Hungry Minds Cliffs Gre_INTRODUCTION TO MATH ABILITY

145
Math Ability questions are multiple-choice math questions that give you five
possible answer choices. You are required to select the best answer.
Ability Tested
Problem Solving questions test your ability to solve mathematical problems in-
volving arithmetic, algebra, and geometry, as well as word problems, by using
problem-solving insight, logic, and the application of basic skills.
Basic Skills Necessary
The basic skills necessary to do well on this section include high school arith-
metic, algebra, and intuitive geometry—no formal trigonometry or calculus is
necessary. These skills, along with logical insight into problem-solving situations,
are covered by the examination.
Directions
Solve each problem in this section by using the information given and your own
mathematical calculations. Select the correct answer of the five choices given.
Use the scratch paper given for any necessary calculations.
Analysis
All scratchwork is to be done on the paper given at the test; get used to referring
back to the screen as you do your calculations and drawings. You are looking for
the one correct answer; therefore, although other answers may be close, there is
never more than one right answer.
Suggested Approach with Samples
Always carefully focus on what you are looking for to ensure that you are an-
swering the right question.
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Part I: Analysis of Exam Areas
Samples
1.
If x + 6 = 9, then 3x + 1 =

A. 3
B. 9
C. 10
D. 34
E. 46
You should first focus on 3x + 1, because this is what you are solving for. Solving
for x leaves x = 3, and then substituting into 3x + 1 gives 3(3) + 1, or 10. The most
common mistake is to solve for x, which is 3, and mistakenly choose A as your an-
swer. But remember, you are solving for 3x + 1, not just x. You should also notice
that most of the other choices would all be possible answers if you made common
or simple mistakes. The correct answer is C. Make sure that you’re answering the
right question.
2.
An employee’s annual salary was increased $15,000. If her new annual
salary now equals $90,000, what was the percent increase?
A. 15%
B. 16
2
⁄
3
%
C. 20%
D. 22%
E. 24%
Focus on what you are looking for. In this case, percent increase.
Percent increase = change/starting point. If the employee’s salary was increased
$15,000 to $90,000, then the starting salary was 90,000 − 15,000 = 75,000.
Therefore,
percent increase = 15,000/75,000 = 1/5 = 20%
The correct answer is C.

“Pulling” information out of the word problem structure can often give you a
better look at what you are working with, and therefore, you gain additional
insight into the problem. Organize this information on your scratch paper.
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Sample
3.
If a mixture is
3
⁄
7
alcohol by volume and
4
⁄
7
water by volume, what is the
ratio of the volume of alcohol to the volume of water in this mixture?
A.
3
⁄
7
B.
4
⁄
7
C.
3
⁄
4
D.
4

⁄
3
E.
7
⁄
4
The first bit of information that you should pull out is what you are looking for:
“ratio of the volume of alcohol to the volume of water.” Rewrite the ration that
you’re looking for as A:W and then rewrite it into its working form: A/W. Next,
pull out the volumes of each; A =
3
⁄
7
and W=
4
⁄
7
. Now you can easily figure the an-
swer by inspection or substitution: Using
3
⁄
7
/
4
⁄
7
, invert the bottom fraction and
multiply to get
3
⁄

7
×
7
/
4
=
3
⁄
4
. The ratio of the volume of alcohol to the volume of
water is 3 to 4. The correct answer is C. When pulling out information, write out
the numbers and/or letters on your scratch paper, putting them into some helpful
form and eliminating some of the wording.
Sometimes combining terms, performing simple operations, or simplifying
the problem in some other way will give you insight and make the problem
easier to solve.
Sample
4.
Which of the following is equal to
1
⁄
5
of 0.02 percent?
A. 0.4
B. 0.04
C. 0.004
D. 0.0004
E. 0.00004
Simplifying this problem first means changing
1

⁄
5
to .2. Next change 0.02 percent
to 0.0002 (that is, .02 × .01 = 0.0002).
Now that you have simplified the problem, multiply .2 × 0.0002, which gives
0.00004. The correct answer is E. Notice that simplifying can make a problem
much easier to solve.
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If you immediately recognize the method or proper formula to solve the
problem, go ahead and do the work. Work forward.
Sample
5.
Which of the following numbers is between
1
⁄
3
and
1
⁄
4
?
A. .45
B. .35
C. .29
D. .22
E. .20
Focus on “between
1

⁄
3
and
1
⁄
4
.” If you know that
1
⁄
3
is .333 . . . and
1
⁄
4
is .25, you
have insight into the problem and should simply work it forward. Since .29 is the
only number between .333 . . . and .25, the correct answer is C. By the way, a quick
peek at the answer choices would tip you off that you should work in decimals.
If you don’t immediately recognize a method or formula, or if using the
method or formula would take a great deal of time, try working backward —
from the answers. Because the answers are usually given in ascending or de-
scending order, almost always start by plugging in choice C first. Then you’ll
know whether to go up or down with your next try. (Sometimes, you may
want to plug in one of the simple answers first.)
Samples
6.
If
x
⁄
2

