Proportions Examples in Math

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

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

Concept Recap

An equation stating that two ratios are equal, used to find an unknown when three of the four values are known.

If 2 candies cost \$1, then 4 candies cost \$2—same proportion.

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: A proportion says two ratios are the same, so a missing fourth value can be solved for.

Common stuck point: The procedure for proportions is the easy part; the trap is setting up the two ratios with units in different positions. Asking "Are two equal ratios set against each other with one unknown to solve?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are two equal ratios set against each other with one unknown to solve?

Worked Examples

Example 1

easy

Solve the proportion

\frac{x}{6} = \frac{10}{15}

Answer

x = 4

First step

Cross-multiply:

15x = 6 \times 10 = 60

Full solution

2
Divide both sides by 15: $x = \frac{60}{15} = 4$ .
3
Check: $\frac{4}{6} = \frac{2}{3}$ and $\frac{10}{15} = \frac{2}{3}$ \checkmark

A proportion states that two ratios are equal. Cross-multiplying converts the proportion into a simple linear equation that you can solve for the unknown.

Example 2

medium

If 5 notebooks cost

\$8.75

, how much do 12 notebooks cost?

Example 3

medium

Solve the proportion:

\frac{x}{12} = \frac{5}{8}

Example 4

medium

A printer prints 12 pages in 30 seconds. How long, in seconds, does it take to print 50 pages at the same rate?

Example 5

medium

A recipe calls for 3 cups of sugar to 8 cups of flour. If you have 20 cups of flour, how many cups of sugar do you need?

Example 6

medium

Two similar triangles have corresponding sides

6

cm and

9

cm. If a different pair of corresponding sides is

8

cm and

x

cm, find

x

Example 7

hard

Solve

\frac{2x+1}{6} = \frac{x-1}{2}

Practice Problems

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

Example 1

easy

Solve:

\frac{3}{7} = \frac{9}{x}

Example 2

hard

A car travels 180 km on 15 litres of fuel. How many litres are needed for a 540 km trip?

Example 3

easy

Solve the proportion

\frac{3}{4}=\frac{x}{8}

Example 4

easy

Solve

\frac{x}{5}=\frac{6}{15}

Example 5

easy

If 2 pencils cost \$1, how much do 4 pencils cost at the same rate?

Example 6

easy

Are

\frac{2}{3}

and

\frac{6}{9}

a true proportion?

Example 7

easy

Solve

\frac{5}{x}=\frac{10}{6}

Example 8

easy

A map scale is

\frac{1\text{ cm}}{50\text{ km}}

. How many km does 3 cm represent?

Example 9

easy

If 3 apples cost \$6, find the cost of 1 apple.

Example 10

easy

Solve

\frac{4}{6}=\frac{x}{9}

Example 11

medium

A recipe uses 2 cups of flour for every 3 eggs. How many cups of flour for 12 eggs?

Example 12

medium

A car travels 90 miles in 2 hours. At the same rate, how far in 5 hours?

Example 13

medium

If 4 workers build a wall in the same time that the ratio predicts, and 4 workers lay 200 bricks, how many bricks do 10 workers lay at the same rate?

Example 14

medium

A photo is enlarged so its 4-inch width becomes 10 inches. If the original height was 6 inches, what is the new height?

Example 15

medium

3 liters of paint cover 24 square meters. How many liters cover 40 square meters?

Example 16

medium

Solve

\frac{x+1}{6}=\frac{2}{3}

Example 17

challenge

A 3 ft fence post casts a 2 ft shadow. At the same time, a tree casts a 16 ft shadow. How tall is the tree?

Example 18

challenge

A recipe for 4 people uses 600 g of pasta. You cook for 7 people. How much pasta, to the nearest gram?

Example 19

challenge

In a proportion

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

, you know

a=8

b=12

d=21

. Find

c

Example 20

medium

Solve

\frac{6}{9}=\frac{8}{x}

Example 21

medium

If 5 notebooks cost

\$12

, how much do 15 notebooks cost?

Example 22

medium

A blueprint uses 2 inches for every 5 feet. How many feet does 7 inches represent?

Example 23

easy

Solve

\frac{x}{10} = \frac{7}{2}

Example 24

easy

Solve

\frac{9}{x} = \frac{3}{4}

Example 25

easy

If 6 pencils cost \$1.50, how much do 2 pencils cost at the same rate?

Example 26

easy

Solve

\frac{2}{5} = \frac{x}{20}

Example 27

medium

A 3:5 ratio of red to blue beads is used to make a necklace with 40 beads total. How many red beads are there?

Example 28

medium

Solve

\frac{x-2}{5} = \frac{3}{10}

Example 29

medium

A car uses 8 gallons of gas to travel 224 miles. How many gallons are needed for 350 miles at the same rate?

Example 30

medium

On a map, 1 inch represents 25 miles. What distance is shown by 4.5 inches?

Example 31

medium

y

is directly proportional to

x

and

y = 18

when

x = 4

, find

y

when

x = 10

Example 32

medium

Solve

\frac{x}{4} = \frac{x+3}{10}

Example 33

medium

A photo enlarged by the same scale has its 5 in width grow to 12 in. What is the new height if the original height is 4 in?

Example 34

medium

If 5 oranges weigh 1.2 kg, how many oranges weigh 3 kg (assuming each orange weighs the same)?

Example 35

hard

\frac{x}{y} = \frac{3}{4}

and

x + y = 28

, find

x

and

y

Example 36

hard

A 6 ft person casts a 4 ft shadow. At the same moment, a flagpole casts a 22 ft shadow. How tall is the flagpole?

Example 37

hard

Three pumps fill a tank in 12 hours. How long would 5 pumps take, working at the same rate?

Example 38

hard

A bag contains red and green marbles in the ratio 5:7. After 6 green marbles are removed, the ratio becomes 5:4. How many red marbles are in the bag?

Example 39

hard

A 250 mL solution contains 30 mL of acid. How much acid is in 750 mL of the same solution?

Example 40

challenge

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

, show that

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

(assuming

b + d \neq 0

← Back to Proportions Formula Explained Practice Mode

Related Concepts

Ratios Equations Similar Figures Unit Rate

Background Knowledge

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

ratiosequations