# Ratio

**Ratio**

**Ratios** is used to compare two amounts or parts against each other. Ratios can also be shown as a fraction, decimal and percentage multipliers.

For example, what is the ratio of green circles to orange?

Orange : green

= 9 : 6

= 3 : 2

Therefore for every three orange circle there is two green circles.

Example 2: What is the ratio of orange circles to yellow circles to green circles?

orange : yellow : green

= 15 : 3 : 12

= 5 : 1 : 4

For every five orange circle, there is one yellow circle and four blue circles.

Point to note:

- As per the example above, ratios can compare more than two parts or quantities

**Simplifying ratios**

It is possible to simplify ratios similar to fractions by dividing each part by the highest common factor.

For example,

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

For example,

**Simplifying ratios with units**

A ratio which is expressed in different units must be written down in the ratio which is the same units before simplifying.

For example: simplify 30p : £9

The first step is to write the ratio using the same units:

30p : 900p

Find a common number to simplify both sides:

30 : 900 (both sides can be divided by 30)

1 : 30

Example 2: Simplify 0.9m : 60 cm : 900 mm

The first step is to write using the same units

900 cm : 60 cm : 90cm

The second step is to find a common number to simplify all sides:

900 cm : 60 cm : 90cm (divide all sides by 30)

30 : 2 : 3

**Simplifying ratios containing decimals**

A ratio which contains a decimal or a fraction is simplified by writing it in its whole number form.

For example, simplify 0.8 : 2

Step 1 – convert the ratio to a whole number

0.8 : 2 (multiply both sides by 10, this is always carried out if the decimal only has one decimal point, if it has two decimal points, then it is multiplied by 100, etc)

= 8 : 20

Step 2 – express the ratio into its simplest form

= 8 : 20 (divide both sides by 4)

= 2 : 5

**Simplifying ratios containing fractions**

The simplification of ratio with fractions is a similar methodology as with decimal points where the ratio is converted to whole numbers first.

For example: simplify : 4

Step 1: convert the fraction into a whole number

= : 4 (multiply both sides by three)

= 2 : 12

Step 2 – express the ratio into its simplest form

= 2 : 12 (divide both sides by two)

= 1 : 6

**Comparing ratios**

Ratios can be compared 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 6.

5 : 8 (divide both sides by 8)

0.625 : 1 (although 0.625 can be written as a fraction, when comparing ratios, it is easier to compare in decimals)

Point to note:

- Decimals in this case should be rounded to two decimal places if there is no other instruction

**Writing ratios as fractions**

Ratios can also be written as a single fraction. For example, when investigating the lengths of the sides in a right angled triangle, the ratio of the length of the opposite to the length go the adjacent would be written in the following way:

opposite : adjacent, however this would be written as a ratio in the following way:

Note, this ratio is also called the tangent of the angle θ.

For example: what is the ratio of the height and width of a picture frame?

Using a ratio notation:

height : width

= 7.5 : 12.5 (divided both sides by 2.5)

= 3 : 5

As a fraction:

height/width = (divide both sides by 2.5)

= (the height is of the width)