1. Column A Column B
The number of fifths in
340%
The number of eighths in 212.5%
(A) The quantity in Column A is greater.
(B) The quantity in Column B is greater.
(C) The quantities are equal.
(D) The relationship cannot be determined from the information given.
The correct answer is (C). First, determine Quantity A. 340% is 3.4, or
3
2
5
.
Since there are five fifths in 1, Quantity A must be (3)(5) + 2, or 17. Next,
determine Quantity B. 212.5% is 2.125, or
2
1
8
. Since there are eight eighths
in 1, Quantity B must be (2)(8) + 1, or 17.
To guard against conversion errors, keep in mind the general magnitude of the
number you’re dealing with. For example, think of .09% as just less than .1%, which is
one-tenth of a percent, or a thousandth (a pretty small valued number). Think of
0.45
5
as just less than
0.5
5
, which is obviously
1
10
, or 10%. Think of 668% as more than
6 times a complete 100%, or between 6 and 7.
To rewrite a fraction as a decimal, simply divide the numerator by the denominator,
using long division. A fraction-to-decimal equivalent might result in a precise value,
an approximation with a repeating pattern, or an approximation with no repeating
pattern:
5
8
5 0.625
The equivalent decimal number is precise after three decimal
places.
5
9
' 0.555
The equivalent decimal number can only be approximated
(the digit 5 repeats indefinitely).
5
7
' 0.714
The equivalent decimal number is expressed here to the nearest
thousandth.
Certain fraction-decimal-percent equivalents show up on theGRE more often than
others. The numbers in the following tables are the test makers’ favorites because
they reward test takers who recognize quick ways to deal with numbers. Memorize
these conversions so that they’re second nature to you on exam day.
Chapter 9: Math Review: Number Forms, Relationships, and Sets 193
ALERT!
You won’t have access to a
calculator during the exam, so
knowing how to convert
numbers from one form to
another is a crucial skill.
www.petersons.com
Percent Decimal Fraction
50% 0.5
1
2
25% 0.25
1
4
75% 0.75
3
4
10% 0.1
1
10
30% 0.3
3
10
70% 0.7
7
10
90% 0.9
9
10
33
1
3
% 0.33
1
3
1
3
66
2
3
% 0.66
2
3
2
3
Percent Decimal Fraction
16
2
3
% 0.16
2
3
1
6
83
1
3
% 0.83
1
3
5
6
20% 0.2
1
5
40% 0.4
2
5
60% 0.6
3
5
80% 0.8
4
5
12
1
2
%
0.125
1
8
37
1
2
%
0.375
3
8
62
1
2
%
0.625
5
8
87
1
2
%
0.875
7
8
SIMPLIFYING AND COMBINING FRACTIONS
AGRE question might ask you to combine fractions using one or more of the four basic
operations (addition, subtraction, multiplication, and division). The rules for com-
bining fractions by addition and subtraction are very different from the ones for
multiplication and division.
Addition and Subtraction and the LCD
To combine fractions by addition or subtraction, the fractions must have a common
denominator. If they already do, simply add (or subtract) numerators. If they don’t,
you’ll need to find one. You can always multiply all of the denominators together to
find a common denominator, but it might be a big number that’s clumsy to work with.
So instead, try to find the least (or lowest) common denominator (LCD) by working
your way up in multiples of the largest of the denominators given. For denominators
of 6, 3, and 5, for instance, try out successive multiples of 6 (12, 18, 24 ),andyou’ll
hit the LCD when you get to 30.
2.
5
3
2
5
6
1
5
2
5
(A)
15
11
(B)
5
2
(C)
15
6
(D)
10
3
(E)
15
3
PART IV: Quantitative Reasoning194
www.petersons.com
The correct answer is (D). To find the LCD, try out successive multiples
of 6 until you come across one that is also a multiple of both 3 and 2. The
LCD is 6. Multiply each numerator by the same number by which you
would multiply the fraction’s denominator to give you the LCD of 6. Place
the three products over this common denominator. Then, combine the
numbers in the numerator. (Pay close attention to the subtraction sign!)
Finally, simplify to lowest terms:
5
3
2
5
6
1
5
2
5
10
6
2
5
6
1
15
6
5
20
6
5
10
3
Multiplication and Division
To multiply fractions, multiply the numerators and multiply the denominators. The
denominators need not be the same. To divide one fraction by another, multiply by the
reciprocal of the divisor (the number after the division sign):
Multiplication
1
2
3
5
3
3
1
7
5
~1!~5!~1!
~2!~3!~7!
5
5
42
Division
2
5
3
4
5
2
5
3
4
3
5
~2!~4!
~5!~3!
