Module 5 (M5) – Algebra - Ratio

Part ofMaths M5: Number & Algebra

What is a ratio?

Ratios are used to show how things are shared. For example, the necklace in the image below has a pattern of two red beads for every three yellow beads.

An image showing fifteen beads on a string necklace, with 6 red beads and 9 yellow beads.

The ratio of red beads to yellow beads is \(2:3\).

Ratios can have more than two numbers for example, \(3:4:2\).

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Simplifying ratios

Ratios can be fully simplified just like fractions.

To a ratio, divide all of the numbers in the ratio by the same number until they cannot be divided any more.

Example

Simplify \(6:12\).

Solution

Divide both numbers by 2:

\(6:12=3:6\)

Divide both numbers by 3:

\(3:6=1:2\)

A quick way of doing this in just one step is to divide by the highest common factor of all the numbers in the ratio. In this example, the highest common factor of 6 and 12 is 6. Dividing each number by the HCF:

\(3:6=1:2\)

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  1. An image showing 18 dots, six of which are red and 12 which are yellow. Ratio reads 6:12

    A ratio of 6:12 with 6 red coloured counters and 12 yellow coloured counters spread out horizontally.

Simplify these rations fully:

  • \(4:6\)
  • \(9:12\)
  • \(5:10:15\)

Simplifying more difficult ratios

Ratios with decimals

To simplify a ratio with a decimal:

  • multiply the numbers to make them all
  • divide both numbers by the

Example

Simplify \(6:1.5\).

Solution

Multiply both numbers by 2:

  • \(6:1.5\times2=12:3\)

Divide both numbers by 3:

  • \(12:3\div3=4:1\)

Ratios with factions

To simplify a ratio with fractions:

  • convert the fractions so they have a
  • multiply both fractions by the common denominator
  • simplify by dividing by the highest common factor

Example

Simplify \(\frac{1}{2}:\frac{3}{4}\)

Solution

Convert so the fractions have a common denominator:

\(\frac{1}{2}:\frac{3}{4}\rightarrow\frac{2}{4}:\frac{3}{4}\)

Multiply by 4:

\(\frac{2}{4}:\frac{3}{4}\times 4=2:3\)

The highest common factor is 1 so this is the simplest form.

Ratios in different units

To simplify ratios that are in different units:

  • convert the larger unit to the smaller unit
  • simplify the ratio as normal

Example

Simplify \(25mm:5cm\).

Solution

Convert centimetres into millimetres by multiplying by 10:

\(5\text{cm}\times 10 = 50\text{mm}\)

\(25:50\)

Simplify by dividing by 25:

\(1:2\)

Ratios as fractions

Ratios can be used to show fractions and of amounts.

Example

A room has to be painted blue and yellow in the ratio \(2:3\). Express the proportion of the room that has to be painted in each colour as a fraction.

Solution

There are five parts in this ratio: \(2 \text{blue} + 3 \text{yellow} = 5 \text{total}\)

Five bars, representing a fraction, where two are coloured blue and three coloured yellow.

The fraction painted blue is \(\frac{2}{5}\) and the fraction painted yellow is \(\frac{3}{5}\).

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Sharing in a given ratio

Lots of things in everyday life are shared in . Money is shared, liquids are mixed and teams are assigned using ratios.

Drawing a diagram to represent the ratio can make these tasks easier.

Example

James and Helen get pocket money in the ratio \(3:5\). The total amount of pocket money they are given is £24. How much money do they each get?

Solution

The amount is divided into 8 equal parts since \(3+5=8\). Draw a rectangle with 8 sections and divide it in the ratio \(3:5\), labelling the two parts with the names James and Helen.

Since James’s name comes first he gets three of the parts as the 3 is the first number in the ratio. Helen gets 5 parts, since her name is second.

Share the £24 between the 8 parts by dividing 24 by 8 and put the amount into each part of the diagram.

An image illustrating a ratio of 3:5, with eight blocks, three of which are blue and labelled James and five which are red and labelled Helen. Each block is labelled £3.

\(24\div8=3\)

The diagram shows that:

  • James gets \(3\times \pounds 3 = \pounds 9\)
  • Helen gets \(5\times \pounds 3 = \pounds 15\)

This can also be done when fractions are involved.

Example

To make pink paint, red and white paint can be mixed in the ratio \(1:2\). If you need to make 4 litres of paint, how much red and white paint would you need?

Solution

The ratio has \(1+2=3\) parts.

An image showing the ratio of making pink paint, using a ratio of red and white paint to 1:2.

4 divided by 3 \(3=\frac{4}{3}\)

Each part is worth \(\frac{4}{3}\) litres.

The diagram shows that:

  • the amount of red paint needed is \(\frac{4}{3} /times 1 = \frac{4}{3} \text{litres}\). This can be written as \(1\frac{1}{3} \text{litres}\)

  • the amount of white paint needed is \(\frac{4}{3} /times 2 = \frac{8}{3} \text{litres}\). This can be written as \(2 \frac{2}{3} \text{litres}\)

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Finding unknown ratios

Using a diagram can help to answer problems when one or more ratios are incomplete or unknown.

Example

Some boys and girls are divided into teams in the ratio \(3:4\). There are 8 girls in one team. How many boys are there in the team?

Solution

There are 8 girls shared into 4 parts. Each part has \(\frac{8}{4}=2\text{girls}.

Fill in the diagram using this information.

An image illustrating a ratio of 3:4, with eight blocks, three of which are blue and labelled Boys and four which are red and labelled Girls. The red blocks are labelled 2.

Now because the amounts are shared equally between all of the parts, the rest of the table can be filled in.

An image illustrating a ratio of 3:4, with eight blocks, three of which are blue and labelled Boys and four which are red and labelled Girls. All the blocks are labelled 2.

From the table it is clear that the number of boys in the team is \(2 \times 3 = 6\)

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Test yourself

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