Inverse Variation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inverse Variation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A relationship where $y = \frac{k}{x}$ : as one quantity doubles, the other halves—their product stays constant.

More workers means less time: if 4 workers take 6 hours, 8 workers take 3 hours.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: In inverse variation $xy=k$ , the two quantities trade off so their product never changes.

Common stuck point: The procedure for inverse variation is the easy part; the trap is finding $k$ as $\frac{y}{x}$ instead of $xy$ . Asking "When $x$ doubles does $y$ halve, keeping the product $xy$ the same?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: When $x$ doubles does $y$ halve, keeping the product $xy$ the same?

Worked Examples

Example 1

medium

y

varies inversely with

x

, and

y = 8

when

x = 3

. Find

k

and the equation. Then find

y

when

x = 6

Answer

k = 24

;

y = 4

when

x = 6

First step

Inverse variation:

y = k/x

, so

k = xy

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Example 2

hard

Speed and travel time are inversely proportional for a fixed distance. At 60 km/h the trip takes 4 hours. How long at 80 km/h? Identify

k

Example 3

easy

y

varies inversely with

x

and

y = 15

when

x = 4

. Write the equation, then find

y

when

x = 10

Example 4

medium

y

varies inversely with

x

. Given two pairs

(2, 12)

and

(a, 4)

, find

a

Example 5

medium

A test pilot finds time-to-cover

1

km varies inversely with average speed. At

40

km/h it takes

90

s. Find time at

60

km/h.

Example 6

hard

The force of gravity

F

between two objects varies inversely with the square of distance

r

. If

F = 100

N at

r = 5

m, find

F

r = 10

Example 7

hard

The intensity

I

of a radio signal varies inversely with the square of distance

d

. If

I = 200

d = 3

km, find

I

d = 12

km.

Example 8

challenge

Two quantities satisfy

xy = k

. Show that if

x

increases by

p\%

, then

y

decreases by

\dfrac{100p}{100 + p}\%

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

y = k/x

and

y = 12

when

x = 5

, find

y

when

x = 15

Example 2

hard

The pressure

P

and volume

V

of a gas at constant temperature satisfy

PV = k

. If

P = 4

atm when

V = 6

L, find

P

when

V = 8

Example 3

easy

y

varies inversely with

x

and

y=6

when

x=2

, find the constant

k

Example 4

easy

y=\frac{k}{x}

with

k=20

. Find

y

when

x=4

Example 5

easy

4

workers finish a job in

6

hours, how long do

8

workers take (same total work)?

Example 6

easy

y=\frac{k}{x}

, what happens to

y

when

x

doubles?

Example 7

easy

y=\frac{5}{x}

a direct or inverse variation?

Example 8

easy

y

varies inversely with

x

, and

y=3

when

x=8

. Find

y

when

x=12

Example 9

easy

Can

x=0

in the inverse variation

y=\frac{k}{x}

Example 10

easy

The product of two inversely varying quantities is

36

. If one is

9

, find the other.

Example 11

medium

Pressure

P

varies inversely with volume

V

. At

V=2\,\text{L}

P=15

. Find

P

V=5\,\text{L}

Example 12

medium

y=\frac{k}{x}

passes through

(3,8)

. Find the equation, then

y

x=2

Example 13

medium

y

varies inversely with

x^2

and

y=4

when

x=3

, find

y

when

x=6

Example 14

medium

Two gears mesh so

\text{teeth}\times\text{speed}

is constant. A

20

-tooth gear spins at

120

rpm. Find the speed of a meshing

30

-tooth gear.

Example 15

medium

A trip takes

4

hours at

60

mph. How fast must you drive to make it in

3

hours?

Example 16

medium

y=\frac{k}{x}

. If

x

is multiplied by

5

, by what factor does

y

change?

Example 17

medium

It takes

6

pipes

8

hours to fill a tank. How many pipes fill it in

4

hours?

Example 18

medium

Distinguish:

y=-2x

y=\frac{2}{x}

. Which is inverse variation, and why?

Example 19

medium

y

varies inversely with

x

, and

y=10

when

x=3

. Find

x

when

y=5

Example 20

challenge

y

varies inversely with

x

. When

x

increases by

50\%

y

decreases by how many percent?

Example 21

challenge

Quantities satisfy

xy=k

. If

x

and

y

are positive integers and

k=24

, how many ordered pairs

(x,y)

exist?

Example 22

challenge

y=\frac{k}{x}

passes through

(2,18)

and

(a,4)

. Find

a

Example 23

easy

y

varies inversely with

x

and

y = 9

when

x = 4

, find

k

Example 24

easy

For

y = \dfrac{30}{x}

, find

y

when

x = 6

Example 25

easy

y = k/x

, if

x

triples, what happens to

y

Example 26

easy

The product

xy

for an inverse variation is

48

. Find

y

when

x = 16

Example 27

easy

Does the equation

xy = -12

describe an inverse variation?

Example 28

medium

y

varies inversely with

x^2

. When

x = 2

y = 18

. Find

y

when

x = 6

Example 29

medium

At constant temperature, a gas has

V = 8

L when

P = 3

atm. Find

V

when

P = 12

atm.

Example 30

medium

y = k/x

and

(x, y) = (1.5, 8)

, find

y

when

x = 4

Example 31

medium

A loud speaker's perceived loudness varies inversely with the square of distance. If it is

80

3

m, find loudness at

6

Example 32

medium

Graph

y = 6/x

has a point at

(2, ?)

and

(-3, ?)

. Fill in the missing

y

-values.

Example 33

medium

y

varies inversely with

x

and

y = -8

when

x = 5

. Find

y

when

x = -4

Example 34

hard

y

varies inversely with

x

. When

x

increases by

25\%

y

decreases by what percent?

Example 35

hard

A joint variation:

z

varies directly with

x

and inversely with

y

. When

x = 6

y = 4

z = 9

. Find

z

when

x = 10

y = 5

Example 36

hard

y

varies inversely with

x

, and the graph passes through

(2, 9)

. Find the equation, then find

x

when

y = 6

Example 37

hard

Show that if

y

varies inversely with

x

, then

\dfrac{1}{y}

varies directly with

x

Example 38

challenge

y

varies inversely with

x

. If

y

values at

x = 2

and

x = 5

differ by

9

, find

k

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Related Concepts

Proportionality Division Rational Functions Hyperbola

Background Knowledge

These ideas may be useful before you work through the harder examples.

proportionalitydivision