Sat math Essentials

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P R O B L E M S O LV I N G
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Va r i a t i o n
Variation
is a term referring to a constant ratio in the change of a quantity.
■
A quantity is said to 
vary directly
with or to be 
directly proportional to
another quantity if they both
change in an equal direction. In other words, two quantities vary directly if an increase in one causes an
increase in the other or if a decrease in one causes a decrease in the other. The ratio of increase or decrease,
however, must be the same.
Example
Thirty elephants drink altogether a total of 6,750 liters of water a day. Assuming each elephant drinks the same
amount, how many liters of water would 70 elephants drink?
Since each elephant drinks the same amount of water, you know that elephants and water vary directly. There-
fore, you can set up a proportion:
ele
w
p
a
h
t
a
e
n
r
ts
6,
3
7
0
50
7
x
0
Find cross products to solve:
6,
3
7
0
50
7
x
0
(6,750)(70) 
30
x
472,500 
30
x
472
3
,
0
500
3
3
0
0
x
15,750 
x
Therefore, 70 elephants would drink 15,750 liters of water.
■
A quantity is said to 
vary inversely
with or to be 
inversely proportional
to another quantity if they change
in opposite directions. In other words, two quantities vary inversely if an increase in one causes a decrease
in the other or if a decrease in one causes an increase in the other.
Example
Three plumbers can install plumbing in a house in six days. Assuming each plumber works at the same rate,
how many days would it take nine plumbers to install plumbing in the same house? 
As the number of plumbers increases, the days needed to install plumbing decreases (because more
plumbers can do more work). Therefore, the relationship between the number of plumbers and the number
of days varies inversely. Because the amount of plumbing to install remains constant, the two expressions can
be set equal to each other:
3 plumbers 
6 days 
9 plumbers 
x
days
3 
6 
9
x
18 
9
x
1
9
8
9
9
x
2 
x
Thus, it would take nine plumbers only two days to install plumbing in the same house.