Sat math Essentials

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P R O B L E M S O LV I N G
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1 5 8

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Practice Question 
The number 
a
is directly proportional to 
b
. If
a
15 when 
b
24, what is the value of
b
when 
a
5?
a.
8
5
b.
2
8
5
c.
8
d.
14
e.
72
Answer
c.
The numbers 
a
and 
b
are directly proportional (in other words, they vary directly), so 
a
increases when
b
increases, and vice versa. Therefore, we can set up a proportion to solve:
1
2
5
4
5
b
Find cross products.
15
b
(24)(5)
15
b
120
1
1
5
5
b
1
1
2
5
0
b
8
Therefore, we know that 
b
8 when 
a
5.
R a t e P r o b l e m s
Rate
is defined as a comparison of two quantities with different units of measure.
Rate 
x
y
u
u
n
n
i
i
t
t
s
s
Examples
d
h
o
o
ll
u
a
r
rs
po
co
u
s
n
t
d
m
ho
il
u
e
r
s
g
m
al
i
l
l
o
e
n
s
There are three types of rate problems you must learn how to solve: cost per unit problems, movement prob-
lems, and work-output problems.
C o s t P e r U n i t
Some rate problems require you to calculate the cost of a specific quantity of items.
Example
If 40 sandwiches cost $298, what is the cost of eight sandwiches?
First determine the cost of one sandwich by setting up a proportion:
40 sa
$
n
2
d
3
w
8
iches
1
x
sandwich
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P R O B L E M S O LV I N G
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1 5 9

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238 
1 
40
x
Find cross products.
238 
40
x
2
4
3
0
8
x
5.95 
x
Now we know one sandwich costs $5.95. To find the cost of eight sandwiches, multiply:
5.95 
8 
$47.60
Eight sandwiches cost $47.60.
Practice Question 
A clothing store sold 45 bandanas a day for three days in a row. If the store earned a total of $303.75 from
the bandanas for the three days, and each bandana cost the same amount, how much did each bandana
cost?
a.
$2.25
b.
$2.75
c.
$5.50
d.
$6.75
e.
$101.25
Answer
a.
First determine how many total bandanas were sold:
45 bandanas per day 
3 days 
135 bandanas
So you know that 135 bandanas cost $303.75. Now set up a proportion to determine the cost of one
bandana:
135
$3
b
0
a
3
n
.
d
7
a
5
nas
1
x
bandana
303.75 
1 
135
x
Find cross products.
303.75 
135
x
30
1
3
3
.
5
75
x
2.25 
x
Therefore, one bandana costs $2.25.
M o v e m e n t
When working with movement problems, it is important to use the following formula:
(Rate)(Time) 
Distance
Example
A boat traveling at 45 mph traveled around a lake in 0.75 hours less than a boat traveling at 30 mph. What was
the distance around the lake?
First, write what is known and unknown.