Maths · Ratio & Proportion
Direct proportion
Solve proportion word problems.
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What is direct proportion?
Two quantities are in direct proportion when they increase or decrease at the same rate. If you double one quantity, the other doubles too. If you triple one, the other triples. The ratio between them stays constant.
Finding the constant of proportionality
To solve proportion problems, first find the constant multiplier (the constant of proportionality). Divide one quantity by its paired quantity. Then use this constant to find unknown values.
The unitary method
Another strategy is the unitary method: find the value for one unit first. For example, if 5 apples cost £2, find the cost of 1 apple. Then multiply to find any other amount.
Setting up proportion equations
You can write proportions as fractions that are equal. If 3 pens cost £6 and you want the cost of 7 pens, write 3/6 = 7/x. Cross-multiply to solve: 3x = 42, so x = 14.
Worked examples
1. A car travels 150 miles using 10 litres of petrol. How many litres are needed for 225 miles?
- Find miles per litre: 150 ÷ 10 = 15 miles per litre.
- Now find litres needed for 225 miles: 225 ÷ 15 = 15 litres.
- Answer: 15 litres.
2. 8 identical books weigh 2.4 kg. What is the weight of 12 books?
- Find the weight of 1 book: 2.4 ÷ 8 = 0.3 kg.
- Multiply by 12: 0.3 × 12 = 3.6 kg.
- Answer: 3.6 kg.
3. 5 workers take 6 hours to build a wall. Working at the same rate, how long for 3 workers?
- This is inverse proportion (fewer workers means more time), not direct proportion.
- Total work = 5 × 6 = 30 worker-hours.
- Time for 3 workers: 30 ÷ 3 = 10 hours.
Try it yourself
6 pencils cost £1.80. How much do 10 pencils cost?
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