Ratio
A ratio compares the sizes of parts of a whole. Almost every ratio question comes down to one move: work out what one part is worth, then scale up.
Part 1 of 3: Learn it
In short
- Simplify a ratio by dividing every part by the same number. Get the units the same first.
- To share in a ratio: add the parts, divide the total by that, then multiply.
- In the ratio a : b, the first share is a/(a + b) of the whole.
Where this is in your specification
Spec points: DfE R3, R4, R5 and R8
| Board | Topic: Ratio and proportion |
|---|---|
| DfE content | N13, R1, R2, R3, R4, R5, R6, R7, R8, R10, R13 |
| AQA 8300 | Ratio, proportion and rates of change |
| Edexcel 1MA1 | Ratio, proportion and rates of change |
| OCR J560 | 5 Ratio, proportion and rates of change |
| Eduqas C300 | HR1, HR3 to HR9, HR11, HR13 |
| Cambridge IGCSE 0580 | E1.11, E2.8 |
| National 5 C847 75 | Numerical skills: direct and inverse variation |
Simplifying ratios
Divide every part by the highest common factor: 45 : 60 = 3 : 4 (÷ 15). A ratio in its simplest form uses whole numbers, so 1.5 : 2 becomes 3 : 4 (× 2).
- Units must match: 2.5 m : 75 cm becomes 250 : 75, which is 10 : 3.
- For the form 1 : n, divide both parts by the first: 4 : 10 = 1 : 2.5.
Sharing in a ratio
- Add the parts to find how many parts there are in total.
- Divide the amount by the total number of parts to find one part.
- Multiply one part by each number in the ratio.
Check that your shares add up to the original amount.
Ratios and fractions
In a class where boys : girls = 3 : 5, there are 8 parts, so 3/8 of the class are boys and 5/8 are girls. The fraction of boys compared with girls is 3/5, which is a different question: read carefully which one is asked.
The unitary method
For direct proportion, find the value of one item and then multiply. If 6 pens cost £4.32, one pen costs £0.72, so 10 pens cost £7.20. Best buy questions work the same way: compare the price per gram, or the grams per penny.
Simplify 2.5 m : 75 cm.
Show the answer
250 : 75 = 10 : 3.
Part 2 of 3: See it worked
Worked examples
Example 1
Share £180 between Asha and Ben in the ratio 4 : 5.
- Total parts: 4 + 5 = 9
- One part: 180 ÷ 9 = 20
- Asha: 4 × 20 = 80, Ben: 5 × 20 = 100
Answer: Asha gets £80 and Ben gets £100.
Example 2
Sweets are shared in the ratio 3 : 7. The second share has 28 more sweets than the first. How many sweets are there altogether?
- The difference is 7 − 3 = 4 parts
- 4 parts = 28, so 1 part = 7
- Total = 10 parts = 70
Answer: 70 sweets (21 and 49).
Common mistakes
- Dividing the amount by one of the numbers in the ratio instead of by the total number of parts.
- Simplifying a ratio with different units, such as 50 cm : 2 m as 25 : 1.
- Mixing up the order: the ratio of red to blue is not the same as blue to red.
- Writing a ratio as a fraction with the wrong denominator: 2 : 3 means 2/5, not 2/3, of the whole.
Share 96 in the ratio 5 : 3.
Show the answer
8 parts, one part is 12, so 60 and 36.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Simplify 2.5 m : 75 cm.
250 : 75 = 10 : 3.
Share 96 in the ratio 5 : 3.
8 parts, one part is 12, so 60 and 36.
Red and blue beads are in the ratio 2 : 5. There are 14 red beads. How many blue beads are there?
2 parts = 14, so 1 part = 7 and blue = 5 × 7 = 35.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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