Ratios Examples in Math

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

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 ratio compares two or more quantities by showing how many times one contains the other, written as $a:b$ or $\frac{a}{b}$ . Unlike fractions, ratios can compare parts to parts, not just parts to wholes.

A recipe that uses 2 cups flour for every 1 cup sugar has a $2:1$ ratio.

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 ratio compares two quantities — often part to part — by how many times one contains the other.

Common stuck point: The procedure for ratios is the easy part; the trap is converting a part-to-part ratio into a part-to-whole fraction by mistake. Asking "Am I comparing two separate amounts to each other, rather than naming a part of one whole?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I comparing two separate amounts to each other, rather than naming a part of one whole?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Fractions Mistakes

Worked Examples

Example 1

easy

A recipe calls for 3 cups of flour and 2 cups of sugar. What is the ratio of flour to sugar, and how much sugar is needed for 12 cups of flour?

Answer

8 \text{ cups of sugar}

First step

The ratio of flour to sugar is

3 : 2

Full solution

2
Set up the proportion: $\frac{3}{2} = \frac{12}{x}$ .
3
Cross-multiply: $3x = 24$ , so $x = 8$ .

A ratio compares two quantities. To scale a ratio, you can set up a proportion and cross-multiply to find the unknown value.

Example 2

medium

The ratio of boys to girls in a class is

5 : 3

. If there are 40 students total, how many boys and how many girls are there?

Example 3

medium

A recipe uses flour and sugar in a 5:2 ratio. If you use 15 cups of flour, how much sugar do you need?

Example 4

medium

Two numbers are in the ratio

3 : 7

and their sum is

50

. Find the two numbers.

Example 5

medium

Simplify the ratio

0.4 : 1.6

Example 6

medium

3 : 5 = x : 20

, find

x

Example 7

medium

Three numbers are in ratio

2 : 3 : 5

and their sum is

80

. Find each number.

Example 8

hard

A class has boys and girls in ratio

4 : 5

. After

3

more boys join, the ratio becomes

1 : 1

. How many girls are in the class?

Example 9

hard

A and B share

\$540

in the ratio

4 : 5

. Then A gives

\$x

to B so the new ratio is

1 : 2

. Find

x

Example 10

hard

If the ratio

x : y = 3 : 4

and

y : z = 6 : 7

, what is

x : y : z

Example 11

challenge

A jar contains red, green, and blue marbles in ratio

2 : 3 : 5

. After

6

red marbles are added the ratio of red to green becomes

4 : 3

. How many blue marbles are in the jar?

Practice Problems

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

Example 1

easy

Simplify the ratio

18 : 24

Example 2

hard

A map uses the scale

1 : 50{,}000

. Two towns are

3.5

cm apart on the map. What is the actual distance in kilometres?

Example 3

easy

Simplify the ratio

12 : 8

Example 4

easy

A class has

15

boys and

10

girls. What is the ratio of boys to girls in simplest form?

Example 5

easy

If the ratio of red to blue marbles is

4 : 7

, and there are

20

red marbles, how many blue?

Example 6

easy

Express the ratio

1 : 4

as a fraction.

Example 7

easy

If a recipe uses flour and sugar in ratio

3 : 1

, and you use

9

cups flour, how much sugar?

Example 8

easy

Simplify the ratio

25 : 75

Example 9

easy

3

apples cost

\$2

, how much do

9

apples cost?

Example 10

easy

Divide

\$50

in the ratio

3 : 2

Example 11

medium

Two numbers are in ratio

5 : 7

. Their sum is

48

. Find the numbers.

Example 12

medium

a : b = 2 : 3

and

b : c = 4 : 5

, find

a : b : c

Example 13

medium

A map has scale

1 : 50{,}000

. If two cities are

4

cm apart on the map, what is the real distance in km?

Example 14

medium

5 : x = 3 : 12

, find

x

Example 15

medium

If a

5

-kg bag of rice costs

\$8

, how much does

12

kg cost?

Example 16

medium

The ratio of cats to dogs in a shelter is

7 : 5

. If there are

48

animals total, how many cats?

Example 17

medium

\frac{a}{b} = \frac{3}{4}

, what is

\frac{a + b}{b}

Example 18

medium

A car travels

180

km in

3

hours. At the same rate, how long for

300

km?

Example 19

medium

A class is

\frac{2}{5}

boys. If there are

20

boys, how many students total?

Example 20

challenge

Three numbers are in the ratio

2 : 3 : 5

. Their sum is

60

. Find the largest.

Example 21

challenge

Two trains leave two stations

300

km apart, traveling toward each other at

40

km/h and

60

km/h. When do they meet?

Example 22

challenge

Salt water is

20\%

salt by weight. How much water do I add to

50

kg of brine to make it

10\%

salt?

Example 23

easy

Simplify the ratio

36 : 48

Example 24

easy

A bag has red and blue marbles in ratio

2 : 5

. If there are

8

red marbles, how many blue?

Example 25

easy

4

pencils cost

\$3

, what is the unit price per pencil?

Example 26

easy

Simplify the ratio

14 : 21

Example 27

easy

5

litres of juice cost

\$12

. What is the price per litre?

Example 28

medium

In a class of

48

students, the ratio of girls to boys is

5 : 7

. How many girls are there?

Example 29

medium

A car travels

180

km in

3

hours. What is its speed in km/h?

Example 30

medium

A drink mixes water and concentrate in ratio

7 : 2

. To make

36

litres of drink, how much concentrate is needed?

Example 31

medium

On a blueprint,

1

cm represents

2.5

m. A wall measured on the blueprint is

8

cm long. What is the actual length?

Example 32

medium

Convert the ratio

\frac{1}{2} : \frac{3}{4}

to a whole-number ratio.

Example 33

hard

At a concert the ratio of adults to children is

5 : 2

. There are

84

more adults than children. How many children are there?

Example 34

hard

A worker can complete a job in

6

days. At the same rate, how many days does it take

3

workers to complete

4

such jobs?

Example 35

hard

A solution contains alcohol and water in ratio

3 : 7

. How much water must be added to

50

mL of solution to make the ratio

1 : 4

← Back to Ratios Formula Explained Common Mistakes Practice Mode

Related Concepts

Fractions Division Proportions Rates Similarity

Background Knowledge

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

fractionsdivision