
4. If a Þ 0or1,then
1
2
2
a
a
−
=
(A)
1
22a −
(B)
2
2a −
(C)
1
2a −
(D)
1
a
(E)
2
21a −
The correct answer is (A). Notice that all of the answer choices here are
fractions in which the denominator contains the variable a. Also, none have
fractions in either the numerator or the denominator. That’s a clue that your job
is to manipulate the expression given in the question so that the result includes
these features. First, place the denominator’s two terms over the common
denominator a. Then cancel a from the denominators of both the numerator
fraction and the denominator fraction (this is a shortcut to multiplying the
numerator fraction by the reciprocal of the denominator fraction):
Don’t Be Lured by Obvious Answer Choices
When attempting multiple-choice Problem Solving questions, expect to be tempted by
wrong-answer choices that are the result of common errors in reasoning, in calcula-
tions, and in setting up and solving equations. Never assume that your solution is
correct just because you see it among the answer choices. The following example is a
variation of the problem on page 155.
5. The average of 6 numbers is 19. When one of those numbers is taken
away, the average of the remaining 5 numbers is 21. What number was
taken away?
(A) 2
(B) 6.5
(C) 9
(D) 11.5
(E) 20
PART IV: Quantitative Reasoning160
www.petersons.com
The correct answer is (C). This question contains two seemingly correct
answer choices that are actually wrong. Choice (A) would be the correct answer to
the question: “What is the difference between 19 and 21?” But this question asks
something entirely different. Choice (E) is the other too-obvious choice. 20 is
simply 19 1 21 divided by 2. If this solution strikes you as too simple, you have
good instincts. You can solve this problem quickly by simply comparing the two
sums. Before the sixth number is taken away, the sum of the numbers is 114 (6 3
19). After taking away the sixth number, the sum of the remaining numbers is
105 (5 3 21). The difference between the two sums is 9, which must be the value
of the number taken away.
Size Up the Question to Narrow Your Choices
If a multiple-choice question asks for a number value, you can probably narrow down
the answer choices by estimating the size and type of number you’re looking for. Use
your common sense and real-world experience to formulate a “ballpark” estimate for
word problems.
6. A container holds 10 liters of a solution that is 20% acid. If 6 liters of pure
acid are added to the container, what percent of the resulting mixture is
acid?
(A) 8
(B) 20
(C)
33
1
3
(D) 40
(E) 50
The correct answer is (E). Common sense should tell you that when you add
more acid to the solution, the percent of the solution that is acid will increase. So,
you’re looking for an answer that’s greater than 20—either choice (C), (D), or (E).
(By the way, notice the too-obvious answer, choice (A); 8 liters is the amount of
acid in the resulting mixture.) If you need to guess at this point, your odds are
one in three. Here’s how to solve the problem: The original amount of acid is
(10)(20%) 5 2 liters. After adding 6 liters of pure acid, the amount of acid
increases to 8 liters, while the amount of total solution increases from 10 to 16
liters. The new solution is
8
16
, or 50%, acid.
Know When to Plug in Numbers for Variables
In multiple-choice questions, if the answer choices contain variables (such as x and y),
the question might be a good candidate for the “plug-in” strategy. Pick simple
numbers (so the math is easy) and substitute them for the variables. You’ll need your
pencil and scratch paper for this strategy.
Chapter 7: Problem Solving 161
TIP
Remember: Number choices
are listed in order of value—
either ascending or
descending. This feature can
help you zero in on the most
viable among the five choices.
www.petersons.com
7. If a train travels r 1 2milesinh hours, which of the following represents
the number of miles the train travels in 1 hour and 30 minutes?
(A)
36
2
r
h
+
(B)
3
2
r
h +
(C)
r
h
+
+
2
3
(D)
r
h + 6
(E)
3
2
~r +2!
The correct answer is (A). This is an algebraic word problem involving rate of
motion (speed). You can solve this problem either conventionally or by using the
plug-in strategy.
The conventional way: Notice that all of the answer choices contain fractions.
This is a tip that you should try to create a fraction as you solve the problem.
Here’s how to do it: Given that the train travels r 1 2milesinh hours, you can
express its rate in miles per hour as
r
h
+ 2
.In
3
2
hours, the train would travel this
distance:
3
2
236
2
⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
=
+r
h
r
h
The plug-in strategy: Pick easy-to-use values for r and h. Let’s try r 5 8 and
h 5 1. Given these values, the train travels 10 miles (8 1 2) in 1 hour. So
obviously, in 1
1
2
hours the train will travel 15 miles. Start plugging these r and h
values into the answer choices. You won’t need to go any further than choice (A):
36
2
38 6
21
30
2
15
r
h
+
=
+
=
()
()
, or
Choice (E) also equals 15,
3
2
(8 + 2) = 15. However, you can eliminate choice (E) out
of hand because it omits h. Common sense should tell you that the correct answer
must include both r and h.
PART IV: Quantitative Reasoning162
ALERT!
The plug-in method can be
time-consuming, so use it only
if you don’t know how to solve
the problem in a conventional
manner.
www.petersons.com
. comparing the two sums. Before the sixth number is taken
away, the sum of the numbers is 114 (6 3 19). After removing the sixth number, the
sum of the remaining. either the numerator or the denominator. That’s a clue that your job
is to manipulate the expression given in the question so that the result includes
these