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Hungry Minds Cliffs Gre_INTRODUCTION TO QUANTITANTIVE ABILITY

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119
INTRODUCTION TO
QUANTITATIVE ABILITY
Prior to starting the exam, you will be carefully walked through a very basic tutor-
ial program explaining how to use the computer for this exam. The computer-
adaptive GRE gives you 45 minutes to answer 28 quantitative questions. These
questions are composed of Quantitative Comparisons and Math Ability (Multiple-
Choice) Questions, and the question types are intermingled. You will be given a
medium difficulty question to start with, and then the computer will adapt the
level of questions you receive based on your responses to all the previous ques-
tions. All of your work will be done on the scratch paper provided, and all of your
answers will be recorded on the computer screen by using a mouse to fill in the
appropriate ovals. You will not be allowed to go back to a previous question, so be
sure to answer each question before you attempt to move to the next question.
The Quantitative Section will generate a score from 200 to 800. Your score will be
based on how well you do on questions presented and also on the number of ques-
tions you answer. You should try to pace yourself so that you have sufficient time
to consider every question. If possible, answer all 28 questions in this section.
Guess if you need to.
In this book — to assist you in understanding explanations and to direct your at-
tention to different questions and answer choices — each question is given a num-
ber, and letters have been placed inside the ovals of the answer choices. Note that
on the actual exam, questions will not have numbers next to them and there
will be no letters in the ovals.
Introduction to Quantitative Comparison
Quantitative Comparison questions require you to make a comparison between
quantities in two columns. You are to decide if one column is greater, if the
columns are equal, or if no comparison can be determined from the information
given.
Ability Tested
Quantitative Comparison tests your ability to use mathematical insight, approxima-


tion, simple calculation, or common sense to quickly compare two given quantities.
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Part I: Analysis of Exam Areas
Basic Skills Necessary
This question type requires twelfth-grade competence in school arithmetic, alge-
bra, and intuitive geometry. Skills in approximating, comparing, and evaluating
are also necessary. No advanced mathematics is necessary.
Directions
You are given two quantities, one in column A and one in column B. You are to
compare the two quantities and choose oval:
A. if the quantity in Column A is greater;
B. if the quantity in Column B is greater;
C. if the two quantities are equal;
D. if the comparison cannot be determined from the information given.
Common Information: Information centered above columns refers to one or
both columns. A symbol that appears in both columns represents the same thing in
each column.
Analysis

The purpose here is to make a comparison; therefore, exact answers are
not always necessary. (Remember that you can tell whether you are taller
than someone in many cases without knowing that person’s height.
Comparisons such as this can be made with only limited or partial infor-
mation—just enough to compare.)

Choice D—the comparison cannot be determined from the information
given—is not a possible answer if there are values in each column, be-
cause you can always compare values.


If you get different relationships, depending on the values you choose for
variables, then the answer is always D. Notice that there are only four pos-
sible choices here.

Note that you can add, subtract, multiply, and divide both columns by the
same value, and the relationship between the columns will not change.
Exception: You should not multiply or divide each column by negative
numbers, because the relationship reverses. Squaring both columns is per-
missible, as long as each side is positive.
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Suggested Approach with
Sample Problems
This section emphasizes shortcuts, insight, and quick techniques. Long and/or
involved mathematical computation is unnecessary and is contrary to the pur-
pose of this section.
Samples
Column A Column B
1.
21 × 43 × 56 44 × 21 × 57
Canceling (or dividing) 21 from each side leaves
43 × 56 44 × 57
The rest of this problem should be done by inspection, because it is obvious that
column B is greater than column A without doing any multiplication. You could
have attained the correct answer by actually multiplying out each column, but you
would then not have enough time to finish the section. The correct answer is B.
Column A Column B
2.
7
3
5