+
3
⁄
4
= 1
1
⁄
4
, what is the value of x?
A. −2
B. −1
C. 0
D. 1
E. 2
You should first focus on “value of x.” If you’ve forgotten how to solve this kind of
equation, work backward by plugging in answers. Start with choice C; plug in 0.
0
⁄
2
+
3
⁄
4
!
1
1
⁄
4
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Part I: Analysis of Exam Areas

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Because this answer is too small, try choice D, a larger number. Plugging in 1
gives you
1
⁄
2
+
3
⁄
4
= 1
1
⁄
4
2
⁄
4
+
3
⁄
4
= 1
1
⁄
4
5
⁄
4
= 1
1

⁄
4
This answer is true, so D is the correct answer. Working from the answers is a
valuable technique.
7.
What is the greatest common factor of the numbers 18, 24, and 30?
A. 2
B. 3
C. 4
D. 6
E. 12
The largest number that divides evenly into 18, 24, and 30 is 6. You could’ve
worked from the answers, but here you should start with the largest answer
choice, because you’re looking for the greatest common factor.
The correct answer is D.
If you don’t immediately recognize a method or formula to solve the prob-
lem, you may want to try a reasonable approach and then work from the an-
swer choices. Try to be reasonable.
Samples
8.
Barney can mow the lawn in 5 hours, and Fred can mow the lawn in 4 hours.
How long will it take them to mow the lawn together?
A. 5 hours
B. 4
1
⁄
2
hours
C. 4 hours
D. 2

2
⁄
9
hours
E. 1 hour
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Suppose that you’re unfamiliar with the type of equation for this problem. Try the
“reasonable” method. Because Fred can mow the lawn in 4 hours by himself, he
will take less than 4 hours if Barney helps him. Therefore, choices A, B, and C are
not sensible. Taking this method a little farther, suppose that Barney could also
mow the lawn in 4 hours. Therefore, together it would take Barney and Fred 2
hours. But, because Barney is a little slower than this, the total time should be
more than 2 hours. The correct answer is D, 2
2
⁄
9
hours.
Using the equation for this problem would give the following calculations:
1
⁄
5
+
1
⁄
4
=
1
⁄

x
In 1 hour, Barney could do
1
⁄
5
of the job, and in 1 hour, Fred could do
1
⁄
4
of the
job; unknown
1
⁄
x
is the part of the job they could do together in 1 hour. Now, solv-
ing, you calculate as follows:
4
⁄
20
+
5
⁄
20
=
1
⁄
x
9
⁄
20

=
1
⁄
x
Cross multiplying gives 9x = 20; therefore, x =
20
⁄
9
, or 2
2
⁄
9
.
9.
Circle O is inscribed in square ABCD as shown above. The area of the
shaded region is approximately
A. 10
B. 25
C. 30
D. 50
E. 75
Using a reasonable approach, you would first find the area of the square:
10 × 10 = 100. Then divide the square into four equal sections as follows:
O
D
C
BA
r
10
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Part I: Analysis of Exam Areas
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Because a quarter of the square is 25, the shaded region must be much less than 25.
The only possible answer is choice A (10). Another approach to this problem is to
first find the area of the square: 10 × 10 = 100. Then subtract the approximate area of
the circle: A =π(r
2
) ≅ 3(5
2
) = 3(25) = 75. Therefore, the total area inside the square,
but outside the circle, is approximately 25. One quarter of that area is shaded.
Therefore,
25
⁄
4
is approximately the shaded area. The closest answer is A (10).
Substituting numbers for variables can often be an aid to understanding a
problem. Remember to substitute simple numbers, because you have to do
the work.
Sample
10 .
If x > 1, which of the following decreases as x decreases?
I. x + x
2
II. 2x
2
− x
III.
x1
1

+
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
This problem is most easily solved by taking each situation and substituting sim-
ple numbers.
However, in the first situation, I, x + x
2
, recognize that this expression will de-
crease as x decreases.
Trying x = 2 gives 2 + (2)
2
, which equals 6.
Now trying x = 3 gives 3 + (3)
2
= 12.
Notice that choices B, C, and E are already eliminated because they don’t contain I.
You should also realize that you now need to try only the values in II; because III is
not paired with I as a possible choice, III cannot be one of the answers.
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Trying x = 2 in the expression 2x
2
− x gives 2(2)
2
− 2, or 2(4) − 2, which leaves 6.
Now trying x = 3 gives 2(3)

2
− 3, or 2(9) − 3 = 18 − 3 = 15. This expression also
decreases as x decreases. Therefore, the correct answer is choice D. Notice again
that III wasn’t attempted because it wasn’t one of the possible choices.
Some problems may deal with percent or percent change. If you don’t see a
simple method for working the problem, try using values of 10 or 100 and see
what you get.
Sample
11.
A corporation triples its annual bonus to 50 of its employees. What percent
of the employees’ new bonus is the increase?
A. 50%
B. 66
2
⁄
3
%
C. 100%
D. 200%
E. 300%
Use $100 for the normal bonus. If the annual bonus was normally $100, tripled it
would be $300. Therefore, the increase ($200) is
2
⁄
3
of the new bonus ($300).
Two-thirds is 66
2
⁄
3

