You've seen that many quantities are related to each other. However, not all of them are directly related. Now you will explore quantities that vary inversely. In inverse variation, one quantity decreases as the other increases.

1. MATHEMATICS 9

2. INVERSE VARIATION
+ Let x and y denote two quantities. y varies
inversely as x, or y is inversely proportional to x,
if there is a nonzero constant k such that:
𝒚 =
𝒌
𝒙
+ The number k is called the constant of variation.

3. INVERSE VARIATION
Example 1:
+ y varies
inversely as x
and 𝑦 = 6
when 𝑥 =
15, find the
equation of
variation.
Solution:
𝑦 =
𝑘
𝑥
6 =
𝑘
15
6(15) = 𝑘
90 = 𝑘
1
Thus the equation of
variation is 𝑦 =
90
𝑥
.
𝑦 =
90
𝑥
2

4. INVERSE VARIATION
Example 2:
+ If y varies
inversely as 𝑥
and y =
1
3
when 𝑥 =
18 , find 𝑦
when 𝑥 = 2.
Solution:
𝑦 =
𝑘
𝑥
1
3
=
𝑘
18
18
3
= 𝑘
6 = 𝑘
1
𝑦 =
6
𝑥
2
𝑦 =
6
𝑥
𝑦 =
6
2
𝑦 = 3
3
Thus, 𝑦 = 3 When x = 2

5. INVERSE VARIATION
Example 3:
+ A crew of 12 can
build a hut in 8
days. How
would it take a
crew of 4 to
build the same
house?
Solution:
Understand the problem.
Let y = number of days to build a hut.
x = the number of crew.
k = the constant of variation
(Common sense tell us that if there were
more crew working the hut, the time
needed to build the hut would be less.
Thus two variables are inversely related.)
1

6. INVERSE VARIATION
Example 3:
+ A crew of 12 can
build a hut in 8
days. How
would it take a
crew of 4 to
build the same
house?
Solution:
Write the equation.
𝑦 =
𝑘
𝑥
8 =
𝑘
12
96 = 𝑘
y =
96
𝑥
2
Solution:
Solve the equation.
𝑦 =
96
4
𝑦 = 24
3
It will take 24
days for a crew
of 4 to build a
hut.

7. INVERSE VARIATION
Example 4:
+ The number of days a
bag of bread last varies
inversely as the number
of people who consume
it. If a bag of bread lasts
three days for six
people, how long will it
lasts for two people?
Solution:
Understand the problem.
Let:
y = number of days
x = number of people
k = the constant of variation
1

8. INVERSE VARIATION
Example 4:
+ The number of days a
bag of bread last varies
inversely as the number of
people who consume it.
If a bag of bread lasts
three days for six people,
how long will it lasts for
two people?
Solution:
Understand the problem.
Let:
y = number of days
x = number of people
k = the constant of variation
1
Solution:
Write the equation.
𝑦 =
𝑘
𝑥
3 =
𝑘
6
3(6) = 𝑘
18 = 𝑘
𝑦 =
18
𝑥
2
Solution:
Solve the equation.
𝑦 =
18
𝑥
𝑦 =
18
2
𝑦 = 9
3
It will take 9 days for
two people to consume
a bag of bread.