T
ᎏ
r
ᎏ
y
ᎏ
ᎏ
a
ᎏ
ᎏ
s
ᎏ
p
ᎏ
e
ᎏ
c
ᎏ
i
ᎏ
f
ᎏ
i
ᎏ
c
ᎏ
ᎏ
n
ᎏ
u
ᎏ

m
ᎏ
b
ᎏ
e
ᎏ
r
ᎏ
.
ᎏ
Let b ϭ 1. Then a ϭ 4b ϭ 4. So the average ϭ
ᎏ
l ϩ
2
4
ᎏ
ϭ
ᎏ
5
2
ᎏ
.
Look at choices where b ϭ 1. The only choice that gives
ᎏ
5
2
ᎏ
is Choice C.
EXAMPLE 5
The sum of three consecutive even integers is P. Find

the sum of the next three consecutive odd integers that
follow the greatest of the three even integers.
(A) P ϩ 9
(B) P ϩ 15
(C) P ϩ 12
(D) P ϩ 20
(E) None of these.
Choice B is correct.
T
ᎏ
r
ᎏ
y
ᎏ
ᎏ
s
ᎏ
p
ᎏ
e
ᎏ
c
ᎏ
i
ᎏ
f
ᎏ
i
ᎏ
c

ᎏ
ᎏ
n
ᎏ
u
ᎏ
m
ᎏ
b
ᎏ
e
ᎏ
r
ᎏ
s
ᎏ
.
ᎏ
Let the three consecutive even integers be 2, 4, 6.
So, 2 ϩ 4 ϩ 6 ϭ P ϭ 12.
The next three consecutive odd integers that follow 6 are:
7, 9, 11
So the sum of
7 ϩ 9 ϩ 11 ϭ 27.
Now, where P ϭ 12, look for a choice that gives you 27:
(A) P ϩ 9 ϭ 12 ϩ 9 ϭ 21—NO
(B) P ϩ 15 ϭ 12 ϩ 15 ϭ 27—YES
EXAMPLE 6
If 3 Ͼ a, which of the following is not true?
(A) 3 Ϫ 3 Ͼ a Ϫ 3

(B) 3 ϩ 3 Ͼ a ϩ 3
(C) 3(3) Ͼ a(3)
(D) 3 Ϫ 3 Ͼ 3 Ϫ a
(E)
ᎏ
3
3
ᎏ
Ͼ
ᎏ
a
3
ᎏ
Choice D is correct.
T
ᎏ
r
ᎏ
y
ᎏ
ᎏ
s
ᎏ
p
ᎏ
e
ᎏ
c
ᎏ
i

ᎏ
f
ᎏ
i
ᎏ
c
ᎏ
ᎏ
n
ᎏ
u
ᎏ
m
ᎏ
b
ᎏ
e
ᎏ
r
ᎏ
s
ᎏ
.
ᎏ
Work backward from Choice E if you wish.
Let a ϭ 1.
Choice E:
ᎏ
3
3

ᎏ
Ͼ
ᎏ
a
3
ᎏ
ϭ
ᎏ
1
3
ᎏ
TRUE STATEMENT
Choice D:
3 Ϫ 3 Ͼ 3 Ϫ a ϭ 3 Ϫ l or 0 Ͼ 2 FALSE STATEMENT
EXAMPLE 7
In the figure of intersecting lines above, which of the fol-
lowing is equal to 180 Ϫ a?
(A) a ϩ d
(B) a ϩ 2d
(C) c ϩ b
(D) b ϩ 2a
(E) c ϩ d
Choice A is correct.
T
ᎏ
r
ᎏ
y
ᎏ
ᎏ

a
ᎏ
ᎏ
s
ᎏ
p
ᎏ
e
ᎏ
c
ᎏ
i
ᎏ
f
ᎏ
i
ᎏ
c
ᎏ
ᎏ
n
ᎏ
u
ᎏ
m
ᎏ
b
ᎏ
e
ᎏ

r
ᎏ
.
ᎏ
Let
Then 2a ϭ 40°
Be careful now—all of the other angles are now deter -
mined, so don’t choose any more.
Because vertical angles are equal, 2a ϭ b,so
.
Now c ϩ b ϭ 180°, so c ϩ 40 ϭ 180 and
.
Thus, (vertical angles are equal).
Now look at the question:
180 Ϫ a ϭ 180 Ϫ 20 ϭ 160
Which is the correct choice?
(A) a ϩ d ϭ 20 ϩ 140 ϭ 160—that’s the one!
d ϭ 140°
c ϭ 140°
b ϭ 40°
a ϭ 20°
STRATEGY SECTION • 89
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 89
EXAMPLE 1
If p is a positive integer, which could be an odd integer?
(A) 2p ϩ 2
(B) p
3
Ϫ p
(C) p

2
ϩ p
(D) p
2
Ϫ p
(E) 7p Ϫ 3
Choice E is correct. Start with Choice E first since you
have to test out the choices.
Method 1: Try a number for p. Let p ϭ 1. Then (starting
with choice E)
7p Ϫ 3 ϭ 7(1) Ϫ 3 ϭ 4. 4 is even, so try another number
for p to see whether 7p Ϫ 3 is odd. Let p ϭ 2.
7p Ϫ 3 ϭ 7(2) Ϫ 3 ϭ 11. 11 is odd. Therefore, Choice E
is correct.
Method 2: Look at Choice E. 7p could be even or odd,
depending on what p is. If p is even, 7p is even. If p is
odd, 7p is odd. Accordingly, 7p Ϫ 3 is either even or odd.
Thus, Choice E is correct.
Note: By using either Method 1 or Method 2, it is not
necessary to test the other choices.
EXAMPLE 2
If y ϭ x
2
ϩ 3, then for which value of x is y divisible by 7?
(A) 10
(B) 8
(C) 7
(D) 6
(E) 5
Choice E is correct. Since you must check all of the

choices, start with Choice E:
y ϭ 5
2
ϩ 3 ϭ 25 ϩ 3 ϭ 28
28 is divisible by 4 (Answer)
If you had started with Choice A, you would have had to
test four choices, instead of one choice before finding the
correct answer.
EXAMPLE 3
Which fraction is greater than
ᎏ
1
2
ᎏ
?
(A)
ᎏ
4
9
ᎏ
(B)
ᎏ
1
3
7
5
ᎏ
(C)
ᎏ
1

6
3
ᎏ
(D)
ᎏ
1
2
2
5
ᎏ
(E)
ᎏ
1
8
5
ᎏ
Choice E is correct.
L
ᎏ
o
ᎏ
o
ᎏ
k
ᎏ
ᎏ
a
ᎏ
t
ᎏ

ᎏ
C
ᎏ
h
ᎏ
o
ᎏ
i
ᎏ
c
ᎏ
e
ᎏ
ᎏ
E
ᎏ
ᎏ
f
ᎏ
i
ᎏ
r
ᎏ
s
ᎏ
t
ᎏ
.
ᎏ
Is

ᎏ
2
l
ᎏ
Ͼ
ᎏ
1
8
5
ᎏ
?
Use the cross-multiplication method.
ᎏ
1
2
ᎏ
ᎏ
1
8
5
ᎏ
15 16
15 Ͻ 16
So,
ᎏ
1
2
ᎏ
Ͻ
ᎏ

1
8
5
ᎏ
You also could have looked at Choice E and said
ᎏ
1
8
6
ᎏ
ϭ
ᎏ
1
2
ᎏ
and realized that
ᎏ
1
8
5
ᎏ
Ͼ
ᎏ
1
2
ᎏ
because
ᎏ
1
8

5
ᎏ
has a smaller
denominator than
ᎏ
1
8
6
ᎏ
.
90 • STRATEGY SECTION
When Each Choice Must Be Tested, Start with Choice E and
Work Backward
If you must check each choice for the correct answer, start with Choice E and work backward.
The reason for this is that the test maker of a question in which each choice must be tested often
puts the correct answer as Choice D or E. In this way, the careless student must check all or most
of the choices before finding the correct one. So if you’re trying all the choices, start with the last
choice, then the next to last choice, etc.
MATH
STRATEGY
8
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 90
EXAMPLE 4
If n is an even integer, which of the following is an odd
integer?
(A) n
2
Ϫ 2
(B) n Ϫ 4
(C) (n Ϫ 4)

2
(D) n
3
(E) n
2
Ϫ n Ϫ 1
Choice E is correct.
L
ᎏ
o
ᎏ
o
ᎏ
k
ᎏ
ᎏ
a
ᎏ
t
ᎏ
ᎏ
C
ᎏ
h
ᎏ
o
ᎏ
i
ᎏ
c

ᎏ
e
ᎏ
ᎏ
E
ᎏ
ᎏ
f
ᎏ
i
ᎏ
r
ᎏ
s
ᎏ
t
ᎏ
.
ᎏ
n
2
Ϫ n Ϫ 1
If n is even
n
2
is even
n is even
1 is odd
So, n
2

Ϫ n Ϫ 1 ϭ even Ϫ even Ϫ odd ϭ odd.
EXAMPLE 5
Which of the following is an odd number?
(A) 7 ϫ 22
(B) 59 Ϫ 15
(C) 55 ϩ 35
(D) 75Ϭ 15
(E) 4
7
Choice D is correct.
L
ᎏ
o
ᎏ
o
ᎏ
k
ᎏ
ᎏ
a
ᎏ
t
ᎏ
ᎏ
C
ᎏ
h
ᎏ
o
ᎏ

i
ᎏ
c
ᎏ
e
ᎏ
ᎏ
E
ᎏ
ᎏ
f
ᎏ
i
ᎏ
r
ᎏ
s
ᎏ
t
ᎏ
.
ᎏ
4
7
is even, since 4 ϫ 4 ϫ 4 is even
So now look at Choice D:
ᎏ
7
5
5

