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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.
Express 750 g to 2.5 kg as a ratio in its simplest form.
Head of Maths note: Always simplify fully. 75:250 or 15:50 would lose the accuracy mark on AQA mark schemes.
Simplify 18:50.
A 400 g tin of beans costs £0.96. A 650 g tin costs £1.43. Which represents the better value?
Equivalently, using a unit ratio approach we can write the cost per 100 g: 496=24p vs 6.5143=22p per 100 g — the same conclusion.
A map uses the scale 1:50000. The real distance between two villages is 4.2 km. What is this distance on the map?
A plan has scale 1:200. A room on the plan has area 30 cm2. What is the real floor area?
Head of Maths note: A frequent error is using 200 instead of 2002 for area. Students must understand that when a length is scaled by k, area scales by k2 and volume by k3.
A cordial is mixed with water in the ratio 1:7 by volume. A jug holds 1.6 litres of the final drink. Work out the volume of cordial required.
In a class, 60% of pupils study French and the rest study Spanish. Express the ratio of French to Spanish learners in its simplest form.
£1 = \1.25.Convert£320todollarsandthen$200$ back into pounds (to the nearest penny).
Exchange rates are quoted as a unit ratio (1:1.25) because a single unit of one currency is easiest to compare.
Exam-style question: "In a bag there are red, blue and green counters in the ratio 2:3:5. Given that there are 12 more green counters than red, work out the total number of counters in the bag. [3 marks]"
Grades 3–4 (Foundation core): The pupil recognises the question involves a ratio of three parts. They may write "total parts =2+3+5=10" and then attempt to divide 12 by 10. A partial answer might state one part =3 by correct reasoning on the difference in parts, but omit the final total. Expect: some method marks for total parts, incomplete final answer.
Grades 5–6 (Foundation top / Higher lower): The pupil correctly identifies that the difference between green and red in the ratio is 5−2=3 parts. They compute 3 parts=12 so 1 part=4. Total =10×4=40 counters. Working is clear: "difference in parts =3; one part =4; total =40." They earn full marks for a standard three-part ratio problem.
Grades 7–9 (Higher confidence): The pupil lays out the solution algebraically, letting one part equal x. They write: "Red =2x, Green =5x, so 5x−2x=12⇒3x=12⇒x=4." They immediately compute total =(2+3+5)x=10×4=40 and verify by re-checking the direct proportion between parts and counter counts. They would extend by stating the proportion as counters∝parts with constant k=4.
Head of Maths note: The language "difference in parts" rather than "difference between ratios" is the precise terminology. Pupils who use proportion, ratio and unit conversion vocabulary correctly consistently reach Grade 7+.
AQA alignment: This content is aligned with AQA GCSE Mathematics (8300) specification — specifically Topic R (R1 Ratio notation and simplification, R2 Using ratios in context, R5 Scales and maps, R7 Percentage as a ratio, R9 Direct proportion reasoning). Assessed on Papers 1, 2, 3.