5
8
15
To simplify the multiplication or division, cancel factors common to a numerator and
a denominator before combining fractions. It’s okay to cancel across fractions. Take,
for instance, the operation
3
4
3
4
9
3
3
2
. Looking just at the first two fractions, you can
cancel out 4 and 3, so the operation simplifies to
1
1
3
1
3
3
3
2
. Now, looking just at the
second and third fractions, you can cancel out 3 and the operation becomes even
simpler:
1
1
3
1
1
3
1
2
5
1
2
.
Apply the same rules in the same way to variables (letters) as to numbers. The
variables a and c do not equal 0.
3.
2
a
3
b
4
3
a
5
3
8
c
5
(A)
ab
4c
(B)
10b
9c
(C)
8
5
(D)
16b
5ac
(E)
4b
5c
The correct answer is (E). Since you’re dealing only with multiplication,
look for factors and variables (letters) in any numerator that are the same
as those in any denominator. Canceling common factors leaves:
2
1
3
b
1
3
1
5
3
2
c
Multiply numerators and denominators and you get
4b
5c
.
Chapter 9: Math Review: Number Forms, Relationships, and Sets 195
ALERT!
On the GRE, pay very close
attention to operation signs.
You can easily flub a question
by reading a plus s ign (+) as a
minus sign (–), or vice versa.
www.petersons.com
Mixed Numbers and Multiple Operations
A mixed number consists of a whole number along with a simple fraction—for
example, the number 4
2
3
. Before combining fractions, you might need to rewrite a
mixed number as a fraction. To do so, follow these three steps:
Multiply the denominator of the fraction by the whole number.
Add the product to the numerator of the fraction.
Place the sum over the denominator of the fraction.
For example, here’s how to rewrite the mixed number 4
2
3
as a fraction:
4
2
3
5
~3!~4!12
3
5
14
3
To perform multiple operations, always perform multiplication and division before you
perform addition and subtraction.
4.
4
1
2
1
1
8
2 3
2
3
is equivalent to what simple fraction?
(A)
1
3
(B)
3
8
(C)
11
6
(D)
17
6
(E)
11
2
Enter an integer in the numerator box,
and enter an integer in the denominator box.
The correct answer is
S
1
3
D
. First, rewrite all mixed numbers as
fractions. Then, eliminate the complex fraction by multiplying the
numerator fraction by the reciprocal of the denominator fraction (cancel
across fractions before multiplying):
9
2
9
8
2
11
3
5
S
9
2
DS
8
9
D
2
11
3
5
S
1
1
DS
4
1
D
2
11
3
5
4
1
2
11
3
Next, express each fraction using the common denominator 3; then
subtract:
4
1
2
11
3
5
12 2 11
3
5
1
3
PART IV: Quantitative Reasoning196
NOTE
In a GRE numeric-entry
question, you don’t need to
reduce a fraction to lowest
terms to receive credit for a
correct answer. So in Question
4, you’d receive credit for
2
6
as
well as for
1
3
.
www.petersons.com
DECIMAL PLACE VALUES AND OPERATIONS
Place value refers to the specific value of a digit in a decimal. For example, in the
decimal 682.793:
• The digit 6 is in the “hundreds” place.
• The digit 8 is in the “tens” place.
• The digit 2 is in the “ones” place.
• The digit 7 is in the “tenths” place.
• The digit 9 is in the “hundredths” place.
• The digit 3 is in the “thousandths” place.
So you can express 682.793 as follows:
600 1 80 12 1
7
10
1
9
100
1
3
1,000
To approximate, or round off, a decimal, round any digit less than 5 down to 0, and
round any digit greater than 5 up to 0 (adding one digit to the place value to the left).
• The value of 682.793, to the nearest hundredth, is 682.79.
• The value of 682.793, to the nearest tenth, is 682.8.
• The value of 682.793, to the nearest whole number, is 683.
• The value of 682.793, to the nearest ten, is 680.
• The value of 682.793, to the nearest hundred, is 700.
Multiplying Decimals
The number of decimal places (digits to the right of the decimal point) in a product
should be the same as the total number of decimal places in the numbers you multiply.
So to multiply decimals quickly, follow these three steps:
Multiply, but ignore the decimal points.
Count the total number of decimal places among the numbers you multiplied.
Include that number of decimal places in your product.