%.
The correct answer is B.
Sketching diagrams or simple pictures can also be very helpful because the dia-
gram may tip off either a simple solution or a method for solving the problem.
Samples
12 .
What is the maximum number of pieces of birthday cake of size 4" by 4"
that can be cut from a cake 20" by 20"?
A. 5
B. 10
C. 16
D. 20
E. 25
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Sketching the cake and marking in as the following figure shows makes this a
fairly simple problem.
Notice that five pieces of cake will fit along each side; therefore, 5 × 5 = 25. The
correct answer is E. Finding the total area of the cake and dividing it by the area
of one of the 4 × 4 pieces would also give you the correct answer, but beware of
this method because it may not work if the pieces don’t fit evenly into the original
area.
13 .
If P lies on
ON
%
such that
OP PN2=
$

$
and Q lies on
OP
$
such that
OQ QP=
%
$
,
what is the relationship of
OQ
%
to
PN
$
?
A.
1
⁄
3
B.
1
⁄
2
C. 1
D..
2
⁄
1
E.

3
⁄
1
A sketch would look like this:
It is evident that
OQ PN=
%
$
, so the ratio is 1/1, or 1. Or, you could assign values
ON
%
such that OP PN2=
$
$
: OP
$
equals 2, and PN
$
equals 1. If Q lies on OP
$
such that
OQ QP=
%
$
, then OP
$
(2) is divided in half. So OQ 1=
%
, and QP 1=
$

. Therefore, the
relationship of
OQ
%
to PN
$
is 1 to 1. The correct answer is C.
Redrawing and marking in diagrams on your scratch paper as you read them
can save you valuable time. Marking can also give you insight into how to
solve a problem because you will have the complete picture clearly in front
of you.
O
N
P
Q
=
=
=
4 44
20″
44
4
420″
4
4
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14.
In the triangle, CD is an angle bisector, angle ACD is 30°, and angle ABC is

a right angle. What is the measurement of angle x in degrees?
A. 80°
B. 75°
C. 60°
D. 45°
E. 30°
After redrawing the diagram on your scratch paper, read the problem and mark as
follows:
In the triangle above,
CD is an angle bisector (stop and mark in the drawing), an-
gle ACD is 30° (stop and mark in the drawing), and angle ABC is a right angle
(stop and mark in the drawing). What is the measurement of angle x in degrees?
(Stop and mark in or circle what you’re looking for in the drawing.)
With the drawing marked in, it is evident that, because angle ACD is 30°, angle
BCD is also 30° because they are formed by an angle bisector (divides an angle
into two equal parts). Because angle ABC is 90° (right angle) and angle BCD is
30°, angle x is 60° because there are 180° in a triangle; 180 − (90 + 30) = 60. The
correct answer is C. After redrawing the diagrams on your scratch paper, always
mark in the diagrams as you read their descriptions and information about them,
including the information you’re looking for.
B
D
A
x°
C
30°
B
D
A
x°

C
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15. If each square in the figure above has a side of length 3, what is the
perimeter?
A. 12
B. 14
C. 21
D. 30
E. 36
Redraw and mark in the information given.
You now have a calculation for the perimeter: 30 plus the darkened parts. Now
look carefully at the top two darkened parts. They will add up to 3. (Notice how
the top square may slide over to illustrate that fact.)
The same is true for the bottom darkened parts. They will add up to 3.
Thus, the total perimeter is 30 + 6 = 36, choice E.
33
3
3
3
33
3
3
3
These together total 3
33
3

3
3
33
3
3
3
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If it appears that extensive calculations are going to be necessary to solve a
problem, check to see how far apart the choices are, and then approximate.
The reason for checking the answers first is to give you a guide for how freely
you can approximate.
Sample
16. The value for (0.889 × 55)/9.97 to the nearest tenth is
A. 0.5
B. 4.63
C. 4.9
D. 7.7
E. 49.1
Before starting any computations, take a glance at the answers to see how far
apart they are. Notice that the only close answers are choices B and C, but B is
not possible because it’s to the nearest hundredth, not tenth. Now, making some
quick approximations, 0.889 ≅ 1 and 9.97 ≅ 10, leaves the problem in this form
.
10
155
10
55
55
#
==

The closest answer is C; therefore, it is the correct answer. Notice that choices A
and E aren’t reasonable.
Some problems may not ask you to solve for a numerical answer or even an
answer including variables. Rather, you may be asked to set up the equation
or expression without doing any solving. A quick glance at the answer choices
will help you know what is expected.
Sample
17. Rick is three times as old as Maria, and Maria is four years older than Leah.
If Leah is z years old, what is Rick’s age in terms of z?
A. 3z + 4
B. 3z − 12
C. 3z + 12
D. (z + 4)/3
E. (z − 4)/3
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