Inverse Quantity Examples in Math

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

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

Concept Recap

The reciprocal or multiplicative inverse of a quantity, where multiplying a number by its inverse yields one. Inverse quantities appear whenever two measurements are inversely related, so that doubling one halves the other.

More workers = less time to finish. Double the workers, halve the time.

Read the full concept explanation →

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: Two quantities are inversely related when their product stays a fixed number.

Common stuck point: The procedure for inverse quantity is the easy part; the trap is holding the ratio constant instead of the product. Asking "Does the product $x\times y$ stay the same when one quantity grows and the other shrinks?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the product $x\times y$ stay the same when one quantity grows and the other shrinks?

Worked Examples

Example 1

easy

5

workers can complete a job in

12

days, how many days will it take

15

workers (assuming equal work rates)?

Answer

It will take

4

days.

First step

Total work

= 5 \times 12 = 60

worker-days.

Full solution

2
With $15$ workers: days $= \dfrac{60}{15} = 4$ days.
3
Alternatively: workers and days are inversely proportional, so $5 \times 12 = 15 \times d$ , giving $d = 4$ .

When two quantities are inversely proportional, their product is constant. More workers means fewer days, and the product (total worker-days) stays the same. The relationship is

w \times d = k

, not

w/d = k

Example 2

medium

The pressure

P

of a gas varies inversely with its volume

V

at constant temperature (Boyle's Law). If

P = 200

kPa when

V = 3

L, find

P

when

V = 5

Example 3

easy

50

km/h a journey takes

6

hours. How long at

75

km/h?

Example 4

medium

A bicycle gear with

48

teeth drives one with

16

teeth. If the

48

-tooth gear turns at

20

rpm, find the rpm of the

16

-tooth gear.

Example 5

medium

A class of

15

students completes a service project in

8

hours. How long would

20

students take, assuming equal work rates?

Example 6

hard

Light intensity varies inversely with the square of distance from the source. If intensity is

400

lux at

2

m, find the intensity at

5

Example 7

hard

Two resistors in parallel satisfy

\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}

. If

R_1 = 12\,\Omega

and

R_2 = 6\,\Omega

, find

R

Example 8

challenge

Three workers can finish a job in

4

hours alone in times

6

8

, and

12

hours respectively. How long together?

Practice Problems

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

Example 1

easy

60

km/h a trip takes

4

hours. How long does the same trip take at

80

km/h?

Example 2

medium

Variables

x

and

y

are inversely proportional. When

x = 4

y = 9

. Find

y

when

x = 12

, and find

x

when

y = 6

Example 3

easy

What is the reciprocal of

5

Example 4

easy

What is the reciprocal of

\frac{3}{4}

Example 5

easy

4

workers take

12

days, do

8

workers take more or fewer days?

Example 6

easy

Multiply

\frac{2}{7}

by its reciprocal. What is the result?

Example 7

easy

In inverse variation

xy = k

, if

x = 2

and

k = 12

, find

y

Example 8

easy

What is the reciprocal of

1

Example 9

easy

Does

0

have a reciprocal?

Example 10

easy

If speed doubles for a fixed distance, what happens to travel time?

Example 11

medium

4

workers take

12

days, how long do

8

workers take (same total work)?

Example 12

medium

xy = 36

, if

x

triples from

3

9

, find the new

y

Example 13

medium

A tank fills in

6

hours with

2

pumps. How long with

3

pumps (same rate each)?

Example 14

medium

y

varies inversely with

x

. When

x = 5

y = 8

. Find

y

when

x = 10

Example 15

medium

A gear with

20

teeth turns at

60

rpm. A meshed gear with

30

teeth turns at what speed?

Example 16

medium

The reciprocal of a number is

\frac{2}{9}

. What is the number?

Example 17

medium

If pressure and volume satisfy

PV = k

at constant temperature, and

P = 4

when

V = 6

, find

V

when

P = 8

Example 18

challenge

Show that if

y

varies inversely with

x

, then

y

varies directly with

\frac{1}{x}

Example 19

challenge

Two workers together finish a job in

4

hours. Alone, the first takes

6

hours. How long does the second take alone?

Example 20

challenge

y = \frac{12}{x}

, by what factor must

x

change to make

y

five times larger?

Example 21

medium

6

identical machines complete a batch in

10

hours. How long would

4

machines take?

Example 22

medium

The reciprocal of

x

added to itself: if

\frac{1}{x} = 0.25

, find

x

Example 23

easy

Find the reciprocal of

\dfrac{7}{8}

Example 24

easy

Find the reciprocal of

10

Example 25

easy

xy = 24

(inverse variation) and

x = 6

, find

y

Example 26

easy

A car travels a fixed route. If it drives twice as fast, the trip time is multiplied by what factor?

Example 27

easy

xy = k

, when

x = 9

and

y = 4

, what is

k

Example 28

medium

10

painters can finish a house in

9

days. How long would

6

painters take?

Example 29

medium

y

varies inversely with

x

. When

x = 6

y = 14

. Find

y

when

x = 21

Example 30

medium

x

is multiplied by

\dfrac{2}{3}

in the relation

xy = k

, by what factor does

y

change?

Example 31

medium

Two numbers multiply to

1

. If one is

\dfrac{5}{12}

, find the other.

Example 32

medium

At constant temperature,

PV = 60

. If

V

is reduced from

5

L to

3

L, what is the new pressure?

Example 33

medium

y \propto \dfrac{1}{x}

and

y = 9

when

x = 4

. Find

y

when

x = 12

Example 34

hard

y

varies inversely with

x

. When

x

decreases by

20\%

, by what percent does

y

change?

Example 35

hard

Pipe A fills a tank in

6

hours; pipe B in

9

hours. Working together, how long to fill the tank?

Example 36

hard

A reciprocal pair: a number plus its reciprocal equals

\dfrac{5}{2}

. Find both possible values of the number.

Example 37

hard

At constant temperature a gas obeys

PV = 240

. Sketch (describe) the graph of

P

V

and state its shape.

Example 38

challenge

y

varies inversely with

\sqrt{x}

. When

x = 16

y = 5

. Find

y

when

x = 100

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Related Concepts

Proportionality Division Inverse Variation Rational Functions

Background Knowledge

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

proportionalitydivision