Here are two simple examples:
1
(23.6)(0.07) Three decimal places altogether
(236)(7) 5 1652 Decimals temporarily ignored
(23.6)(0.07) 5 1.652 Decimal point inserted
2
(0.01)(0.02)(0.03) Six decimal places altogether
(1)(2)(3) 5 6 Decimals temporarily ignored
(0.01)(0.02)(0.03) 5 0.000006 Decimal point inserted
Chapter 9: Math Review: Number Forms, Relationships, and Sets 197
TIP
Eliminate decimal points from
fractions, as well as f rom
percents, to help you see
more clearly the magnitude
of the quantity you’re
dealing with.
www.petersons.com
Dividing Decimal Numbers
When you divide (or compute a fraction), you can move the decimal point in both
numbers by the same number of places either to the left or right without altering the
quotient (value of the fraction). Here are three related examples:
11.4 4 0.3 5
11.4
0.3
5
114
3
5 38
1.14 4 3 5
1.14
3
5
114
300
5 0.38
114 4 0.003 5
114
0.003
5
114,000
3
5 38,000
GRE questions involving place value and decimals usually require a bit more from you
than just identifying a place value or moving a decimal point around. Typically, they
require you to combine decimals with fractions or percents.
5.
1
3
3 0.3 3
1
30
3 0.03 =
(A)
1
10,000
(B)
33
100,000
(C)
99
100,000
(D)
33
10,000
(E)
99
10,000
The correct answer is (A). There are several ways to convert and
combine the four numbers provided in the question. One method is to
combine the two fractions:
1
3
3
1
30
5
1
90
. Then, combine the two decimals:
0.3 3 0.03 5 0.009 5
9
1,000
. Finally, combine the two resulting fractions:
1
90
3
9
1,000
5
9
90,000
5
1
10,000
which is choice (A).
SIMPLE PERCENT PROBLEMS
On the GRE, a simple problem involving percent might ask you to perform any one of
these four tasks:
Find a percent of a percent.
Find a percent of a number.
Find a number when a percent is given.
Find what percent one number is of another.
The following examples show you how to handle these four tasks (task 4 is a bit
trickier than the others):
Finding a percent of a percent
What is 2% of 2%?
PART IV: Quantitative Reasoning198
www.petersons.com
Rewrite 2% as 0.02, then multiply:
0.02 3 0.02 5 0.0004 or 0.04%
Finding a percent of a number
What is 35% of 65?
Rewrite 35% as 0.35, then multiply:
0.35 3 65 5 22.75
Finding a number when a percent is given
7 is 14% of what number?
Translate the question into an algebraic equation, writing the percent as
either a fraction or decimal:
7 5 14% of x
7 5 0.14x
x 5
7
0.14
5
1
0.02
5
100
2
5 50
Finding what percent one number is of another
90 is what % of 1,500?
Set up an equation to solve for the percent:
90
1,500
5
x
100
1,500x 5 9,000
15x 5 90
x 5
90
15
or 6
PERCENT INCREASE AND DECREASE
In the fourth example above, you set up a proportion. (90 is to 1,500 as x is to 100.)
You’ll need to set up a proportion for other types of GRE questions as well, including
questions about ratios, which you’ll look at in the next section.
The concept of percent change is one of the test makers’ favorites. Here’s the key to
answering questions involving this concept: Percent change always relates to the
value before the change. Here are two simple illustrations:
10 increased by what percent is 12?
1. The amount of the increase is 2.
2. Compare the change (2) to the original number (10).
3. The change in percent is
S
2
10
D
~100!520, or 20%.
Chapter 9: Math Review: Number Forms, Relationships, and Sets 199
www.petersons.com
12 decreased by what percent is 10?
1. The amount of the decrease is 2.
2. Compare the change (2) to the original number (12).
3. The change is
1
6
,or16
2
3
%, or approximately 16.7%.
Notice that the percent increase from 10 to 12 (20%) is not the same as the percent
decrease from 12 to 10
S
16
2
3
%
D
. That’s because the original number (before the
change) is different in the two questions.
A typical GRE percent-change problem will involve a story about a type of quantity
such as tax, profit or discount, or weight, in which you need to calculate successive
changes in percent. For example:
• An increase, then a decrease (or vice versa)
• Multiple increases or decreases
Whatever the variation, just take the problem one step at a time and you’ll have no
trouble handling it.
6. A stereo system originally priced at $500 is discounted by 10%, then by
another 10%. If a 20% tax is added to the purchase price, how much
would a customer pay who is buying the system at its lowest price,
including tax?
(A) $413
(B) $480
(C) $486
(D) $500
(E) $512
The correct answer is (C). After the first 10% discount, the price is $450 ($500
minus 10% of $500). After the second discount, which is calculated based on the
$450 price, the price of the stereo is $405 ($450 minus 10% of $450). A 20% tax on
$405 is $81. Thus, the customer has paid $405 1 $81 5 $486.
RATIOS AND PROPORTION
A ratio expresses proportion or comparative size—the size of one quantity relative to
the size of another. As with fractions, you can simplify ratios by dividing common
factors. For example, given a class of 28 students—12 freshmen and 16 sophomores:
• The ratio of freshmen to sophomores is 12:16, or 3:4.
• The ratio of freshmen to the total number of students is 12:28, or 3:7.
• The ratio of sophomores to the total number of students is 16:28, or 4:7.