2
8
5
##
5
2
11
4
8
5
##
Because both sides have the factors
2

5
and
5

8
, you may eliminate them from each
column. Now compare
3

7
and
4

11
by cross-multiplying upward, and you get
Because 33 is greater than 28,

3

7
>
4

11.
The correct answer is A.
Always keep the columns in perspective before starting any calculations. Take
a good look at the value in each column before starting to work on one
column.
3
7
4
11
33 28
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Introduction to Quantitative Ability
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Samples
Column A Column B
3.
40% of 60 60% of 40
There is no need to do any calculations for this problem. Column A can be written
(
40

100
) × 60. Column B can be written (
60


100
) × 40. You should note that both
columns have (40 × 60)/100.
The correct answer is C.
Column A Column B
4.
7
6
3
2
8
After looking at each column (note that the answer could not be D because there
are values in each column), compute the value on the left. Because you are taking
a cube root, simply divide the power of 7 by 3 leaving 7
2
, or 49. There is no need
to take 2 out to the 8th power; just do as little as necessary:
2
2
= 4
2
3
= 8
2
4
= 16
2
5
= 32

STOP
It is evident that 2
8
is much greater than 49; the correct answer is B.
Approximating can also be valuable while remembering to keep the columns in
perspective.
As you keep the columns in perspective, check to see if the value in each col-
umn increases or decreases from the starting point.
Sample
Column A Column B
5.
(.9)
8
(1.01)
4
In Column A, a fractional value (a value less than 1) is multiplied by itself many
times. So its value becomes increasingly smaller. (For example,
1

2
×
1

2
=
1

4
;
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Part I: Analysis of Exam Areas
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1

4
×
1

2
=
1

8
, and so forth). In Column B, a number greater than 1 is multiplied by
itself; its value grows larger. So Column B is greater.
The correct answer is B.
As you keep the columns in perspective, notice if the signs (+, −) in each col-
umn are different. If they are, you don’t need to work out the problem.
Samples
Column A Column B
6.
(−10)
100
(−10)
101
A negative number multiplied an even number of times will yield a positive prod-
uct. A negative number multiplied an odd number of times will yield a negative
product. Since Column A will be positive and Column B will be negative, A is
greater.
The correct answer is A.

Column A Column B
7.
.05 − .125 .1
Subtracting in Column A, you get .05 − .125 =−.075. Our difference is a negative
number. Thus, the positive value in Column B must be greater.
The correct answer is B.
Column A Column B
8.
x + ya+ b
Because coordinates (x,y) are in quadrant III, they are both negative, so their sum
is negative. Because coordinates (a, b) are in quadrant I, they are both positive, so
their sum is positive. Therefore, Column B is greater than Column A.
The correct answer is B.
• (a,b)
• (x,y)
(0,0)
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Introduction to Quantitative Ability
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The use of partial comparisons can be valuable in giving you insight into find-
ing a comparison. If you cannot simply make a complete comparison, look at
each column part by part.
Sample
Column A Column B
9.
57
1
65
1
-

58
1
63
1
-
Because finding a common denominator would be too time consuming, you
should first compare the first fraction in each column (partial comparison). Notice
that
1

57
is greater than
1

58
. Now compare the second fractions and notice that
1

65
is
less than
1

63
. Using some common sense and insight, if you start with a larger
number and subtract a smaller number, it must be greater than starting with a
smaller number and subtracting a larger number, as pointed out below.
The correct answer is A.
Often, simplifying one or both columns can make an answer evident.
Samples

Column A Column B
10 .
a, b, c, all greater than 0
a(b + c) ab + ac
Using the distributive property on Column A to simplify gives ab and ac; there-
fore, the columns are equal.
The correct answer is C.
1
57
1
65

1
58
1
63

Larger
Smaller
Larger
Smaller
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Column A Column B
11.
a > 0
b > 0
c > 0
(3a)(3b)(3c) 3abc

