© Boardworks Ltd 2004 1 of 47 N8 Ratio and proportion KS3 Mathematics.

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N8 Ratio and proportion

KS3 Mathematics

N8.1 Ratio

Contents

N8.5 Ratio and proportion problems

N8.3 Direct proportion

N8.2 Dividing in a given ratio

N8.4 Using scale factors

Stacking blocks

A ratio compares the sizes of parts or quantities to each other.

For example,

What is the ratio of red counters to blue counters?

red : blue

= 9 : 3

= 3 : 1

For every three red counters there is one blue counter.

A ratio compares the sizes of parts or quantities to each other.

The ratio of blue counters to red counters is not the same as the ratio of red counters to blue counters.

blue : red

= 3 : 9

= 1 : 3

For every blue counter there are three red counters.

For example,

What is the ratio of blue counters to red counters?

What is the ratio of red counters to yellow counters to blue counters?

red : yellow : blue

= 12 : 4 : 8

= 3 : 1 : 2

For every three red counters there is one yellow counter and two blue counters.

Simplifying ratios

Ratios can be simplified like fractions by dividing each part by the highest common factor.

For example,21 : 35

= 3 : 5÷ 7 ÷ 7

For a three-part ratio all three parts must be divided by the same number.

For example,6 : 12 : 9

= 2 : 4 : 3÷ 3 ÷ 3

Equivalent ratio spider diagrams

When a ratio is expressed in different units, we must write the ratio in the same units before simplifying.

Simplify the ratio 90p : £3

First, write the ratio using the same units.

90p : 300p

When the units are the same we don’t need to write them in the ratio.

90 : 300÷ 30 ÷ 30

= 3 : 10

Simplifying ratios with units

Simplify the ratio 0.6 m : 30 cm : 450 mm

First, write the ratio using the same units.

60 cm : 30 cm : 45 cm

60 : 30 : 45

÷ 15 ÷ 15

= 4 : 2 : 3

Simplifying ratios with units

When a ratio is expressed using fractions or decimals we can simplify it by writing it in whole-number form.

Simplify the ratio 0.8 : 2

We can write this ratio in whole-number form by multiplying both parts by 10.

0.8 : 2

= 8 : 20

× 10 × 10

÷ 4 ÷ 4

= 2 : 5

Simplifying ratios containing decimals

Simplifying ratios containing fractions

Simplify the ratio : 4 23

We can write this ratio in whole-number form by multiplying both parts by 3.

23 : 4

× 3 × 3

= 2 : 12

÷ 2 ÷ 2

= 1 : 6

Comparing ratios

We can compare ratios by writing them in the form 1 : m or m : 1, where m is any number.

For example, the ratio 5 : 8 can be written in the form 1 : m by dividing both parts of the ratio by 5.

5 : 8÷ 5 ÷ 5

= 1 : 1.6

The ratio 5 : 8 can be written in the form m : 1 by dividing both parts of the ratio by 8.

5 : 8÷ 8 ÷ 8

= 0.625 : 1

Comparing ratios

The ratio of boys to girls in class 9P is 4:5.The ratio of boys to girls in class 9G is 5:7.Which class has the higher proportion of girls?

The ratio of boys to girls in 9P is 4 : 5÷ 4 ÷ 4

= 1 : 1.25

The ratio of boys to girls in 9G is 5 : 7÷ 5 ÷ 5

= 1 : 1.4

9G has a higher proportion of girls.

Contents

N8.1 Ratio

Mixing paint

Ratios and proportions on a metre rule

Dividing in a given ratio

A ratio is made up of parts.

We can write the ratio 2 : 3 as

2 parts : 3 parts

The total number of parts is

2 parts + 3 parts = 5 parts

Divide £40 in the ratio 2 : 3.

£40 ÷ 5 = £8

We need to divide £40 by the total number of parts.

Divide £40 in the ratio 2 : 3.

Each part is worth £8 so

2 parts = 2 × £8 = £16

and 3 parts = 3 × £8 = £24

£40 divided in the ratio 2 : 3 is

£16 : £24

Always check that the parts add up to the original amount.

£16 + £24 = £40

A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1.How much of each type of juice is contained in 750 ml of the cocktail?

