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Open Math›Year 7›Ratios and rates›Ratios

Ratios

Simplify ratios and divide quantities in a given ratio.

Worked examples

Simplifying a ratio

Straightforward

Problem

Simplify the ratio

18:24

.

1

Find the highest common factor (HCF) of 18 and 24.

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

HCF

= 6

2

Divide both parts of the ratio by the HCF.

18 \div 6 = 3, \quad 24 \div 6 = 4

3

Write the simplified ratio.

18:24 = 3:4

Answer

3:4

Dividing a quantity in a given ratio

Moderate

Problem

Divide

\$75

in the ratio

2:3

.

1

Find the total number of parts.

2 + 3 = 5 \text{ parts}

2

Find the value of one part.

\$75 \div 5 = \$15 \text{ per part}

3

Multiply to find each share.

First share:

2 \times \$15 = \$30

Second share:

3 \times \$15 = \$45

4

Check that the shares add to the original amount.

\$30 + \$45 = \$75 \checkmark

Answer

\$30

and

\$45

Finding an unknown in an equivalent ratio

Challenging

Problem

Find

x

if

4:x

is equivalent to

12:21

.

1

Write the equivalent ratios as equal fractions.

\frac{4}{x} = \frac{12}{21}

2

Simplify the right-hand fraction by dividing by the HCF of 12 and 21.

HCF of 12 and 21 is 3:

\frac{12}{21} = \frac{4}{7}

3

The fractions are equal, so the denominators must match.

\frac{4}{x} = \frac{4}{7} \implies x = 7

Answer

x = 7

Practise

Q1·Straightforward

Simplify the ratio

6:9

.

$2:3$$3:2$$3:6$$1:2$

Explanation

The HCF of 6 and 9 is 3. Divide both parts by 3:

6 \div 3 = 2, \quad 9 \div 3 = 3

Simplified ratio:

2:3

Q2·Straightforward

Simplify the ratio

15:20

.

$3:4$$4:3$$5:4$$3:5$

Explanation

The HCF of 15 and 20 is 5. Divide both parts by 5:

15 \div 5 = 3, \quad 20 \div 5 = 4

Simplified ratio:

3:4

Q3·Straightforward

Simplify the ratio

12:8

.

$3:2$$2:3$$4:3$$6:4$

Explanation

The HCF of 12 and 8 is 4. Divide both parts by 4:

12 \div 4 = 3, \quad 8 \div 4 = 2

Simplified ratio:

3:2

Q4·Straightforward

Simplify the ratio

10:25

.

$2:5$$5:2$$1:3$$2:3$

Explanation

The HCF of 10 and 25 is 5. Divide both parts by 5:

10 \div 5 = 2, \quad 25 \div 5 = 5

Simplified ratio:

2:5

Q5·Moderate

Divide

\$60

in the ratio

1:2

. How much is the smaller share, in dollars?

Explanation

Total parts:

1 + 2 = 3

Value of one part:

\$60 \div 3 = \$20

Smaller share (

1

part):

1 \times \$20 = \$20

Q6·Moderate

40 lollies are shared in the ratio

3:5

. How many lollies does the first person receive?

Explanation

Total parts:

3 + 5 = 8

Value of one part:

40 \div 8 = 5

lollies

First person's share (

3

parts):

3 \times 5 = 15

lollies

Q7·Moderate

A 56 cm length of ribbon is divided in the ratio

3:4

. What is the length of the shorter piece, in centimetres?

Explanation

Total parts:

3 + 4 = 7

Value of one part:

56 \div 7 = 8

cm

Shorter piece (

3

parts):

3 \times 8 = 24

cm

Q8·Moderate

90 minutes is divided in the ratio

2:1

. How long is the longer time, in minutes?

Explanation

Total parts:

2 + 1 = 3

Value of one part:

90 \div 3 = 30

minutes

Longer time (

2

parts):

2 \times 30 = 60

minutes

Q9·Challenging

Find

x

if the ratio

3:x

is equivalent to

9:15

.

Explanation

Write the ratios as equal fractions:

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

Cross-multiply:

3 \times 15 = 9 \times x

45 = 9x

x = 5

Q10·Challenging

Find

n

if the ratio

n:8

is equivalent to

6:24

.

Explanation

Write the ratios as equal fractions:

\frac{n}{8} = \frac{6}{24}

Simplify the right side:

\dfrac{6}{24} = \dfrac{1}{4}

So

\dfrac{n}{8} = \dfrac{1}{4}

, which gives

n = 2

.

Q11·Challenging

A map uses a scale of

1:50\,000

. Two towns are

4

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

Explanation

Scale

1:50\,000

means 1 cm on the map represents 50 000 cm in real life.

Actual distance:

4 \times 50\,000 = 200\,000

cm

Convert to kilometres:

200\,000 \div 100\,000 = 2

km

Q12·Challenging

Amira and Ben share some money in the ratio

3:7

. Amira receives

\$120

. How many dollars does Ben receive?

Explanation

Amira's share is

3

parts and equals

\$120

.

Value of one part:

\$120 \div 3 = \$40

Ben's share (

7

parts):

7 \times \$40 = \$280

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