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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.
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 |
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.
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 a ratio works exactly like simplifying a fraction — you divide every part by the highest common factor (HCF).
Worked Example 2: Simplify 24 : 36.
Worked Example 3: Simplify 15 : 25 : 45.
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.
Worked Example 5: Simplify 1/3 : 1/2.
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.
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?
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.
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.
Worked Example 8: Write 12 : 5 in the form n : 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.
Ratios appear in many real-world GCSE problems including recipes, map scales, best-buy problems, and mixing solutions.
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?
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?
Exam Tip: Show every step when simplifying ratios — especially the unit conversion. Marks are awarded for method even if your final answer is wrong.