First, find the total number of parts in the ratio.

6 parts + 3 parts + 1 part = 10 parts

Next, divide 750 ml by the total number of parts.

750 ml ÷ 10 = 75 ml

A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1.How much of each type of juice is contained in 750 ml of the cocktail?

Each part is worth 75 ml so,

6 parts of orange juice = 6 × 75 ml = 450 ml

3 parts of lemon juice = 3 × 75 ml = 225 ml

1 part of lime juice = 75 ml

Check that the parts add up to 750 ml.

450 ml + 225 ml + 75 ml = 750 ml

Dividing in a given ratio spider diagram

Contents

N8.1 Ratio

Proportion

There are many ways to express a proportion. For example,

We can express this proportion as:

12 out of 16 3 in every 434

0.75 or 75%

Proportion compares the size of a part to the size of a whole.

What proportion of these counters are red?

Proportional sets

Direct proportion problems

3 packets of crisps weigh 90 g.How much do 6 packets weigh?

3 packets weigh 90 g.× 2

6 packets weigh× 2

120 g.

If we double the number of packets then we double the weight.

The number of packets and the weights are in direct proportion.

3 packets weigh 90 g.÷ 3

1 packet weighs÷ 3

If we divide the number of packets by three then divide the weight by three.

Once we know the weight of one packet we can work out the weight of any number of packets.

3 packets weigh 90 g.÷ 3 ÷ 3

1 packet weighs 30 g.× 7 × 7

7 packets weigh 210 g.

This is called using a unitary method.

Contents

N8.1 Ratio

Using scale factors

How can we get from 4 to 5 using only multiplication and division?

We could divide 4 by 4 to get 1 and then multiply by 5.

(4 ÷ 4) × 5 = 5

We could also multiply 4 by 5 to get 20 and then divide by 4.

(4 × 5) ÷ 4 = 5

Using scale factors

How can we divide by 4 and multiply by 5 in a single step?

Dividing by 4 and multiplying by 5 is equivalent to × 54

4 × 54 5=

We call the 54

a multiplier or scale factor.

We can write the scale factor as a decimal,54

= 1.25

We can also write it as a percentage,54

= 125%

Using a diagram to represent scale factors

We can represent the scaling from 4 to 5 using a diagram:

1÷ 4 or

Using scale factors

How can we get from 5 to 4 using only multiplication and division?

We could divide 5 by 5 to get 1 and then multiply by 4.

(5 ÷ 5) × 4 = 4

We could also multiply 5 by 4 to get 20 and then divide by 5.

(5 × 4) ÷ 5 = 4

Using scale factors

How can we divide by 5 and multiply by 4 in a single step?

Dividing by 5 and multiplying by 4 is equivalent to × 45

5 × 45 4=

We call the 45

a multiplier or scale factor.

We can write the scale factor as a decimal,45

We can also write it as a percentage,45

Using a diagram to represent scale factors

We can represent the scaling from 5 to 4 using a diagram:

÷ 5 or

Inverse scale factors

To scale from 4 to 5 we multiply by

To scale from 5 to 4 we multiply by

When we scale from a smaller number to a larger number the scale factor must be more than 1.

When we scale from a larger number to a number smaller the scale factor must be less than 1.

Scale factor diagrams

Using scale factors

To scale a to b we multiply byba

To scale b to a we multiply byab

For example,

Using scale factors and direct proportion

We can use scale factors to solve problems involving direct proportion.

To scale from £8 to £2 we × 14

or × 0.25

£8 is worth 13€

£2 is worth

or × 0.25

or × 0.25(13 ÷ 4)€

= 3.25€

£8 is worth 13 euros.How much is £2 worth?

£8 is worth 13€

× 138

or × 1.625

£2 is worth (2 × 1.625)€ = 3.25€

× 138

or × 1.625 Alternatively, to scale from 8 to 13 we

× 138

or × 1.625

× 138

or × 1.625

We can convert between any number of pounds or euros using

× 813

or × 0.615 (to 3 dp)

pounds euros

Contents

N8.1 Ratio

Direct proportion spider diagrams

Cog wheels

© Boardworks Ltd 2004 1 of 47 N8 Ratio and proportion KS3 Mathematics.

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