Division in a given ratio - KS3 Maths

Key points

Division in a given is also known as sharing in a given ratio.
Similar processes are used when dividing an amount in a ratio and finding a fraction of an amount.
To divide in a given ratio:
- can be used to support understanding
- a numerical method without bar models can also be used
Understanding the link between ratios and fractions is a valuable skill to practise.

Understanding the link between ratios and fractions

Examples

To understand the link between a and a fraction, it is important to know that:

the ratio represents the parts that make up the whole
the total of the parts gives the of the fractions
each part of the ratio gives the for that fractional part

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There are nine circles. 4 circles are orange and 5 circles are blue. Find the fraction of orange circles, the fraction of blue circles and the ratio of orange to blue circles.
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4 out of the 9 circles are orange. The fraction of orange circles is 4⁄9
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5 out of the 9 circles are blue. The fraction of blue circles is 5⁄9
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There are 4 orange circles for every 5 blue circles. The ratio of orange circles to blue circles is therefore 4 : 5
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What fraction of the bar is blue? What fraction of the bar is orange?
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There are 7 equal parts. 2 parts are blue. 2⁄7 of the bar is blue. 5 parts are orange. 5⁄7 of the bar is orange.
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What is the ratio of orange to blue parts? There are 5 orange parts for every 2 blue parts. The ratio of orange to blue parts is 5 : 2. The first number in the ratio is 5 because the orange parts are mentioned first. The numbers are in the same order as the description.
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What fraction of the bar is blue? What is the ratio of blue to orange parts?
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1 of the 4 parts is blue. ¼ of the shape is blue. There is 1 blue part for every 3 orange parts. The ratio of blue to orange parts is 1 : 3

Question

The ratio of the number of green to orange to blue circles is 4 : 2 : 1

What fraction of the circles are orange?

Using bar models to divide in a given ratio

To divide in a using :

Draw a to represent the whole and split it into the total number of parts.
Find the value of one part by dividing the whole by the total number of parts.
Calculate the value of each share of the ratio by multiplying the number of parts in each share of the ratio by the value of one part.

Examples

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The bar represents the whole. Divide 40 in the ratio 5 : 3
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The whole (40) is in the ratio 5 : 3. The total number of parts (5 + 3) is 8. Split the whole (40) into 8 equal parts.
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To find the value of one part, divide the whole (40) by the total number of parts (8). 40 ÷ 8 = 5. The value of one part is 5
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Multiply the number of parts in each share of the ratio by the value of one part (5) - this calculates each share of 40. The first share is 5 × 5 which is 25. The second share is 3 × 5 which is 15. 40 divided in the ratio 5 : 3 is 25 : 15
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Tariq and Annie share £90 in the ratio 2 : 3. How much money do they each receive?
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Draw bars to represent the parts Tariq and Annie receive. Tariq will receive 2 parts and Annie will receive 3 parts. The total of the 5 parts is £90
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To find the value of one part, divide the whole (90) by the total number of parts (5). 90 ÷ 5 = 18. The value of one part is 18. One part is worth £18
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Multiply the number of parts each person receives (2 and 3) by the value of one part (18) to calculate how much they receive. Tariq has 2 parts - he receives 2 × 18. Annie has 3 parts - she receives 3 × 18. £90 is divided between Tariq and Annie in the ratio 2 : 3. Tariq receives £36 and Annie receives £54. To check the calculation, Tariq’s and Annie’s shares should add up to £90. £36 + £54 = £90

Question

There are 30 dogs (German Shepherds, Golden Retrievers and Labradors) at a doggy day care centre in the ratio 1 : 5 : 4. How many Labradors are there?

How to divide in a given ratio – numerical method

To divide an amount in a given :

the parts of the ratio to get the total number of parts.
Find the value of one part by dividing the amount by the total number of parts.
Find the value of each share in the ratio by multiplying the number of parts in each share by the value of one part.

Examples

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Divide 600 in the ratio 4 : 3 : 5
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Sum the parts of the ratio. 4 + 3 + 5 = 12
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To find the value of one part, divide the amount (600) by the total number of parts (12). 600 ÷ 12 = 50. The value of one part is 50
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To find the value of each share, multiply the number of parts of each share (4, 3 and 5) by the value of one part (50). 600 shared in the ratio 4 : 3 : 5 is 200 : 150 : 250
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Dai, Jim and Kate share £300 in the ratio 8 : 5 : 2. How much money do they each receive?
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Sum the parts of the ratio (8, 5 and 2) to get the total number of parts. 8 + 5 + 2 = 15. The total number of parts is 15
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To find the value of one part, divide the amount (300) by the total number of parts (15). 300 ÷ 15 = 20. The value of each part is 20
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To find the value of each share in the ratio, multiply each ratio share (8, 5 and 2) by the value of one part (20). The calculations are 8 × 20, 5 × 20 and 2 × 20
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8 × 20 = 160, 5 × 20 = 100 and 2 × 20 = 40. Dai, Jim and Kate share £300 in the ratio 8 : 5 : 2. Dai receives £160, Jim receives £100 and Kate receives £40

Question

Divide 72 in the ratio 7 : 3 : 8

Game - Sharing ratios given a total

Complete this puzzle on sharing ratios given a total from our Divided Islands game.

Play the full Divided Islands game.

Practise division in a given ratio

Quiz

Practise division in a given ratio in this quiz. You may need a pen and paper to complete these questions.

Real-world maths

Video

Ratios are often used in jobs where liquids must be mixed in the correct proportions.

Watch the video to see how a hairdresser needs to mix a hair colour dye solution and a developer in a specific ratio to achieve a particular shade safely. The ratio can vary depending on the types of products being mixed.