Inverse Variation
Two variables, x and y, vary inversely if there is a non-zero number k made by
multiplication:
k
y
x
y  x  k  k  xy or k  yx
k is the constant of the inverse variation
(constant = number without a variable)
Inverse variation always uses multiplication to get a number by itself. In a problem,
x will always be attached to k with division.
Example: Find the constant of the following inverse variation:
3
1) y 
3 is attached to x by division
x
xy  3
This is inverse variation and the constant k is 3
2)
y  3x  4
6
x
3) y =
6 is attached to x by division
xy = 6
4) y 
This is not inverse variation.
This is inverse variation and the constant k is 6
2
x
3
Move x to the other side by division
y 2

x 3
This is not inverse variation
In the above examples, notice that the final answer has xy , and the numbers or
constants are by themselves on the other side. To show a relationship of inverse
variation in a problem, write as k = xy
Ex: Write an equation that relates x and y with inverse variation.
x = 36
y=9
k  (9)(36)
Plug numbers into x and y
k  324
The constant k is 324
If given two sets of numbers with a missing variable, use the formula
xy  xy
Ex: Assume that y varies inversely as x. If x = -2 when y = 14, find x when y = 21.
21 x  (14)( 2)
21x  28
x
28
21
x
4
3
Reduce
A table has inverse variation if xy gives the same answer for each part:
x 1 2 3 4
5
y 48 24 16 12 9.6
x
y
xy  48 for each part
1 2 3 4 5
2 4 12 16 20
xy  ? does not match for any
table is inverse
table is not inverse
Use the xy  xy pattern when given a word problem:
Ex: When setting up for a party, the hostess noticed the number of apples varied
inversely as the number of oranges on each table. One table had 4 apples and 5
oranges. If another table had 7 apples, how many oranges would be needed to keep
the same proportions?
Make apples y and oranges x:
(apples )(oranges )  (apples )(oranges )
7 x  (4)(5)
7x = 20
divide by 7
x = 2.86 oranges (either cut 1 up or add another to the table)
Inverse Variation Worksheet
Write an inverse variation equation that relates x and y. Assume
y varies inversely as x. Then solve.
1. If y = 10 when x = 7, find y when x = 5.
2. If y = 21 when x = 10, find y when x = 4.
3. If y = 17.5 when x = 12, find y when x = 8.
4. If y = 5 when x = 5, find x when y = 2.
5. If y = 13 when x = 3, find x when y = 3.9.
6. In the following charts, does one variable vary inversely with the other?
a.
x
y
1
2 3 4 6
48 24 16 12 8
b.
x
y
1 2 3 4 5
64 32 28 16 12
7. In a particular experiment, the pressure varied inversely as the volume.
When the pressure was 15 pounds per square inch, the volume was 20 liters.
What was the pressure when the volume was reduced to 12 liters?
Tell whether the equation represents direct variation, inverse variation or neither.
8. y = -4x
11.
y
=4
x
14. x = -8
9. xy = 7
10. x = 7y
12. 3x + 4y = 8
13. 2xy = 20
15. 3 x 
12
y
16. y = x + 7
Inverse and Direct Variation
For each problem find a) the direct variation
b) inverse variation
1. If y = 21 when x = 10, find y when x = 4.
1)a.______________
b.______________
2. If y = 17.5 when x = 12, find y when x = 8.
2)a.______________
b.______________
3. The number of girls in a class varied directly as the number of boys. One
class had 6 boys and 24 girls. If another class had 7 boys, how many girls
were in this class?
3)______________
4. To travel a fixed distance, the rate is inversely proportional to the time
required. When the rate is 60 miles per hour, the time required is 4
hours. What time would be required for the same distance if the rate
were increased to 80 miles per hour?
4)______________
5. Tell if the graph is direct, inverse, or neither?
x
y
1 2 3 4 5
3 6 9 12 15
x 1 2 3 4 5
y 2 4 9 16 25
__________________
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x
y
1 2 3 4 6
24 12 8 6 4
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