Two quantities are in direct proportion if when one doubles, the other doubles too — they increase or decrease at the same rate. For example, if you buy twice as many apples, you pay twice as much.
In plain words: more of one always means proportionally more of the other. The cost of one item stays the same, so you can always work out the price of one and then scale up or down.
💡 Key fact: Direct proportion means as one quantity doubles, so does the other. If you draw a graph, it is always a straight line that starts at zero (buy nothing, pay nothing).
✓ Quick Check 1 — What is Direct Proportion?
Question 1 of 10
The Unitary Method
Find the value for 1 unit, then scale up or down.
Example 1: 5 pens cost £3.20. How much do 8 pens cost?
Step 1
Cost of 1 pen: \(£3.20 \div 5 = £0.64\)
Step 2
Cost of 8 pens: \(£0.64 \times 8 = \mathbf{£5.12}\)
Example 2: A car uses 12 litres of fuel to travel 96 km. How far can it travel on 20 litres?
Step 1
Distance per litre: \(96 \div 12 = 8\text{ km}\)
Step 2
Distance on 20 litres: \(8 \times 20 = \mathbf{160\text{ km}}\)
Scaling with Proportion
Proportion is used whenever you need to scale quantities up or down — for example, adjusting a recipe.
Identifying Direct Proportion
To check if two quantities are in direct proportion, work out the cost (or amount) for one in each row. If it is the same every time, the quantities are in direct proportion.
⚠ Watch out for inverse proportion: More of one thing = LESS of the other (for example, more workers means fewer days). That is the opposite of direct proportion — it has its own lesson.
Example 3: A market stall sells 3 oranges for 90p. At the same rate, how much for 8 oranges?
🔑 Quick test: in direct proportion, more of one always means more of the other, and the price (or amount) for one stays the same throughout.
✓ Quick Check 2 — Unitary Method & Scaling
Question 1 of 10
Practice Problems
Type your answer and press Check. Watch out for the trick question!
Practice: Direct Proportion — watch out for the trick question!
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Question
Answer
Result
1
5 pens cost £3. How much do 8 pens cost?
2
3m of wire costs £7.50. How much does 5m cost?
3
A car travels 240 miles in 4 hours. What is its speed in mph?
4
A recipe for 6 servings uses 300g of flour. How much flour for 10 servings?
5
£1 = $1.25. How many dollars is £80?
6
⚠ 4 workers paint a room in 3 hours. How long would 1 worker take? (Think carefully!)
7
A wall needs 15 bricks per metre. How many bricks are needed for a wall 7 metres long?
8
12 items cost £8.40. What is the unit price?
9
4kg of rice costs £6. How much does 7kg cost?
10
A machine produces 45 parts in 3 minutes. How many parts in 8 minutes?
✓ Quick Check 3 — Mixed Direct Proportion
Question 1 of 10
⚠ Things to Watch Out For
Confusing direct and inverse proportion: Direct: as one quantity increases, the other increases. Inverse: as one increases, the other decreases. Identify which type before calculating.
Not finding the unit rate first: If 5 items cost £8.50, find 1 item (÷5 = £1.70) then multiply for any quantity. Skipping the unit step leads to errors.
Proportional graphs must go through the origin: A direct proportion graph is a straight line through (0,0). If it doesn't pass through the origin, the relationship is not direct proportion.
Scaling by the wrong factor: If 3 workers build a wall in 12 hours, 6 workers (double) take 6 hours (half) — that's inverse, not direct. Direct proportion: 3 workers, 3 times as many bricks in the same time.
Units in the ratio: Keep units consistent throughout. Convert to the same unit before setting up the proportion.