Simplifying and Using Ratios

Ratios are one of the most important and widely tested topics in the AQA GCSE Mathematics specification. A ratio compares the size of two or more quantities relative to each other. This lesson covers how to write, simplify, and use ratios in a range of contexts — from recipe problems to map scales.

What Is a Ratio?

A ratio compares two or more quantities and shows the relative size of each. Ratios are written using a colon, for example 3 : 5, and they have no units — they simply tell us how many times bigger or smaller one quantity is compared to another.

Concept	Explanation	Example
Ratio	A comparison of two or more quantities	Boys to girls = 3 : 5
Parts	Each number in the ratio represents a "part"	3 parts and 5 parts = 8 parts total
Order	The order of the numbers matters	3 : 5 is NOT the same as 5 : 3
Units	Both quantities must be in the same unit before writing the ratio	Convert cm and m to the same unit first

Writing a Ratio

When writing a ratio, you must ensure the quantities are in the same units.

Worked Example 1: Write the ratio 40 cm to 2 m in its simplest form.

Convert to the same units: 2 m = 200 cm
Write as a ratio: 40 : 200
Simplify by dividing both by the HCF (40): 1 : 5

Exam Tip: Always check the units before writing a ratio. A very common mistake is to write 40 : 2 instead of converting metres to centimetres first.

Simplifying Ratios

Simplifying a ratio works exactly like simplifying a fraction — you divide every part by the highest common factor (HCF).

Method

Find the HCF of all the numbers in the ratio.
Divide each part by the HCF.
The result is the ratio in its simplest form.

Worked Example 2: Simplify 24 : 36.

HCF of 24 and 36 = 12
24 / 12 = 2
36 / 12 = 3
Simplified ratio = 2 : 3

Worked Example 3: Simplify 15 : 25 : 45.

HCF of 15, 25 and 45 = 5
15 / 5 = 3, 25 / 5 = 5, 45 / 5 = 9
Simplified ratio = 3 : 5 : 9

Simplifying Ratios Involving Fractions and Decimals

If a ratio contains fractions, multiply every part by the lowest common denominator. If it contains decimals, multiply by a power of 10 to eliminate the decimal places.

Worked Example 4: Simplify 0.6 : 1.5.

Multiply both by 10: 6 : 15
Divide both by HCF (3): 2 : 5

Worked Example 5: Simplify 1/3 : 1/2.

LCD of 3 and 2 is 6
Multiply both by 6: 2 : 3

Exam Tip: Exam questions often mix fractions and whole numbers in ratios (e.g. 2 : 1/4). Multiply through by the denominator to clear the fraction, then simplify.

Equivalent Ratios

Two ratios are equivalent if one can be obtained by multiplying (or dividing) every part of the other by the same number. This is identical to the concept of equivalent fractions.

Original Ratio	Multiply by	Equivalent Ratio
2 : 3	x 4	8 : 12
5 : 2	x 3	15 : 6
12 : 8	/ 4	3 : 2

Worked Example 6: The ratio of red beads to blue beads is 3 : 7. If there are 21 blue beads, how many red beads are there?

7 parts = 21, so 1 part = 3
Red beads = 3 parts = 3 x 3 = 9 red beads

Unit Ratios (1 : n and n : 1)

A unit ratio expresses the ratio in the form 1 : n or n : 1. This is useful for comparison and is commonly used for map scales and exchange rates.

Method

To write a ratio in the form 1 : n, divide both parts by the left-hand number. To write it in the form n : 1, divide both parts by the right-hand number.

Worked Example 7: Write 5 : 8 in the form 1 : n.

Divide both by 5: 1 : 1.6
Answer: 1 : 1.6

Worked Example 8: Write 12 : 5 in the form n : 1.

Divide both by 5: 2.4 : 1
Answer: 2.4 : 1

graph LR
    A[Original Ratio 5 : 8] --> B[Divide both by 5]
    B --> C[1 : 1.6]
    D[Original Ratio 12 : 5] --> E[Divide both by 5]
    E --> F[2.4 : 1]

Exam Tip: When a question says "write in the form 1 : n", give your answer to a reasonable degree of accuracy. If no rounding instruction is given, use exact decimals or round to 2 decimal places.

Using Ratios in Context

Ratios appear in many real-world GCSE problems including recipes, map scales, best-buy problems, and mixing solutions.

Recipe Problems

Worked Example 9: A recipe for 12 biscuits uses 150 g of flour, 80 g of butter, and 50 g of sugar. How much of each ingredient is needed to make 30 biscuits?

Scale factor = 30 / 12 = 2.5
Flour: 150 x 2.5 = 375 g
Butter: 80 x 2.5 = 200 g
Sugar: 50 x 2.5 = 125 g

Map Scales

Worked Example 10: A map has a scale of 1 : 25 000. Two towns are 8 cm apart on the map. What is the real distance in kilometres?

Real distance = 8 x 25 000 = 200 000 cm
Convert to km: 200 000 / 100 000 = 2 km

Summary

A ratio compares two or more quantities in the same units.
Simplify ratios by dividing each part by the HCF.
Ratios with fractions or decimals can be simplified by multiplying through.
Equivalent ratios are found by multiplying or dividing all parts by the same number.
Unit ratios (1 : n or n : 1) are useful for comparisons, map scales and conversions.
Always convert to the same units before writing a ratio.

Exam Tip: Show every step when simplifying ratios — especially the unit conversion. Marks are awarded for method even if your final answer is wrong.