Proportional Relationships Examples in Math

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

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 two quantities maintain a constant ratio: doubling one always doubles the other, giving $y = kx$ .

If you double one, you double the other. Triple one, triple the other.

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 proportional when one is always a fixed number times the other.

Common stuck point: The procedure for proportionality is the easy part; the trap is calling any growing pair proportional. Asking "Is $y/x$ the same number for every pair, and is $y=0$ when $x=0$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is $y/x$ the same number for every pair, and is $y=0$ when $x=0$ ?

Worked Examples

Example 1

easy

A car travels

150

miles in

3

hours at constant speed. How far will it travel in

5

hours?

Answer

The car travels

250

miles in

5

hours.

First step

Find the unit rate (speed):

\dfrac{150 \text{ miles}}{3 \text{ hours}} = 50

mph.

Full solution

2
Distance in $5$ hours: $50 \times 5 = 250$ miles.
3
Alternatively, set up a proportion: $\dfrac{150}{3} = \dfrac{d}{5}$ , so $d = \dfrac{150 \times 5}{3} = 250$ miles.

Two quantities are proportional when their ratio is constant. Here, distance and time have a constant ratio (speed). Setting up a proportion or multiplying by the unit rate both give the same result.

Example 2

medium

The table shows:

x = 2, y = 8

;

x = 5, y = 20

;

x = 9, y = 36

. Determine whether

y

is proportional to

x

, and if so write the proportionality equation.

Example 3

medium

A car uses

4

L of fuel per

50

km. Assuming proportional fuel use, how much fuel does it use for a

325

km trip?

Example 4

medium

Determine whether each table is proportional. Table A:

(x, y) = (1, 3), (2, 6), (3, 9)

. Table B:

(x, y) = (1, 4), (2, 7), (3, 10)

Example 5

medium

200

g serving of cereal contains

24

g of sugar. How much sugar is in

325

g of the same cereal, assuming proportionality?

Example 6

hard

The mass

m

of a metal cube is proportional to its volume

V

. A cube of side

2

cm has mass

72

g. Find the mass of a cube of side

5

cm of the same metal.

Example 7

hard

y

is proportional to

x

. When

x

increases from

4

9

y

increases by

15

. Find the constant of proportionality.

Example 8

hard

A relationship satisfies

y = 4x + 0

. Verify it is proportional and state

k

. Then check whether

y = 4x + 1

is proportional.

Example 9

challenge

At a science fair, the brightness of a bulb is proportional to the cube of the current through it. When

I = 2

A, brightness is

40

units. Find brightness when

I = 3

A and verify the proportional model.

Practice Problems

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

Example 1

easy

12

apples cost

\$3.60

, how much do

20

apples cost?

Example 2

medium

Is the relationship

y = 3x + 2

proportional? Is

y = 5x

proportional? Explain.

Example 3

easy

y = kx

with

k = 4

, find

y

when

x = 3

Example 4

easy

3

apples cost

\$6

. What is the cost per apple?

Example 5

easy

y

is proportional to

x

and

y = 10

when

x = 2

, find

k

Example 6

easy

y = 3x + 5

a proportional relationship?

Example 7

easy

Solve

\frac{x}{4} = \frac{6}{8}

Example 8

easy

A car travels

60

km in

1

hour at constant speed. How far in

3

hours?

Example 9

easy

4

pens cost

\$12

, what do

6

pens cost at the same rate?

Example 10

easy

A graph passes through the origin and

(2, 8)

. What is

k

Example 11

medium

5

workers build a wall in the same time pattern and pay is proportional,

5

workers earn

\$200

. What do

8

earn?

Example 12

medium

On a map,

2

cm represents

5

km. How many km does

7

cm represent?

Example 13

medium

Solve the proportion

\frac{3}{x} = \frac{9}{15}

Example 14

medium

A printer prints

24

pages in

3

minutes. How long for

40

pages?

Example 15

medium

y \propto x

and

y

increases from

6

9

, by what factor does

x

change?

Example 16

medium

3

kg bag of rice costs

\$7.50

. At the same rate, what is the cost of

5

kg?

Example 17

medium

Two quantities satisfy

\frac{a}{b} = \frac{c}{d}

with

a=6, b=9, c=10

. Find

d

Example 18

challenge

a

is proportional to

b

and

b

is proportional to

c

, show

a

is proportional to

c

Example 19

challenge

Quantity

y

is proportional to

x^2

. If

y = 12

when

x = 2

, find

y

when

x = 5

Example 20

challenge

Three friends split a cost in the ratio

2:3:5

. If the total is

\$120

, how much does each pay?

Example 21

medium

A recipe needs

3

eggs for

12

cookies. How many eggs for

20

cookies?

Example 22

medium

7

meters of cloth cost

\$21

, how many meters can you buy with

\$15

Example 23

easy

y = 7x

, find

y

when

x = 5

Example 24

easy

5

notebooks cost $20, what is the cost of

1

notebook at the same rate?

Example 25

easy

y

is proportional to

x

. When

x = 7

y = 21

. Find

k

Example 26

easy

At a constant rate,

4

workers paint

2

rooms in

3

hours. How many rooms can

4

workers paint in

6

hours?

Example 27

medium

The cost of

c

cans of soup is proportional to

c

. If

3

cans cost $4.50, what do

11

cans cost?

Example 28

medium

y

is proportional to

x

. If

y = 18

when

x = 12

, find

y

when

x = 30

Example 29

medium

A printer prints

24

pages in

3

minutes. At the same rate, how long to print

200

pages?

Example 30

medium

Solve the proportion

\dfrac{x}{18} = \dfrac{5}{6}

for

x

Example 31

medium

On a map,

2.5

cm represents

30

km. How many km does

7

cm represent?

Example 32

hard

In a recipe, the ratio of flour to sugar is

3:2

. If you use

7.5

cups of flour, how much sugar do you need?

Example 33

hard

The graph of a proportional relationship passes through

(6, 9)

. Find the value of

y

when

x = 14

Example 34

hard

At a constant exchange rate, $50 USD = €45. How many euros do you get for $280 USD?

Example 35

hard

Two tanks fill proportionally. Tank A fills

3

L in

4

minutes; Tank B fills

5

L in

4

minutes. How long for Tank A to fill what Tank B fills in

10

minutes?

← Back to Proportional Relationships Formula Explained Practice Mode

Related Concepts

Ratios Multiplication Direct Variation Linear Functions

Background Knowledge

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

ratiosmultiplication