ᎏ
ϭ 5, which is odd.
EXAMPLE 6
3 Ե 2
ϫ 8
28
ଙ
6
If Ե and
ଙ
are different digits in the correctly calculated
multiplication problem above, then Ե could be
(A) 1
(B) 2
(C) 3
(D) 4
(E) 6
Choice E is correct.
T
ᎏ
r
ᎏ
y
ᎏ
ᎏ
C
ᎏ
h
ᎏ
o

ᎏ
i
ᎏ
c
ᎏ
e
ᎏ
ᎏ
E
ᎏ
ᎏ
f
ᎏ
i
ᎏ
r
ᎏ
s
ᎏ
t
ᎏ
.
ᎏ
3 Ե 232
ϫ 8 ϫ 8
28
ଙ
9289
9 and 6 are different numbers, so Choice E is correct.
EXAMPLE 7

Which choice describes a pair of numbers that are
un equal?
(A)
ᎏ
1
6
ᎏ
,
ᎏ
1
6
1
6
ᎏ
(B) 3.4,
ᎏ
3
1
4
0
ᎏ
(C)
ᎏ
1
7
5
5
ᎏ
,
ᎏ

1
5
ᎏ
(D)
ᎏ
3
8
ᎏ
, 0.375
(E)
ᎏ
8
2
6
4
ᎏ
,
ᎏ
4
1
2
0
ᎏ
Choice E is correct.
L
ᎏ
o
ᎏ
o
ᎏ

k
ᎏ
ᎏ
a
ᎏ
t
ᎏ
ᎏ
C
ᎏ
h
ᎏ
o
ᎏ
i
ᎏ
c
ᎏ
e
ᎏ
ᎏ
E
ᎏ
ᎏ
f
ᎏ
i
ᎏ
r
ᎏ

s
ᎏ
t
ᎏ
.
ᎏ
ᎏ
8
2
6
4
ᎏ
?
ᎏ
4
1
2
0
ᎏ
Cross multiply:
ᎏ
8
2
6
4
ᎏᎏ
4
1
2
0

ᎏ
860 ends in 0 24 ϫ 42 ends in 8
Thus, the numbers must be different and unequal.
EXAMPLE 8
ଙ 3
4
ଙ
ଙ 1
6
ଙ
ଙ 3
2
ෆ
0
ෆ
3
ෆ
In the above addition problem, the symbol
ଙ
describes a
particular digit in each number. What must
ଙ
be in order
to make the answer correct?
(A) 7
(B) 6
(C) 5
(D) 4
(E) 3
Choice E is correct.

Try substituting the number in Choice E first for the
ଙ
.
ଙ 333
4
ଙ 43
ଙ 131
6
ଙ 63
ଙ 333
2
ෆ
0
ෆ
3
ෆ
2
ෆ
0
ෆ
3
ෆ
Since you get 203 for the addition, Choice E is correct.
6
9
STRATEGY SECTION • 91
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 91
92 • STRATEGY SECTION
EXAMPLE 1
The diagram below shows two paths: Path 1 is 10 miles

long, and Path 2 is 12 miles long. If Person X runs along
Path 1 at 5 miles per hour and Person Y runs along Path 2
at y miles per hour, and if it takes exactly the same
amount of time for both runners to run their whole path,
then what is the value of y?
(A) 2
(B) 4
1
/
6
(C) 6
(D) 20
(E) 24
Choice C is correct. Let T ϭ Time (in hours) for either
runner to run the whole path.
Using R ϫ T ϭ D, for Person X, we have
(5 mi/hr)(T hours) ϭ 10 miles
or 5T ϭ 10 or
T ϭ 2
For Person Y, we have
( y mi/hr)(T hours) ϭ 12 miles
or yT ϭ 12
Using y(2) ϭ 12 or y ϭ 6
EXAMPLE 2
A car traveling at 50 miles per hour for two hours travels
the same distance as a car traveling at 20 miles per hour
for x hours. What is x?
(A)
ᎏ
4

5
ᎏ
(B)
ᎏ
5
4
ᎏ
(C) 5
(D) 2
(E)
ᎏ
1
2
ᎏ
Choice C is correct.
U
ᎏ
s
ᎏ
e
ᎏ
ᎏ
R
ᎏ
ᎏ
ϫ
ᎏ
ᎏ
T
ᎏ

ᎏ
ϭ
ᎏ
ᎏ
D
ᎏ
.
ᎏ
Call distance both cars travel, D (since
distance is same for both cars).
So we get:
50 ϫ 2 ϭ D(ϭ100)
20 ϫ x ϭ D(ϭ100)
Solving you can see that x ϭ 5.
EXAMPLE 3
John walks at a rate of 4 miles per hour. Sally walks at a
rate of 5 miles per hour. If both John and Sally both start
at the same starting point, how many miles is one person
from the other after T hours of walking? (Note: Both
are walking on the same road in the same direction.)
(A)
ᎏ
2
t
ᎏ
(B) t
(C) 2t
(D)
ᎏ
4

5
ᎏ
t
(E)
ᎏ
5
4
ᎏ
t
Choice B is correct.
2
2
1
1
1
Know How to Solve Problems Using the Formula R ϫ T ϭ D
Almost every problem involving motion can be solved using the formula
R ϫ T ϭ D
or
rate ϫ elapsed time ϭ distance
MATH
STRATEGY
9
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 92
Draw a diagram:
John (4 mph)
Sally (5 mph)
Let D
J
be distance that John walks in t hours.

Let D
S
be distance that Sally walks in t hours.
Then, using R ϫ T ϭ D,
for John: 4 ϫ t ϭ D
J
for Sally: 5 ϫ t ϭ D
S
The distance between Sally and John after T hours of
walking is:
D
S
Ϫ D
J
ϭ 5t Ϫ 4t ϭ t
EXAMPLE 4
A man rode a bicycle a straight distance at a speed of 10
miles per hour and came back the same distance at a
speed of 20 miles per hour. What was the man’s total
number of miles for the trip back and forth, if his total
traveling time was 1 hour?
(A) 15
(B) 7
1
/
2
(C) 6
1
/
3

(D) 6
2
/
3
(E) 13
1
/
3
Choice E is correct.
Always use R ϫ T ϭ D (Rate ϫ Time ϭ Distance) in
problems like this. Call the first distance D and the time
for the first part, T
1
. Since he rode at 10 mph:
10 ϫ T
1
ϭ D
Now for the trip back. He rode at 20 mph. Call the time it
took to go back, T
2
. Since he came back the same dis-
tance, we can call that distance D also. So for the trip back
using R ϫ T ϭ D, we get:
20 ϫ T
2
ϭ D
Since it was given that the total traveling time was 1
hour, the total traveling time is:
T
1

ϩ T
2
ϭ 1
Now here’s the trick: Let’s make use of the fact that T
1
ϩ
T
2
ϭ 1. Dividing Equation by 10 we get:
T
1
ϭ
ᎏ
1
D
0
ᎏ
Dividing Equation by 20 we get:
T
2
ϭ
ᎏ
2
D
0
ᎏ
Now add T
1
ϩ T
2

and we get:
T
1
ϩ T
2
ϭ 1 ϭ
ᎏ
1
D
0
ᎏ
ϩ
ᎏ
2
D
0
ᎏ
Factor D:
1 ϭ D
΂
ᎏ
1
1
0
ᎏ
ϩ
ᎏ
2
1
0

ᎏ
΃
Add
ᎏ
1
1
0
ᎏ
ϩ
ᎏ
2
1
0
ᎏ
. Remember the fast way of adding
fractions?
ᎏ
1
1
0
ᎏ
ϩ
ᎏ
2
1
0
ᎏ
ϭ
ᎏ
2

2
0
0
ϩ
ϫ
1
1
0
0
ᎏ
ϭ
ᎏ
2
3
0
0
0
ᎏ
So:
1 ϭ (D)
ᎏ
2
3
0
0
0
ᎏ
Multiply by 200 and divide by 30 and we get:
ᎏ
2

3
0
0
0
ᎏ
ϭ D; D ϭ 6
ᎏ
2
3
ᎏ
Don’t forget, we’re looking for 2D: 2D ϭ 13
ᎏ
1
3
ᎏ
EXAMPLE 5
What is the average rate of a bicycle traveling at 10 mph
a distance of 5 miles and at 20 mph the same distance?
(A) 15 mph
(B) 20 mph
(C) 12
1
/
2
mph
(D) 13
1
/
3
mph

(E) 16 mph
Choice D is correct.
Ask yourself, what does average rate mean? It does not
mean the average of the rates! If you thought it did, you
would have selected Choice A as the answer (averaging
10 and 20 to get 15)—the “lure” choice.
Average is a word that modifies the word rate in this case.
So you must define the word rate first, before you do
anything with averaging. Since Rate ϫ Time ϭ Distance,
Rate ϭ
ᎏ
D
T
is
i
t
m
an
e
ce
ᎏ
2
1
2
1
STRATEGY SECTION • 93
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 93
94 • STRATEGY SECTION
Then average rate must be:
Average rate ϭ

ᎏ
TO
T
T
O
A
T
L
AL
dis
ti
t
m
an
e
ce
ᎏ
The total distance is the distance covered on the whole
trip, which is 5 ϩ 5 ϭ 10 miles.
The total time is the time traveled the first 5 miles at 10
mph added to the time the bicycle traveled the next 5
miles at 20 mph.
Let t
1
be the time the bicycle traveled first 5 miles.
Let t
2
be the time the bicycle traveled next 5 miles.
Then the total time ϭ t
1

ϩ t
2
.
Since R ϫ T ϭ D,
for the first 5 miles: 10 ϫ t
1
ϭ 5
for the next 5 miles: 20 ϫ t
2
ϭ 5
Finding t
1
: t
1
ϭ
ᎏ
1
5
0
ᎏ
Finding t
2
: t
2
ϭ
ᎏ
2
5
0
ᎏ

So, t
1
ϩ t
2
ϭ
ᎏ
1
5
0
ᎏ
ϩ
ᎏ
2
5
0
ᎏ
ϭ
ᎏ
1
2
ᎏ
ϩ
ᎏ
1
4
ᎏ
ϭ
ᎏ
4 ϩ
8

2
ᎏ
(remembering how to quickly add
fractions)
ϭ
ᎏ
6
8
ᎏ
ϭ
ᎏ
3
4
ᎏ
Average rate ϭ
ϭ
ϭ (5 ϩ 5) ϫ
ᎏ
4
3
ᎏ
ϭ 10 ϫ
ᎏ
4
3
ᎏ
ϭ
ᎏ
4
3

0
ᎏ
ϭ 13
ᎏ
1
3
ᎏ
(Answer)
Here’s a formula you can memorize:
If a vehicle travels a certain distance at a mph and trav-
els the same distance at b mph, the average rate is
ᎏ
a
2
ϩ
ab
b
ᎏ
.
Try doing the problem using this formula:
ᎏ
a
2
ϩ
ab
b
ᎏ
ϭ
ᎏ
2 ϫ

1
(
0
10
ϩ
) ϫ
20
(20)
ᎏ
ϭ
ᎏ
4
3
0
0
0
ᎏ
ϭ 13
ᎏ
1
3
ᎏ
Caution: Use this formula only when you are looking for
average rate and when the distance is the same for both
speeds.
5 ϩ 5
ᎏ
ᎏ
3
4

ᎏ
TOTAL DISTANCE
ᎏᎏᎏ
TOTAL TIME
Know How to Use Units of Time, Distance, Area, or Volume
to Find or Check Your Answer
EXAMPLE 1
What is the distance in miles covered by a car that trav-
eled at 50 miles per hour for 5 hours?
(A) 10
(B) 45
(C) 55
(D) 200
(E) 250
Choice E is correct. Although this is an easy “R ϫ T ϭ D”
problem, it illustrates this strategy very well.
Recall that
rate ϫ time ϭ distance
(50 mi./hr.)(5 hours)ϭ distance
Notice that when I substituted into R ϫ T ϭ D, I kept the
units of rate and time (miles/hour and hours). Now I will
treat these units as if they were ordinary variables. Thus,
distance ϭ (50 mi./hr.)(5 hours)
By knowing what the units in your answer must be, you will often have an easier time finding or
checking your answer. A very helpful thing to do is to treat the units of time or space as variables
(like “x” or “y”). Thus, you should substitute, multiply, or divide these units as if they were ordi-
nary variables. The following examples illustrate this idea.
MATH
STRATEGY
10

1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:52 PM Page 94
I have canceled the variable “hour(s)” from the numerator
and denominator of the right side of the equation. Hence,
distance ϭ 250 miles
The distance has units of “miles” as I would expect. In
fact, if the units in my answer had been “miles/hour” or
“hours,” then I would have been in error.
Thus, the general procedure or problems using this
strategy is:
Step 1. K
ᎏ
e
ᎏ
e
ᎏ
p
ᎏ
ᎏ
t
ᎏ
h
ᎏ
e
ᎏ
ᎏ
u
ᎏ
n
ᎏ
i

ᎏ
t
ᎏ
s
ᎏ
ᎏ
g
ᎏ
i
ᎏ
v
ᎏ
e
ᎏ
n
ᎏ
ᎏ
i
ᎏ
n
ᎏ
ᎏ
t
ᎏ
h
ᎏ
e
ᎏ
ᎏ
q

ᎏ
u
ᎏ
e
ᎏ
s
ᎏ
t
ᎏ
i
ᎏ
o
ᎏ
n
ᎏ
.
ᎏ
Step 2. T
ᎏ
r
ᎏ
e
ᎏ
a
ᎏ
t
ᎏ
ᎏ
t
ᎏ

h
ᎏ
e
ᎏ
ᎏ
u
ᎏ
n
ᎏ
i
ᎏ
t
ᎏ
s
ᎏ
ᎏ
a
ᎏ
s
ᎏ
ᎏ
o
ᎏ
r
ᎏ
d
ᎏ
i
ᎏ
n

ᎏ
a
ᎏ
r
ᎏ
y
ᎏ
ᎏ
v
ᎏ
a
ᎏ
r
ᎏ
i
ᎏ
a
ᎏ
b
ᎏ
l
ᎏ
e
ᎏ
s
ᎏ
.
ᎏ
Step 3. M
ᎏ

a
ᎏ
k
ᎏ
e
ᎏ
ᎏ
s
ᎏ
u
ᎏ
r
ᎏ
e
ᎏ
ᎏ
t
ᎏ
h
ᎏ
e
ᎏ
ᎏ
a
ᎏ
n
ᎏ
s
ᎏ
w

ᎏ
e
ᎏ
r
ᎏ
ᎏ
h
ᎏ
a
ᎏ
s
ᎏ
ᎏ
u
ᎏ
n
ᎏ
i
ᎏ
t
ᎏ
s
ᎏ
ᎏ
t
ᎏ
h
ᎏ
a
ᎏ

t
ᎏ
ᎏ
y
ᎏ
o
ᎏ
u
ᎏ
ᎏ
w
ᎏ
o
ᎏ
u
ᎏ
l
ᎏ
d
ᎏ
e
ᎏ
x
ᎏ
p
ᎏ
e
ᎏ
c
ᎏ

t
ᎏ
.
ᎏ
EXAMPLE 2
How many inches is equivalent to 2 yards, 2 feet, and 7
inches?
(A) 11
(B) 37
(C) 55
(D) 81
(E) 103
Choice E is correct.
Remember that
1 yard ϭ 3 feet
1 foot ϭ 12 inches
Treat the units of length as variables! Divide by 1
yard, and by 1 foot, to get
1 ϭ
ᎏ
1
3
y
fe
a
e
rd
t
ᎏ
1 ϭ

ᎏ
12
1
in
fo
c
o
h
t
es
ᎏ
We can multiply any expression by 1 and get the same
value. Thus, 2 yards ϩ 2 feet ϩ 7 inches ϭ
(2 yards)(1)(1) ϩ (2 feet)(1) ϩ 7 inches
Substituting and into , 2 yards ϩ 2 feet ϩ 7
inches
ϭ 2 yards
΋
΂
ᎏ
3
ya
fe
r
e
d
΋
t
΋
ᎏ

΃΂
ᎏ
12
f
i
o
n
o
c
t
΋
hes
ᎏ
΃
ϩ2 feet
΋
΂
ᎏ
12
f
i
o
n
o
c
t
΋
hes
ᎏ
΃

ϩ7
inches
ϭ 72 inches ϩ 24 inches ϩ 7 inches
ϭ 103 inches
Notice that the answer is in “inches” as I expected. If the
answer had come out in “yards” or “feet,” then I would
have been in error.
EXAMPLE 3
A car wash cleans x cars per hour, for y hours at z dollars
per car. How much money in cents did the car wash
receive?
(A)
ᎏ
10
xy
0z
ᎏ
(B)
ᎏ
1
x
0
yz
0
ᎏ
(C) 100xyz
(D)
ᎏ
10
yz

0x
ᎏ
(E)
ᎏ
10
yz
0x
ᎏ
Choice C is correct.
Use units:
΂
ᎏ
x
h
c
o
a
u
r
r
΋
s
΋
ᎏ
΃
( y hours
΋
)
΂
ᎏ

z d
c
o
a
ll
r
΋
ars
ᎏ
΃
ϭ xyz dollars
Multiply by 100. We get
100xyz cents.
EXAMPLE 4
There are 3 feet in a yard and 12 inches in a foot. How
many yards are there altogether in 1 yard, 1 foot, and 1
inch?
(A) 1
ᎏ
1
3
ᎏ
(B) 1
ᎏ
1
3
3
6
ᎏ
(C) 1

ᎏ
1
1
1
8
ᎏ
(D) 2
ᎏ
1
5
2
ᎏ
(E) 4
ᎏ
1
1
2
ᎏ
Choice B is correct. Know how to work with units.
Given: 3 feetϭ 1 yard
12 inches ϭ 1 foot
Thus,
1 yard ϩ 1 foot ϩ 1 inch ϭ
1 yard ϩ 1 foot
΂
ᎏ
1
3
y
fe

a
e
rd
t
ᎏ
΃
ϩ
1 inch
΂
ᎏ
12
1
in
fo
c
o
h
t
es
ᎏ
΃
ϭ
΂
ᎏ
1
3
y
fe
a
e

rd
t
ᎏ
΃
ϭ
1 ϩ
ᎏ
1
3
ᎏ
ϩ
ᎏ
3
1
6
ᎏ
yards ϭ
1 ϩ
ᎏ
1
3
2
6
ᎏ
ϩ
ᎏ
3
1
6
ᎏ

yards ϭ
1
ᎏ
1
3
3
6
ᎏ
yards
5
1
1
5
43
4
3
2
1
2
1
STRATEGY SECTION • 95
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96 • STRATEGY SECTION
EXAMPLE 1
If the symbol ␾ is defined by the equation
a ␾ b ϭ a Ϫ b Ϫ ab
for all a and b, then
΂
Ϫ
ᎏ

1
3
ᎏ
΃
␾(Ϫ3) ϭ
(A)
ᎏ
5
3
ᎏ
(B)
ᎏ
1
3
1
ᎏ
(C) Ϫ
ᎏ
1
3
3
ᎏ
(D) Ϫ4
(E) Ϫ5
Choice A is correct. All that is required is substitution:
a ␾ b ϭ a Ϫ b Ϫ ab
΂
Ϫ
ᎏ
1

3
ᎏ
΃
␾ (Ϫ3)
Substitute Ϫ
ᎏ
1
3
ᎏ
for a and
Ϫ3 for b in a Ϫ b Ϫ ab:
΂
Ϫ
ᎏ
1
3
ᎏ
΃
␾(Ϫ3) ϭϪ
ᎏ
1
3
ᎏ
Ϫ (Ϫ3) Ϫ
΂
Ϫ
ᎏ
1
3
ᎏ

΃
(Ϫ3)
ϭϪ
ᎏ
1
3
ᎏ
ϩ 3 Ϫ 1
ϭ 2 Ϫ
ᎏ
1
3
ᎏ
ϭ
ᎏ
5
3
ᎏ
(Answer)
EXAMPLE 2
Let ϭ
Ά
ᎏ
5
2
ᎏ
(x ϩ 1) if x is an odd integer
ᎏ
5
2

ᎏ
x if x is an even integer
Find , where y is an integer.
(A)
ᎏ
5
2
ᎏ
y (B) 5y (C)
ᎏ
5
2
ᎏ
y ϩ 1
(D) 5y ϩ
ᎏ
5
2
ᎏ
(E) 5y ϩ 5
Choice B is correct. All we have to do is to substitute 2y
into the definition of . In order to know which defini-
tion of to use, we want to know if 2y is even. Since y is
an integer, then 2y is an even integer. Thus,
ϭ
ᎏ
5
2
ᎏ
(2y)

or ϭ 5y (Answer)
2y
2y
x
x
2y
x
Use New Definitions and Functions Carefully
Some SAT questions use new symbols, functions, or definitions that were created in the question.
At first glance, these questions may seem difficult because you are not familiar with the new
symbol, function, or definition. However, most of these questions can be solved through simple sub-
stitution or application of a simple definition.
MATH
STRATEGY
11
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 96
EXAMPLE 3
As in the previous Example 1, ø is defined as
a ø b ϭ a Ϫ b Ϫ ab.
If, a ø 3ϭ 6, a ϭ
(A)
ᎏ
9
2
ᎏ
(B)
ᎏ
9
4
ᎏ

(C) Ϫ
ᎏ
9
4
ᎏ
(D) Ϫ
ᎏ
4
9
ᎏ
(E) Ϫ
ᎏ
9
2
ᎏ
Choice E is correct.
a ø b ϭ a Ϫ b Ϫ ab
a ø3ϭ 6
Substitute a for a, 3 for b:
a ø 3 ϭ a Ϫ 3 Ϫ a(3) ϭ 6
ϭ a Ϫ 3 Ϫ 3a ϭ 6
ϭϪ2a Ϫ 3 ϭ 6
2a ϭϪ9
a ϭϪ
ᎏ
9
2
ᎏ
EXAMPLE 4
The symbol is defined as the greatest integer

less than or equal to x.
(A) 16
(B) 16.6
(C) 17
(D) 17.6
(E) 18
Choice C is correct.
is defined as the greatest integer less than or
equal to Ϫ3.4. This is Ϫ4, since Ϫ4 ϽϪ3.4.
is defined as the greatest integer less than or equal
to 21. That is just 21, since 21 ϭ 21.
Thus, Ϫ4 ϩ 21 ϭ 17
EXAMPLE 5
is defined as xz Ϫ yt
ϭ
Choice E is correct.
ϭ xz Ϫ yt; ϭ ?
Substituting 2 for x, 1 for z, 1 for y, and 1 for t,
ϭ (2)(1) Ϫ (1)(1)
ϭ 1
Now work from Choice E:
(E) ϭ xz Ϫ yt ϭ (3)(1) Ϫ (1)(2)
ϭ 3 Ϫ 2 ϭ 1
EXAMPLE 6
If for all numbers a, b, c the operation ᭹ is defined as
a
᭹ b ϭ ab Ϫ a
then
a
᭹ (b ᭹ c) ϭ

(A) a(bc Ϫ b Ϫ 1)
(B) a(bc ϩ b ϩ 1)
(C) a(bc Ϫ c Ϫ b Ϫ 1)
(D) a(bc Ϫ b ϩ 1)
(E) a(b Ϫ a ϩ c)
Choice A is correct.
STRATEGY SECTION • 97
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 97
98 • STRATEGY SECTION
a ᭹ b ϭ ab Ϫ a
a
᭹ (b ᭹ c) ϭ ?
Find (b
᭹ c) first. U
ᎏ
s
ᎏ
e
ᎏ
ᎏ
s
ᎏ
u
ᎏ
b
ᎏ
s
ᎏ
t
ᎏ

i
ᎏ
t
ᎏ
u
ᎏ
t
ᎏ
i
ᎏ
o
ᎏ
n
ᎏ
:
ᎏ
a ᭹ b ϭ ab Ϫ a
↑↑
b
᭹ c
Substitute b for a and c for b:
b
᭹ c ϭ b(c) Ϫ b
Now, a
᭹ (b ᭹ c) ϭ a ᭹ (bc Ϫ b)
Use definition a
᭹ b ϭ ab Ϫ a
Substitute a for a and bc Ϫ b for b:
a
᭹ b ϭ ab Ϫ a

a
᭹ (bc Ϫ b) ϭ a (bc Ϫ b) Ϫ a
ϭ abc Ϫ ab Ϫ a
ϭ a(bc Ϫ b Ϫ 1)
In many of the examples given in these strategies, it has been explicitly stated that one should
not calculate complicated quantities. In some of the examples, we have demonstrated a fast and
a slow way of solving the same problem. On the actual exam, if you find that your solution to a
problem involves a tedious and complicated method, then you are probably doing the problem
in a long, hard way.* Almost always there will be an easier way.
Examples 3, 7, and 8 can also be solved with the aid of a calculator and some with the aid of a
calculator allowing for exponential calculations. However, to illustrate the effectiveness of Math
Strategy 12, we did not use the calculator method of solving these examples.
Try Not to Make Tedious Calculations Since There Is Usually
an Easier Way
EXAMPLE 1
If y
8
ϭ 4 and y
7
ϭ
ᎏ
3
x
ᎏ
,
what is the value of y in terms of x?
(A)
ᎏ
4
3

x
ᎏ
(B)
ᎏ
3
4
x
ᎏ
(C)
ᎏ
4
x
ᎏ
(D)
ᎏ
4
x
ᎏ
(E)
ᎏ
1
x
2
ᎏ
Choice A is correct.
Don’t solve for the value of y first, by finding y ϭ 4
ᎏ
1
8
ᎏ

*Many times, you can DIVIDE, MULTIPLY, ADD, SUBTRACT, or
FACTOR to simplify.
Just divide the two equations:
(Step 1) y
8
ϭ 4
(Step 4) y ϭ 4 ϫ
ᎏ
3
x
ᎏ
(Step 2) y
7
ϭ
ᎏ
3
x
ᎏ
(Step 5) y ϭ
ᎏ
4
3
x
ᎏ
(Answer)
(Step 3)
ᎏ
y
y
8

7
ᎏ
ϭ
EXAMPLE 2
If x ϭ 1 ϩ 2 ϩ 2
2
ϩ 2
3
ϩ 2
4
ϩ 2
5
ϩ 2
6
ϩ 2
7
ϩ 2
8
ϩ 2
9
and y ϭ 1 ϩ 2x, then y Ϫ x ϭ
(A) 2
7
(B) 2
8
(C) 2
9
(D) 2
10
(E) 2

11
Choice D is correct. I hope you did not calculate 1 ϩ 2
ϩ.
2
9
. If you did, then you found that x ϭ 1,023 and
y ϭ 2,047 and y Ϫ x ϭ 1,024.
4
ᎏ
ᎏ
3
x
ᎏ
MATH
STRATEGY
12
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 98
Here is the FAST method. Instead of making these
te dious calculations, observe that since
x ϭ 1 ϩ 2 ϩ 2
2
ϩ 2
3
ϩ 2
4
ϩ 2
5
ϩ 2
6
ϩ 2

7
ϩ 2
8
ϩ 2
9
then 2x ϭ 2 ϩ 2
2
ϩ 2
3
ϩ 2
4
ϩ 2
5
ϩ 2
6
ϩ 2
7
ϩ 2
8
ϩ 2
9
ϩ 2
10
and y ϭ 1 ϩ 2x ϭ 1 ϩ 2 ϩ 2
2
ϩ 2
3
ϩ 2
4
ϩ 2

5
ϩ 2
6
ϩ 2
7
ϩ 2
8
ϩ 2
9
ϩ 2
10
Thus, calculating Ϫ , we get
y Ϫ x ϭ 1 ϩ 2 ϩ 2
2
ϩ 2
3
ϩ 2
4
ϩ 2
5
ϩ 2
6
ϩ 2
7
ϩ 2
8
ϩ 2
9
ϩ 2
10

Ϫ(1 ϩ 2 ϩ 2
2
ϩ 2
3
ϩ 2
4
ϩ 2
5
ϩ 2
6
ϩ 2
7
ϩ 2
8
ϩ 2
9
)
ϭ 2
10
(Answer)
EXAMPLE 3
U
ᎏ
s
ᎏ
e
ᎏ
ᎏ
f
ᎏ

a
ᎏ
c
ᎏ
t
ᎏ
o
ᎏ
r
ᎏ
i
ᎏ
n
ᎏ
g
ᎏ
ᎏ
t
ᎏ
o
ᎏ
ᎏ
m
ᎏ
a
ᎏ
k
ᎏ
e
ᎏ

ᎏ
p
ᎏ
r
ᎏ
o
ᎏ
b
ᎏ
l
ᎏ
e
ᎏ
m
ᎏ
s
ᎏ
ᎏ
s
ᎏ
i
ᎏ
m
ᎏ
p
ᎏ
l
ᎏ
e
ᎏ

r
ᎏᎏ
.
͙(88)
2
ϩ
ෆ
(88)
2
ෆ
(3)
ෆ
ϭ
(A) 88 (B) 176 (C) 348 (D) 350 (E) 352
Choice B is correct. Factor:
(88)
2
ϩ (88)
2
(3) ϭ 88
2
(1 ϩ 3) ϭ 88
2
(4)
So:
͙(88)
2
ϩ
ෆ
(88)

2
ෆ
(3)
ෆ
ϭ ͙88
2
(4)
ෆ
ϭ ͙88
2
ෆ
ϫ ͙4
ෆ
ϭ 88 ϫ 2
ϭ 176
EXAMPLE 4
If 16r Ϫ 24q ϭ 2, then 2r Ϫ 3q ϭ
(A)
ᎏ
1
8
ᎏ
(B)
ᎏ
1
4
ᎏ
(C)
ᎏ
1

2
ᎏ
(D) 2
(E) 4
Choice B is correct.
D
ᎏ
i
ᎏ
v
ᎏ
i
ᎏ
d
ᎏ
e
ᎏ
ᎏ
b
ᎏ
y
ᎏ
ᎏ
8
ᎏ
:
ᎏ
ᎏ
16r Ϫ
8

24q
ᎏ
ϭ
ᎏ
2
8
ᎏ
2r Ϫ 3q ϭ
ᎏ
1
4
ᎏ
EXAMPLE 5
If (a
2
ϩ a)
3
ϭ (a ϩ 1)
3
x, where a ϩ 1  0, then x ϭ
(A) a
(B) a
2
(C) a
3
(D)
ᎏ
a ϩ
a
1

ᎏ
(E)
ᎏ
a ϩ
a
1
ᎏ
Choice C is correct.
Isolate x first:
x ϭ
ᎏ
(
(
a
a
2
ϩ
ϩ
1
a
)
)
3
3
ᎏ
Now use the fact that
΂
ᎏ
x
y

3
3
ᎏ
΃
ϭ
΂
ᎏ
x
y
ᎏ
΃
3
:
ᎏ
(
(
a
a
2
ϩ
ϩ
1
a
)
)
3
3
ᎏ
ϭ
΂

ᎏ
a
a
2
ϩ
ϩ
1
a
ᎏ
΃
3
Now f
ᎏ
a
ᎏ
c
ᎏ
t
ᎏ
o
ᎏ
r
ᎏ
a
2
ϩ a ϭ a(a ϩ 1)
So:
΂
ᎏ
a

a
2
ϩ
ϩ
1
a
ᎏ
΃
3
ϭ
΄
ᎏ
a(
a
a
ϩ
ϩ
1
1)
ᎏ
΅
3
ϭ
΄
ᎏ
a(
a
a
ϩ
ϩ

1
΋
1)
΋
ᎏ
΅
3
ϭ a
3
EXAMPLE 6
If
ᎏ
p
r ϩ
ϩ
1
1
ᎏ
ϭ 1 and p, r are nonzero, and p is not equal to
Ϫ1, and r is not equal to Ϫ1, then
(A) 2 Ͼ p/r Ͼ l always
(B) p/r Ͻ 1 always
(C) p/r ϭ 1 always
(D) p/r can be greater than 2
(E) p/r ϭ 2 always
Choice C is correct.
Get rid of the fraction. M
ᎏ
u
ᎏ

l
ᎏ
t
ᎏ
i
ᎏ
p
ᎏ
l
ᎏ
y
ᎏ
both sides of the equation
ᎏ
p
r ϩ
ϩ
1
1
ᎏ
ϭ 1 by r ϩ 1!
΂
ᎏ
p
r
ϩ
ϩ
1
1
΋

ᎏ
΃
r ϩ 1
΋
ϭ r ϩ 1
p ϩ 1 ϭ r ϩ 1
Cancel the 1’s:
p ϭ r
So:
ᎏ
p
r
ᎏ
ϭ 1
13
3
2
1
STRATEGY SECTION • 99
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100 • STRATEGY SECTION
EXAMPLE 7
ᎏ
2
4
50
ᎏ
ϭ
(A) 0.16
(B) 0.016

(C) 0.0016
(D) 0.00125
(E) 0.000125
Choice B is correct.
D
ᎏ
o
ᎏ
n
ᎏ
’
ᎏ
t
ᎏ
ᎏ
d
ᎏ
i
ᎏ
v
ᎏ
i
ᎏ
d
ᎏ
e
ᎏ
4 into 250! M
ᎏ
u

ᎏ
l
ᎏ
t
ᎏ
i
ᎏ
p
ᎏ
l
ᎏ
y
ᎏ
:
ᎏ
ᎏ
2
4
50
ᎏ
ϫ
ᎏ
4
4
ᎏ
ϭ
ᎏ
1,
1
0

6
00
ᎏ
Now
ᎏ
1
1
0
6
0
ᎏ
ϭ .16, so
ᎏ
1,
1
0
6
00
ᎏ
ϭ .016.
EXAMPLE 8
(3 ϫ 4
14
) Ϫ 4
13
ϭ
(A) 4
(B) 12
(C) 2 ϫ 4
13

(D) 3 ϫ 4
13
(E) 11 ϫ 4
13
Choice E is correct.
Factor
4
13
from
(3 ϫ 4
14
) Ϫ 4
13
We get 4
13
[(3 ϫ 4
1
) Ϫ 1]
or 4
13
[12 Ϫ 1] ϭ 4
13
[11]
You will see more of the technique of dividing, multiply-
ing, adding, and subtracting in the next strategy, MATH
STRATEGY 13.
Know How to Find Unknown Expressions by Adding,
Subtracting, Multiplying, or Dividing Equations or Expressions
When you want to calculate composite quantities like x ϩ 3y or m Ϫ n, often you can do it by
adding, subtracting, multiplying, or dividing the right equations or expressions.

EXAMPLE 1
If 4x ϩ 5y ϭ 10 and x ϩ 3y ϭ 8,
then
ᎏ
5x ϩ
3
8y
ᎏ
ϭ
(A) 18
(B) 15
(C) 12
(D) 9
(E) 6
Choice E is correct. Don’t solve for x, then for y.
Try to get the quantity
ᎏ
5x ϩ
3
8y
ᎏ
by adding or subtracting
the equations. In this case, a
ᎏ
d
ᎏ
d
ᎏ
equations.
4x ϩ 5y ϭ 10

ϩ x ϩ 3y ϭ 8
5x ϩ 8y ϭ 18
Now divide by 3:
ᎏ
5x ϩ
3
8y
ᎏ
ϭ
ᎏ
1
3
8
ᎏ
ϭ 6 (Answer)
EXAMPLE 2
If 25x ϩ 8y ϭ 149 and 16x ϩ 3y ϭ 89, then
ᎏ
9x ϩ
5
5y
ᎏ
ϭ
(A) 12
(B) 15
(C) 30
(D) 45
(E) 60
Choice A is correct. We are told
25x ϩ 8y ϭ 149

16x ϩ 3y ϭ 89
The long way to do this problem is to solve and
for x and y, and then substitute these values into
ᎏ
9x ϩ
5
5y
ᎏ
The fast way to do this problem is to subtract from
and get
9x ϩ 5y ϭ 60
Now all we have to do is to divide by 5
ᎏ
9x ϩ
5
5y
ᎏ
ϭ 12 (Answer)
3
3
1
2
1 2
2
1
MATH
STRATEGY
13
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 100
EXAMPLE 3

If 21x ϩ 39y ϭ 18, then 7x ϩ 13y ϭ
(A) 3
(B) 6
(C) 7
(D) 9
(E) It cannot be determined from the information given.
Choice B is correct. We are given
21x ϩ 39y ϭ 18
Divide by 3:
7x ϩ 13y ϭ 6 (Answer)
EXAMPLE 4
If x ϩ 2y ϭ 4, then 5x ϩ 10y Ϫ 8 ϭ
(A) 10
(B) 12
(C) Ϫ10
(D) Ϫ12
(E) 0
Choice B is correct.
Multiply
x ϩ 2y ϭ 4 by 5 to get:
5x ϩ 10y ϭ 20
Now subtract 8:
5x ϩ 10y Ϫ 8 ϭ 20 Ϫ 8
ϭ 12
EXAMPLE 5
If 6x
5
ϭ y
2
and x ϭ

ᎏ
1
y
ᎏ
, then y ϭ
(A) x
6
(B)
ᎏ
x
6
5
ᎏ
(C) 6x
6
(D)
ᎏ
6
5
x
5
ᎏ
(E)
ᎏ
x
5
5
ᎏ
Choice C is correct.
Multiply

6x
5
ϭ y
2
by x ϭ
ᎏ
1
y
ᎏ
to get:
6x
6
ϭ y
2
ϫ
ᎏ
1
y
ᎏ
ϭ y
EXAMPLE 6
If x Ͼ 0, y Ͼ 0 and x
2
ϭ 27 and y
2
ϭ 3, then
ᎏ
x
y
3

3
ᎏ
ϭ
(A) 9
(B) 27
(C) 36
(D) 48
(E) 54
Choice B is correct.
DD
ᎏ
i
ᎏ
v
ᎏ
i
ᎏ
d
ᎏ
e
ᎏ
:
ᎏ
ᎏ
x
y
2
2
ᎏ
ϭ

ᎏ
2
3
7
ᎏ
ϭ 9
Take square root:
ᎏ
x
y
ᎏ
ϭ 3
So
΂
ᎏ
x
y
ᎏ
΃
3
ϭ
ᎏ
x
y
3
3
ᎏ
ϭ 3
3
ϭ 27

EXAMPLE 7
If
ᎏ
m
n
ᎏ
ϭ
ᎏ
3
8
ᎏ
and
ᎏ
m
q
ᎏ
ϭ
ᎏ
4
7
ᎏ
, then
ᎏ
n
q
ᎏ
ϭ
(A)
ᎏ
1

1
2
5
ᎏ
(B)
ᎏ
1
5
2
6
ᎏ
(C)
ᎏ
5
1
6
2
ᎏ
(D)
ᎏ
3
2
2
1
ᎏ
(E)
ᎏ
2
3
1

2
ᎏ
Choice D is correct.
First get rid of fractions!
Cross-multiply
ᎏ
m
n
ᎏ
ϭ
ᎏ
3
8
ᎏ
to get 8m ؍ 3n.
Now cross-multiply
ᎏ
m
q
ᎏ
ϭ
ᎏ
4
7
ᎏ
to get 7m ؍ 4q.
Now divide equations and :
ᎏ
8
7

m
m
ᎏ
ϭ
ᎏ
3
4
n
q
ᎏ
The m’s cancel and we get:
ᎏ
8
7
ᎏ
ϭ
ᎏ
3
4
n
q
ᎏ
4
3
1 2
2
1
1
1
STRATEGY SECTION • 101

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102 • STRATEGY SECTION
Multiply Equation by 4 and divide by 3 to get
ᎏ
8
7
ϫ
ϫ
4
3
ᎏ
ϭ
ᎏ
n
q
ᎏ
.
Thus
ᎏ
n
q
ᎏ
ϭ
ᎏ
3
2
2
1
ᎏ
.

EXAMPLE 8
If
ᎏ
a ϩ b ϩ
4
c ϩ d
ᎏ
ϭ 20
And
ᎏ
b ϩ
3
c ϩ d
ᎏ
ϭ 10
Then a ϭ
(A) 50
(B) 60
(C) 70
(D) 80
(E) 90
Choice A is correct.
We have
ᎏ
a ϩ b ϩ
4
c ϩ d
ᎏ
ϭ 20
ᎏ

b ϩ
3
c ϩ d
ᎏ
ϭ 10
Multiply Equation by 4:
We get: a ϩ b ϩ c ϩ d ϭ 80
Now multiply equation by 3:
We get: b ϩ c ϩ d ϭ 30
Now subtract
Equation from Equation :
a ϩ b ϩ c ϩ d ϭ 80
Ϫ (b ϩ c ϩ d ϭ 30)
We get a ϭ 50.
4
3
4
3
4 3
2
1
2
1
4
Draw or Extend Lines in a Diagram to Make a Problem Easier;
Label Unknown Quantities
EXAMPLE 1
The circle with center A and radius AB is inscribed in the
square to the left. AB is extended to C. What is the ratio
of AB to AC ?

(A)
͙2
ෆ
(B)
ᎏ
͙
4
2
ෆ
ᎏ
(C)
ᎏ
͙
2
ෆ
2
Ϫ 1
ᎏ
(D)
ᎏ
͙
2
2
ෆ
ᎏ
(E) None of these.
Choice D is correct. Always draw or extend lines to get
more information. Also label unknown lengths, angles,
or arcs with letters.
Label AB ϭ a and BC ϭ b.

Draw perpendicular AD. Note it is just the radius, a. CD
also ϭ a, because each side of the square is length 2a (the
diameter) and CD is
ᎏ
1
2
ᎏ
the side of the square.
We want to find
ᎏ
A
A
B
C
ᎏ
ϭ
ᎏ
a ϩ
a
b
ᎏ
Now ⌬ADC is an isosceles right triangle so
AD ϭ CD ϭ a.
By the Pythagorean Theorem,
a
2
ϩ a
2
ϭ (a ϩ b)
2

where a ϩ b is hypotenuse of right
triangle.
We get: 2a
2
ϭ (a ϩ b)
2
Divide by (a ϩ b)
2
:
ᎏ
(a
2
ϩ
a
2
b)
2
ᎏ
ϭ 1
MATH
STRATEGY
14
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 102
Divide by 2:
ᎏ
(a ϩ
a
2
b)
2

ᎏ
ϭ
ᎏ
1
2
ᎏ
Take square roots of both sides:
ᎏ
(a ϩ
a
b)
ᎏ
ϭ
ᎏ
͙
1
2
ෆ
ᎏ
ϭ
ϭ
ᎏ
͙
1
2
ෆ
ᎏ
΂
ᎏ
͙

͙
2
ෆ
2
ෆ
ᎏ
΃
ϭ
ᎏ
͙
2
2
ෆ
ᎏ
(Answer)
EXAMPLE 2
What is the perimeter of the above figure if B and C are
right angles?
(A) 14
(B) 16
(C) 18
(D) 20
(E) Cannot be determined.
Choice C is correct.
Draw perpendicular AE. Label side BC ϭ h. You can see
that AE ϭ h.
ABCE is a rectangle, so CE ϭ 3. This makes ED ϭ 3 since
the whole DC ϭ 6.
Now use the Pythagorean Theorem for triangle AED:
h

2
ϩ 3
2
ϭ 5
2
h
2
ϭ 5
2
Ϫ 3
2
h
2
ϭ 25 Ϫ 9
h
2
ϭ 16
h ϭ 4
So the perimeter is 3 ϩ h ϩ 6 ϩ 5 ϭ 3 ϩ 4 ϩ 6 ϩ 5 ϭ 18
(Answer)
EXAMPLE 3
In the figure above, O is the center of a circle with a
radius of 6, and AOCB is a square. If point B is on the cir-
cumference of the circle, the length of AC ϭ
(A) 6
͙2
ෆ
(B) 3 ͙2
ෆ
(C) 3

(D) 6
(E) 6
͙3
ෆ
Choice D is correct.
This is tricky if not impossible if you don’t draw OB. S
ᎏ
o
ᎏ
draw OB:
Since AOCB is a square, OB ϭ AC; and since OB ϭ
radius ϭ 6, AC ϭ 6.
EXAMPLE 4
Lines ᐉ
1
and ᐉ
2
are parallel. AB ϭ
ᎏ
1
3
ᎏ
AC.
ϭ
The area of triangle ABD
ᎏᎏᎏ
The area of triangle DBC
STRATEGY SECTION • 103
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104 • STRATEGY SECTION

(A)
ᎏ
1
4
ᎏ
(B)
ᎏ
1
3
ᎏ
(C)
ᎏ
3
8
ᎏ
(D)
ᎏ
1
2
ᎏ
(E) Cannot be determined.
Choice D is correct.
AB ϭ
ᎏ
1
3
ᎏ
AC
Ask yourself, what is the area of a triangle? It is
1

⁄
2
(height ϫ base). So let’s get the heights and the bases of
the
triangles ABD and DBC. First d
ᎏ
r
ᎏ
a
ᎏ
w
ᎏ
ᎏ
t
ᎏ
h
ᎏ
e
ᎏ
ᎏ
a
ᎏ
l
ᎏ
t
ᎏ
i
ᎏ
t
ᎏ

u
ᎏ
d
ᎏ
e
ᎏ
(call it h).
AB ϩ BC ϭ AC
Thus the area of ⌬ABD ϭ
ᎏ
1
2
ᎏ
h (AB) ϭ
ᎏ
1
2
ᎏ
h
΂
ᎏ
1
3
ᎏ
AC
΃
Area of ⌬DBC ϭ
ᎏ
1
2

ᎏ
h (BC) ϭ
ᎏ
1
2
ᎏ
h
΂
ᎏ
2
3
ᎏ
AC
΃
ϭ
ϭϭ
ᎏ
1
3
ᎏ
ϫ
ᎏ
3
2
ᎏ
ϭ
ᎏ
1
2
ᎏ

EXAMPLE 5
(Note: Figure is not drawn to scale.)
The area of the above figure ABCD
(A) is 36
(B) is 108
(C) is 156
(D) is 1,872
(E) Cannot be determined.
Choice A is correct.
DD
ᎏ
r
ᎏ
a
ᎏ
w
ᎏ
BD. BCD is a 3-4-5 right triangle, so BD ϭ 5. Now
remember that a 5-12-13 triangle is also a right triangle,
so angle ABD is a right angle. The area of triangle BCD is
(3 ϫ 4)/2 ϭ 6 and the area of triangle BAD is (5 ϫ 12)/2 ϭ
30, so the total area is 36.
EXAMPLE 6
In the above figure, two points, B and C, are placed to the
right of point A such that 4AB ϭ 3AC. The value of
ᎏ
A
B
B
C

ᎏ
(A) equals
ᎏ
1
3
ᎏ
(B) equals
ᎏ
2
3
ᎏ
(C) equals
ᎏ
3
2
ᎏ
(D) equals 3
(E) Cannot be determined.
Choice A is correct.
ᎏ
1
3
ᎏ
ᎏ
ᎏ
2
3
ᎏ
Area of ABD
ᎏᎏ

Area of DBC
ᎏ
1
2
ᎏ
h
΂
ᎏ
1
3
ᎏ
AC
΃
ᎏᎏ
ᎏ
1
2
ᎏ
h
΂
ᎏ
2
3
ᎏ
AC
΃
Now label AB ϭ
ᎏ
1
3

ᎏ
AC (given)
ᎏᎏᎏᎏ
This makes BC ϭ
ᎏ
2
3
ᎏ
AC, since
1FM-pg116.qxd:22678_0000-FM.qxd 5/1/08 3:53 PM Page 104
Place B and C to the right of A:
Now label AB ϭ a and BC ϭ b:
ᎏ
A
B
B
C
ᎏ
ϭ
ᎏ
a
b
ᎏ
΂
ᎏ
a
b
ᎏ
is what we want to find
΃

We are given 4AB ϭ 3AC
So, 4a ϭ 3(a ϩ b).
Expand: 4a ϭ 3a ϩ 3b
Subtract 3a: a ϭ 3b
Divide by 3 and a:
ᎏ
1
3
ᎏ
ϭ
ᎏ
a
b
ᎏ
But remember
ᎏ
A
B
B
C
ᎏ
ϭ
ᎏ
a
b
ᎏ
, so
ᎏ
A
B

B
C
ᎏ
ϭ
ᎏ
1
3
ᎏ
EXAMPLE 7
In the figure above, ABCDE is a pentagon inscribed in
the circle with center at O. ЄDOC ϭ 40°. What is the
value of x ϩ y?
(A) 80
(B) 100
(C) 180
(D) 200
(E) Cannot be determined.
Choice D is correct.
Label degrees in each arc.
ր
Єx is measured by
1
⁄
2
arc it cuts.
So, x ϭ
ᎏ
1
2
ᎏ

(b ϩ a ϩ 40)
Likewise, y ϭ
ᎏ
1
2
ᎏ
(c ϩ d ϩ 40)
You want to find x ϩ y, so add:
x ϭ
ᎏ
1
2
ᎏ
(b ϩ a ϩ 40)
y ϭ
ᎏ
1
2
ᎏ
(c ϩ d ϩ 40)
x ϩ y ϭ
ᎏ
1
2
ᎏ
(b ϩ a ϩ 40 ϩ c ϩ d ϩ 40)
But what is a ϩ b ϩ c ϩ d ϩ 40? It is the total number of
degrees around the circumference, which is 360.
So, x ϩ y ϭ
ᎏ

1
2
ᎏ
((b ϩ a ϩ c+ϩ d ϩ40ϩ 40)
ϭ
ᎏ
1
2
ᎏ
(360ϩ 40 )
ϭ
ᎏ
1
2
ᎏ
(400)ϭ 200
EXAMPLE 8
In the above figure, if ЄABE ϭ 40°, ЄDBC ϭ 60°, and
ЄABC ϭ 90°, what is the measure of ЄDBE?
(A) 10°
(B) 20°
(C) 40°
(D) 100°
(E) Cannot be determined.
Choice A is correct.
Label angles first.
Now ЄABE ϭ 40, so a ϩ b ϭ 40
ЄDBC ϭ 60, so b ϩ c ϭ 60
ЄABC ϭ 90, so a ϩ b ϩ c ϭ 90
Ά

Ά
STRATEGY SECTION • 105
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106 • STRATEGY SECTION
You want to find ЄDBE. ЄDBE ϭ b and you want to get
the value of b from:
a ϩ b ϭ 40
b ϩ c ϭ 60
a ϩ b ϩ c ϭ 90
Add and : a ϩ b ϭ 40
ϩ b ϩ c ϭ 60
a ϩ 2b ϩ c ϭ 100
Subtract Ϫ (a ϩ b ϩ c ϭ 90)
b ϭ 10
EXAMPLE 9
In the figure above, three lines intersect at the points
shown. What is the value of A ϩ B ϩ C ϩ D ϩ E ϩ F?
(A) 1,080
(B) 720
(C) 540
(D) 360
(E) Cannot be determined.
Choice B is correct.
(B) R
ᎏ
e
ᎏ
l
ᎏ
a

ᎏ
b
ᎏ
e
ᎏ
l
ᎏ
,
ᎏ
using the fact that vertical angles are equal.
Now use the fact that a straight angle has 180° in it:
Now use the fact that the sum of the angles of a triangle ϭ
180°:
180 Ϫ a ϩ 180 Ϫ b ϩ 180 Ϫ c ϭ 180
540 Ϫ a Ϫ b Ϫ c ϭ 180
540 Ϫ 180 ϭ a ϩ b ϩ c
360 ϭ a ϩ b ϩ c
Now remember what we are looking to find (the sum):
a ϩ a ϩ b ϩ b ϩ c ϩ c ϭ 2a ϩ 2b ϩ 2c
But this is just 2(a ϩ b ϩ c) ϭ 2(360) ϭ 720
3
2
1
3
2
1
Know How to Eliminate Certain Choices
Instead of working out a lot of algebra, you may be able to eliminate several of the choices at
first glance. In this way you can save yourself a lot of work. The key is to remember to use
pieces of the given information to eliminate several of the choices at once.

EXAMPLE 1
The sum of the digits of a three-digit number is 15. If
this number is not divisible by 2 but is divisible by 5,
which of the following is the number?
(A) 384
(B) 465
(C) 635
(D) 681
(E) 780
MATH
STRATEGY
15
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Choice B is correct. Use pieces of the given information
to eliminate several of the choices.
Which numbers are divisible by 2? Choices A and E
are divisible by 2 and, thus, can be eliminated. Of Choices
B, C, and D, which are not divisible by 5? Choice D can
be eliminated. We are left with Choices B and C.
Only Choice B (465) has the sum of its digits equal
to 15. Thus, 465 is the only number that satisfies all the
pieces of the given information.
If you learn to use this method well, you can save
loads of time.
EXAMPLE 2
Which of the following numbers is divisible by 5 and 9,
but not by 2?
(A) 625
(B) 639
(C) 650

(D) 655
(E) 675
Choice E is correct. Clearly, a number is divisible by 5
if, and only if, its last digit is either 0 or 5. A number is
also divisible by 2 if, and only if, its last digit is divisible
by 2. Certain choices are easily eliminated. Thus we can
eliminate Choices B and C.
Method 1: To eliminate some more choices, remember
that a number is divisible by 9 if, and only if, the sum of
its digits is divisible by 9. Thus, Choice E is the only
correct answer.
Method 2: If you did not know the test for divisibility
by 9, divide the numbers in Choices A, D, and E by 9 to
find the answer.
EXAMPLE 3
If the last digit and the first digit are interchanged in
each of the numbers below, which will result in the num-
ber with the largest value?
(A) 5,243
(B) 4,352
(C) 4,235
(D) 2,534
(E) 2,345
Choice E is correct.
The numbers with the largest last digit will become the
largest numbers after interchanging.
Certain choices are easily eliminated.
Using , we see that Choices B and E each end in 5. All
others end in digits less than 5 and may be eliminated.
Starting with Choice E (See Strategy 8).

Choice E, 2,345, becomes 5,342.
Choice B, 4,235, becomes 5,234.
is larger than .
EXAMPLE 4
Which of the following could be the value of 3
x
where x is
an integer?
(A) 339,066
(B) 376,853
(C) 411,282
(D) 422,928
(E) 531,441
Choice E is correct. Let’s look at what 3
x
looks like for
integral values of x:
3
1
ϭ 3
3
2
ϭ 9
3
3
ϭ 27
3
4
ϭ 81
3

5
ϭ 243
3
6
ϭ . . . 9
3
7
ϭ . . . 7
3
8
ϭ . . . 1
Note that 3
x
always has the units digit ending in 3, 9, 7,
or 1. So we can eliminate choices A, C, and D since the
units digits in those choices end in other numbers than
3, 9, 7, or 1. We are left with Choices B and E. The num-
ber in the correct choice must be exactly divisible by 3
since it is of the form 3
x
(ϭ 3 ϫ 3 ϫ 3 . . .) where x is an
integer. This is a good time to use your calculator.
Divide the number in choice B by 3: You get 125,617.66.
That’s not an integer. So the only remaining choice is
Choice E.
32
3
2
1
1

STRATEGY SECTION • 107
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108 • STRATEGY SECTION
Watch Out for Questions That Seem Very Easy But That Can
Be Tricky—Beware of Choice A as a “Lure Choice”
EXAMPLE 1*
The diagram above shows a 12-hour digital clock whose
hour digit is the same as the minutes digit. Consider each
time when the same number appears for both the hour and
the minutes as a “double time” situation. What is the short-
est elapsed time period between the appearance of one
double time and an immediately succeeding double time?
(A) 61 minutes
(B) 60 minutes
(C) 58 minutes
(D) 50 minutes
(E) 49 minutes
Choice E is correct. Did you think that just by subtracting
something like 8:08 from 9:09 you would get the answer
(1 hour and 1 minute ϭ 61 minutes)? That’s Choice A,
which is wrong. So beware, because your answer came
too easily for a test like the SAT. You must realize that
there is another possibility of double time occurrence—
12:12 and 1:01, whose difference is 49 minutes. This is
Choice E, the correct answer.
EXAMPLE 2
The letters d and m are integral digits in a certain num-
ber system. If 0 Յ d Յ m, how many different possible
values are there for d ?
(A) m

(B) m Ϫ 1
(C) m Ϫ 2
(D) m ϩ 1
(E) m ϩ 2
Choice D is correct. Did you think that the answer was m?
Do not be careless! The list 1,2,3, , m contains m
*Note: This problem also appears in Strategy 1 of the 5 General
Strategies on page 60.
elements. If 0 is included in the list, then there are m ϩ 1
elements. Hence, if 0 Յ d Յ m where d is integral, then d
can have m ϩ 1 different values.
EXAMPLE 3
There are some flags hanging in a horizontal row. Starting
at one end of the row, the U.S. flag is 25th. Starting at
the other end of the row, the U.S. flag is 13th. How many
flags are in the row?
(A) 36
(B) 37
(C) 38
(D) 39
(E) 40
Choice B is correct. The obvious may be tricky!
Method 1: Given:
The U.S. flag is 25th from one end.
The U.S. flag is 13th from the other end.
At first glance it may appear that adding and ,
25 ϩ 13 ϭ 38, will be the correct answer. This is WRONG!
The U.S. flag is being counted twice: Once as the
25th and again as the 13th from the other end. The cor-
rect answer is