PART IV: Quantitative Reasoning200
NOTE
GRE problems involving
percent and percent change
are often accompanied by a
chart, graph, or table.
www.petersons.com
Finding a Ratio
A GRE question might ask you to determine a ratio based on given quantities. This is
the easiest type of GRE ratio question.
7. A class of 56 students contains only freshmen and sophomores. If 21 of the
students are sophomores, what is the ratio of the number of freshmen to
the number of sophomores in the class?
(A) 3:5
(B) 5:7
(C) 5:3
(D) 7:4
(E) 2:1
The correct answer is (C). Since 21 of 56 students are sophomores, 35 must be
freshmen. The ratio of freshmen to sophomores is 35:21. To simplify the ratio to
simplest terms, divide both numbers by 7, giving you a ratio of 5:3.
Determining Quantities from a Ratio (Part-to-Whole Analysis)
You can think of any ratio as parts adding up to a whole. For example, in the ratio 5:6,
5 parts 1 6 parts 5 11 parts (the whole). If the actual total quantity were 22, you’d
multiply each element by 2: 10 parts 1 12 parts 5 22 parts (the whole). Notice that
the ratios are the same: 5:6 is the same ratio as 10:12.
You might be able to solve a GRE ratio question using this part-to-whole approach.
8. A class of students contains only freshmen and sophomores. If 18 of the
students are sophomores, and if the ratio of the number of freshmen to the
number of sophomores in the class is 5:3, how many students are in the
class?
students
Enter a number in the box.
The correct answer is (48). Using a part-to-whole analysis, look first at
the ratio and the sum of its parts: 5 (freshmen) 1 3 (sophomores) 5 8
(total students). These aren’t the actual quantities, but they’re proportion-
ate to those quantities. Given 18 sophomores altogether, sophomores
account for 3 parts—each part containing 6 students. Accordingly, the total
number of students must be 6 3 8 5 48.
Determining Quantities from a Ratio (Setting Up a Proportion)
Since you can express any ratio as a fraction, you can set two equivalent, or propor-
tionate, ratios equal to each other, as fractions. So the ratio 16:28 is proportionate to
the ratio 4:7 because
16
28
5
4
7
. If one of the four terms is missing from the equation (the
proportion), you can solve for the missing term using algebra. So if the ratio 3:4 is
Chapter 9: Math Review: Number Forms, Relationships, and Sets 201
www.petersons.com
proportionate to 4:x, you can solve for x in the equation
3
4
5
4
x
. Using the cross-product
method, equate products of numerator and denominator across the equation:
~3!~x!5~4!~4!
3x 5 16
x 5
16
3
or 5
1
3
If the quantities in a proportion problem strike you as “unround,” it’s a good bet that
doing the math will be easier than you might first think.
9. If 3 miles are equivalent to 4.8 kilometers, then 14.4 kilometers are
equivalent to how many miles?
(A) 18.2
(B) 12.0
(C) 10.6
(D) 9.0
(E) 4.8
The correct answer is (B). The question essentially asks, “3 is to 4.83 as what
is to 14.4?” Set up a proportion, then solve for x by the cross-product method:
3
4.8
5
x
14.4
~4.8!~x!5~3!~14.4!
x 5
~3!~14.4!
4.8
5
14.4
1.2
=12
Notice that, despite all the intimidating decimal numbers, the solution turns out
to be a tidy number. That’s typical of the GRE.
Altering Fractions and Ratios
An average test taker might assume that adding the same positive quantity to a
fraction’s numerator (p) and to its denominator (q) leaves the fraction’s value
S
p
q
D
unchanged. But this is true if and only if the original numerator and denominator
were equal to each other. Otherwise, the fraction’s value will change. Remember the
following three rules, which apply to any positive numbers x, p, and q:
If p 5 q, then
p
q
5
p 1 x
q 1 x
. (The fraction’s value remains unchanged and is
always 1.)
If p . q, then
p
q
.
p 1 x
q 1 x
. (The fraction’s value will decrease.)
If p , q, then
p
q
,
p 1 x
q 1 x
. (The fraction’s value will increase.)
As you might suspect, this concept makes great fodder for GRE Quantitative Com-
parison questions.
PART IV: Quantitative Reasoning202
TIP
On the GRE, what look like
unwieldy numbers typically boil
down to simple ones. In fact,
your ability to recognize this
feature is one of the skills being
tested on the GRE.
www.petersons.com
. Ratio (Part- to-Whole Analysis)
You can think of any ratio as parts adding up to a whole. For example, in the ratio 5:6,
5 parts 1 6 parts 5 11 parts (the. whole). If the actual total quantity were 22, you’d
multiply each element by 2: 10 parts 1 12 parts 5 22 parts (the whole). Notice that
the ratios are the same: