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Ratios are one of the most frequently examined topics in the Edexcel GCSE Mathematics (1MA1) specification. They appear across all three papers — Paper 1 (non-calculator), Paper 2 (calculator) and Paper 3 (calculator). This lesson covers how to write, simplify and use ratios, including dividing quantities in a given ratio and combining ratios.
A ratio compares two or more quantities and shows their relative sizes. 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.
Edexcel 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. On Paper 1 (non-calculator) you must do all simplifying by hand, so look for common factors.
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 (LCD). If it contains decimals, multiply by a power of 10 to eliminate the decimal places, then simplify.
Worked Example 4: Simplify 0.6 : 1.5.
Worked Example 5: Simplify 1/3 : 1/2.
Worked Example 6: Simplify 2 : 3/4.
Edexcel Exam Tip: On Paper 1 (non-calculator), ratio-with-fractions questions appear regularly. Multiply through by the LCD to clear all fractions in one step.
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 7: 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 one part of the ratio as 1. This is useful for comparing and converting scales.
Worked Example 8: Express 8 : 5 in the form 1 : n.
Worked Example 9: Express 8 : 5 in the form n : 1.
Edexcel Exam Tip: Map scale questions often require you to express a ratio in the form 1 : n. You may also be asked to find a real-life distance given a map scale — always check the units of your final answer.
This is one of the most common ratio question types in the Edexcel exam.
Worked Example 10: Divide 180 in the ratio 2 : 3 : 4.
Check: 40 + 60 + 80 = 180
Worked Example 11: Alex and Beth share 240 pounds in the ratio 5 : 7. How much does Beth receive?
Worked Example 12: Two numbers are in the ratio 3 : 8. The difference between them is 30. Find both numbers.
Check: 48 - 18 = 30
Sometimes you need to combine two separate ratios into a single ratio.
Worked Example 13: A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C.
Worked Example 14: X : Y = 5 : 3 and Y : Z = 6 : 7. Find X : Y : Z.
Edexcel Exam Tip: Combining ratios questions are typically worth 3-4 marks and appear on both Foundation and Higher papers. Always make the common term equal before combining.
| Mistake | How to Avoid It |
|---|---|
| Forgetting to convert to the same units | Always check units before writing a ratio |
| Dividing by different numbers when simplifying | Always use the same HCF for all parts |
| Getting the order wrong in the ratio | Read the question carefully — the order matters |
| Using the difference instead of the total when dividing in a ratio | Re-read whether the question gives a total amount or a difference |
| Not simplifying fully | Check there is no further common factor |
Edexcel Exam Tip: Ratio questions can appear as part of bigger problems involving fractions, percentages or algebra. Be ready to use ratio skills alongside other topics.