25 ϩ 13 Ϫ 1 ϭ 37.
Method 2:
24 ϩ 12 ϩ U.S. flag ϭ 36 ϩ U.S. flag ϭ 37
1 2
2
1
6:06
When questions appear to be solved very easily, think again! Watch out especially for the “lure”
Choice A.
MATH
STRATEGY
16
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STRATEGY SECTION • 109
EXAMPLE 4
OR ϭ RQ in the figure above. If the coordinates of Q are
(5,m), find the value of m.
(A) Ϫ5
(B) Ϫ͙5
ෆ
(C) 0
(D) ͙5
ෆ
(E) 5
Choice A is correct.
Given: OR ϭ RQ
Coordinates of Q ϭ (5,m)
From , we get RQ ϭ 5
Substitute into . We get
OR ϭ 5

The obvious may be tricky! Since Q is below the x-
axis, its y-coordinate is negative. Thus m ϭϪ5.
3 1
2 3
2
1
Use the Given Information Effectively (and Ignore Irrelevant
Information)
EXAMPLE 1
(Note: Figure is not drawn to scale.)
In the figure above, side BC of triangle ABC is extended
to D. What is the value of a?
(A) 15
(B) 17
(C) 20
(D) 24
(E) 30
Choice C is correct.
Use the piece of information that will give you something
definite. You might have first thought of using the fact
that the sum of the angles of a triangle ϭ 180°. However,
that will give you
a ϩ 2y ϩ 6y ϭ 180
That’s not very useful. However, if you use the fact that
the sum of the angles in a straight angle is 180 we get:
6y ϩ 3y ϭ 180
and we get 9y ϭ 180
y ϭ 20
Now we have gotten something useful. At this point, we
can use the fact that the sum of the angles in a triangle is

180.
a ϩ 2y ϩ 6y ϭ 180
Substituting 20 for y, we get
a ϩ 2(20) ϩ 6(20) ϭ 180
a ϭ 20 (Answer)
You should always use first the piece of information that tells you the most, or gives you a use-
ful idea, or that brings you closest to the answer.
MATH
STRATEGY
17
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110 • STRATEGY SECTION
EXAMPLE 2
(Note: Figure is not drawn to scale.)
Which of the above angles has a degree measure that
can be determined?
(A) ЄWOS
(B) ЄSOU
(C) ЄWOT
(D) ЄROV
(E) ЄWOV
Choice C is correct.
U
ᎏ
s
ᎏ
e
ᎏ
ᎏ
i

ᎏ
n
ᎏ
f
ᎏ
o
ᎏ
r
ᎏ
m
ᎏ
a
ᎏ
t
ᎏ
i
ᎏ
o
ᎏ
n
ᎏ
ᎏ
t
ᎏ
h
ᎏ
a
ᎏ
t
ᎏ

ᎏ
w
ᎏ
i
ᎏ
l
ᎏ
l
ᎏ
ᎏ
g
ᎏ
e
ᎏ
t
ᎏ
ᎏ
y
ᎏ
o
ᎏ
u
ᎏ
ᎏ
s
ᎏ
o
ᎏ
m
ᎏ

e
ᎏ
t
ᎏ
h
ᎏ
i
ᎏ
n
ᎏ
g
ᎏ
ᎏ
u
ᎏ
s
ᎏ
e
ᎏ
f
ᎏ
u
ᎏ
l
ᎏ
.
ᎏ
4a ϩ 2b ϭ 360 (sum of all angles ϭ 360°)
Divide by 2 to simplify:
2a ϩ b ϭ 180

Now try all the choices. You could work backward from
Choice E, but we’ll start with Choice A:
(A) ЄWOS ϭ 2a—You know that 2a ϩ b ϭ 180 but
don’t know the value of 2a.
(B) ЄSOU ϭ b ϩ a—You know 2a ϩ b ϭ 180 but don’t
know the value of b ϩ a.
(C) ЄWOT ϭ b ϩ 2a—You know that 2a ϩ b ϭ 180, so
you know the value of b ϩ 2a.
Choice C is correct.
EXAMPLE 3
If a ranges in value from 0.003 to 0.3 and b ranges in
value from 3.0 to 300.0, then the minimum value of
ᎏ
a
b
ᎏ
is
(A) 0.1
(B) 0.01
(C) 0.001
(D) 0.0001
(E) 0.00001
Choice E is correct.
Start by using the definition of minimum and maximum.
The minimum value of
ᎏ
a
b
ᎏ
is when a is minimum and b

is maximum.
The minimum value of a ϭ .003
The maximum value of b ϭ 300
So the minimum value of
ᎏ
a
b
ᎏ
ϭᎏ
.
3
0
0
0
0
3
ᎏϭᎏ
.
1
0
0
0
0
1
ᎏϭ.00001.
EXAMPLE 4
If xry ϭ 0, yst ϭ 0, and rxt ϭ 1, then which must be 0?
(A) r
(B) s
(C) t

(D) x
(E) y
Choice E is correct.
Use information that will give you something to work
with.
rxt ϭ 1 tells you that r  0, x  0, and t  0.
So if xry ϭ 0 then y must be 0.
EXAMPLE 5*
On a street with 25 houses, 10 houses have fewer than 6
rooms, 10 houses have more than 7 rooms, and 4 houses
have more than 8 rooms. What is the total number of
houses on the street that are either 6-, 7-, or 8-room
houses?
(A) 5
(B) 9
(C) 11
(D) 14
(E) 15
Choice C is correct.
There are three possible situations:
(a) Houses that have fewer than 6 rooms (call the num-
ber a)
(b) Houses that have 6, 7, or 8 rooms (call the number b)
(c) Houses that have more than 8 rooms (call the num-
ber c)
a ϩ b ϩ c must total 25 (given).
a is 10 (given).
c is 4 (given).
Substituting and in we get 10 ϩ b ϩ 4 ϭ 25.
b must therefore be 11.

*This problem also appears in the 14 Questions That Can
Determine Top College Eligibility, Part 3.
2 3 1
3
2
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EXAMPLE 6
In a room, there are 5 blue-eyed blondes. If altogether
there are 14 blondes and 8 people with blue eyes in the
room, how many people are there in the room? (Assume
that everyone in the room is blonde, has blue eyes, or is
a blue-eyed blonde.)
(A) 11
(B) 17
(C) 22
(D) 25
(E) 27
Choice B is correct.
Method 1:
Draw two intersecting circles.
Above, subtracting: all blondes (14) – blue-eyed blondes
(5), we get 9.
Above, subtracting: all blue-eyed people (8) – blue-eyed
blondes (5), we get 3.
So the number of people in room are 9 ϩ 5 ϩ 3 ϭ 17.
Method 2:
Total number of people are:
(a) blondes without blue eyes
(b) blue-eyed people who are not blonde

(c) blue-eyed blondes
(a) There are 14 blondes and 5 blue-eyed blondes, so,
subtracting, there are 9 blondes without blue eyes.
(b) There are 8 people with blue eyes and 5 blue-eyed
blondes, so, subtracting, there are 3 blue-eyed peo-
ple who are not blonde.
(c) The number of blue-eyed blondes is 5 (given).
Adding the number of people in a, b, and c, we get
9 ؉ 3 ؉ 5 ؍ 17.
STRATEGY SECTION • 111
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112 • STRATEGY SECTION
I.
If a ϭ b, then x ϭ y
The base angles of an isosceles triangle are equal
If x ϭ y, then a ϭ b
If the base angles of a triangle are equal, the triangle is
isosceles
II.
ᐉ is a straight line.
Then, x ϭ y ϩ z
The measure of an exterior angle is equal to the sum of
the measures of the remote interior angles
III.
If a Ͻ b, then y Ͻ x
If y Ͻ x, then a Ͻ b
In a triangle, the greatest angle lies opposite the great-
est side
IV. Similar Triangles
If ⌬ABC ϳ ⌬DEF, then

m Մ A ϭ m Մ D
m Մ B ϭ m Մ E
m Մ C ϭ m Մ F
and
ᎏ
a
d
ᎏ
ϭ
ᎏ
b
e
ᎏ
ϭ
ᎏ
c
f
ᎏ
V.
m Մ A ϩ m Մ Bϩ m Մ Cϭ 180°
The sum of the interior angles of a triangle is 180 degrees
VI.
Area of ⌬ABC ϭ
ᎏ
AD ϫ
2
BC
ᎏ
The area of a triangle is one-half the product of the alti-
tude to a side and the side.

Note: If m Մ A ϭ 90°,
Area also ϭ
ᎏ
AB ϫ
2
AC
ᎏ
Know and Use Facts About Triangles
By remembering these facts about triangles, you can often save yourself a lot of time and trouble.
MATH
STRATEGY
18
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STRATEGY SECTION • 113
VII. In a right triangle,
c
2
ϭ a
2
ϩ b
2
and x ° ϩ y° ϭ 90°
VIII. Memorize the following standard triangles:
EXAMPLE 1
In the diagram below, what is the value of x?
(A) 20
(B) 25
(C) 26
(D) 45
(E) 48

Choice C is correct.
Method 1: Use VII above. Then,
x
2
ϭ 24
2
ϩ 10
2
ϭ 576 ϩ 100
ϭ 676
Thus, x 5 26 (Answer)
Method 2: Look at VIII in left column. Notice that ⌬MNP
is similar to one of the standard triangles:
This is true because
ᎏ
1
2
2
4
ᎏ
ϭ
ᎏ
1
5
0
ᎏ
(Look at IV).
Hence,
ᎏ
1

2
2
4
ᎏ
ϭ
ᎏ
1
x
3
ᎏ
or x ϭ 26 (Answer)
EXAMPLE 2
If Masonville is 50 kilometers due north of Adamston
and Elvira is 120 kilometers due east of Adamston, then
the minimum distance between Masonville and Elvira is
(A) 125 kilometers
(B) 130 kilometers
(C) 145 kilometers
(D) 160 kilometers
(E) 170 kilometers
Choice B is correct. Draw a diagram first.
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