Multiplying column A gives (3a)(3b)(3c) = 27abc. Because a, b, and c are all posi-
tive values, 9abc will always be greater than 3abc.
The correct answer is A.
Column A Column B
12 .
Number of prime numbers 5
between 3 and 19
The prime numbers between 3 and 19 are 5, 7, 11, 13, and 17. The correct answer
is C, since there are 5 primes.
If a problem involves variables (without an equation), substitute in the num-
bers 0, 1, and -1. Then try 1⁄2, and 2 if necessary. Using 0, 1, and -1 will often
tip off the answer.
Samples
Column A Column B
13 .
a + bab
Substituting 0 for a and 0 for b gives the following:
0 + 0 (0)
Therefore, 0 = 0.
Using these values for a and b gives the answer C. But when you multiply two
numbers, you don’t always get the same result as when you add them, so try some
other values. Substituting 1 for a and −1 for b gives the following:
1 + (−1) 1(−1)
Therefore, 0 >−1
and the answer is now A.
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Introduction to Quantitative Ability
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Anytime you get more than one comparison (different relationships), depend-
ing on the values you choose, the correct answer must be D, the relationship

cannot be determined. Notice that if you had substituted the values a = 4, b = 5;
or a = 6, b = 7; or a = 7, b = 9; and so on, you would repeatedly get the answer B
and may have chosen the incorrect answer.
The correct answer is D.
Column A Column B
14 .
x < y < z
x + y + z xyz
Substituting 0 for x, 1 for y, and 2 for z, gives
(0) + (1) + (2) (0)(1)(2)
Therefore, 3 > 0.
Now substituting −1 for x, 0 for y, and 1 for z gives
(−1) + (0) + (1) (−1)(0)(1)
Therefore, 0 = 0.
Because different values give different comparisons, the correct answer is D.
Column A Column B
15 .
x > y > 0
x and y are integers
x
xy
x
+
_i
y
xy
y
+
_i
Plug in values for x and y such that x > y > 0, and x and y are integers. For exam-

ple, let y = 1 and x = 2. This gives
2
21
2
+
^h
1
12
1
+
^h
2
3
2
^h
1
3
1
^h
2
9
>
1
3
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Part I: Analysis of Exam Areas
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Using these values,
9


2
, or 4
1

2
, is greater than 3, so Column A is greater. Using
other values such that x > y > 0 will always give the same relationship. Column A
is greater.
The correct answer is A.
Sometimes you can solve for a column directly, in one step, without solving
and substituting. If you have to solve an equation or equations to give the
columns values, take a second and see if there is a very simple way to get an
answer before going through all of the steps.
Sample
Column A Column B
16 .
4x + 2 = 10
2x + 14
Hopefully, you would spot that the easiest way to solve for 2x + 1 is directly by
dividing 4x + 2 = 10 by 2, leaving 2x + 1 = 5. Therefore,
5 > 4
Solving for x first in the equation and then substituting would also have worked
but would have been more time consuming. The correct answer is A.
Redrawing and marking diagrams and figures can be very helpful for giving
insight into a problem. If you are given a diagram or figure on the screen,
quickly redraw it on your scratch paper. Remember that diagrams and figures
are meant for positional information only. Just because something “looks” a
certain way is not enough reason to choose an answer.
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Introduction to Quantitative Ability

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Sample
Column A Column B
XZ = YZ
17.
xy
Even though x appears larger, this is not enough. Mark in the diagram as shown.
Notice that you should mark things of equal measure with the same markings,
and since angles opposite equal sides in a triangle are equal, x = y. The correct
answer is C.
If you are given a description of a diagram or a geometry problem without a
diagram, you should make a sketch. When in doubt, “draw.” This may tip off a
simple solution.
Sample
Column A Column B
18 .
Perimeter of an equilateral Perimeter of a square with side
triangle with side length 5x length of 4x
Simply sketch and label each geometric figure as follows:
5x 4x
4x
5x5x 4x4x
XY
Z
X
X° y°
Y
Z
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Part I: Analysis of Exam Areas

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