Two quantities are in direct proportion when they increase and decrease at the same rate. If one doubles, the other doubles; if one halves, the other halves. The unitary method finds the value of one unit first, then scales to any amount.
What is direct proportion?
Two quantities are directly proportional if their ratio stays constant.
If y is directly proportional to x, then:
y = kx (where k is the constant of proportionality)
This means:
- When x doubles, y doubles.
- When x is multiplied by any number, y is multiplied by the same number.
- The graph of y against x is a straight line through the origin.
Examples of directly proportional situations
| Situation | Two quantities | Why they are proportional |
|---|---|---|
| Shopping | Number of items, total cost | Buying twice as many costs twice as much |
| Speed | Distance, time (at constant speed) | Twice the time at the same speed = twice the distance |
| Recipe scaling | Ingredients, number of servings | Double the servings requires double each ingredient |
| Exchange rates | Pounds, euros | Double the pounds converts to double the euros |
The unitary method
The unitary method is the standard KS3 strategy for solving direct proportion problems:
- Find the value of one unit (by dividing).
- Multiply by the number of units you need.
This two-step approach works for any directly proportional situation.
Worked examples
Example 1 — basic unitary method
5 pens cost £3.75. How much do 8 pens cost?
Step 1 — find the cost of 1 pen: £3.75 ÷ 5 = £0.75 per pen
Step 2 — find the cost of 8 pens: £0.75 × 8 = £6.00
Example 2 — recipe scaling
A recipe for 4 people needs 320 g of flour. How much flour is needed for 7 people?
Step 1 — flour per person: 320 ÷ 4 = 80 g per person
Step 2 — flour for 7 people: 80 × 7 = 560 g
Example 3 — currency exchange
£1 = €1.18. How many euros do you get for £65?
Step 1 — already know 1 unit: €1.18 per £1
Step 2 — for £65: 1.18 × 65 = €76.70
Example 4 — finding the constant of proportionality
y is directly proportional to x. When x = 4, y = 20. Find y when x = 9.
Step 1 — find k (value when x = 1): k = y ÷ x = 20 ÷ 4 = 5
So y = 5x
Step 2 — find y when x = 9: y = 5 × 9 = 45
Example 5 — working backwards
8 workers can build a wall in 6 days. How many workers are needed to build the same wall in 4 days?
Note: this is NOT direct proportion — more workers means fewer days. It is inverse proportion. For direct proportion, both quantities must move in the same direction.
(8 × 6 = 48 worker-days total; 48 ÷ 4 = 12 workers needed — but this is inverse proportion, shown here to highlight the contrast.)
Recognising direct proportion on a graph
The graph of two directly proportional quantities is always:
- A straight line
- That passes through the origin (0, 0)
If the straight line does not pass through the origin, the quantities are linearly related but not directly proportional.
| Feature | Direct proportion | Linear but not proportional |
|---|---|---|
| Graph type | Straight line | Straight line |
| Passes through origin? | Yes | No |
| Example | y = 3x | y = 3x + 5 |
Direct proportion in the national curriculum
The DfE's KS3 mathematics programme of study requires pupils to use ratio and proportion to solve problems, and to understand the connection between two directly proportional quantities. The statutory national curriculum for Key Stage 3 and 4 confirms that problems involving the unitary method and the constant of proportionality are regularly examined at both KS3 and GCSE, including on non-calculator papers.
Frequently asked questions
How is direct proportion different from inverse proportion?
In direct proportion, both quantities increase together (more of one → more of the other). In inverse proportion, as one quantity increases the other decreases (more of one → less of the other). For example, speed and journey time at a fixed distance are inversely proportional: double the speed halves the time.
Does the unitary method always work for proportion problems?
Yes, whenever two quantities are directly proportional. The method works because the ratio between the two quantities is constant. If 3 books cost £12, then 1 book costs £4, and any number of books at the same price can be found by multiplying £4 by that number. The method is reliable as long as the proportionality assumption holds.
How do I write a direct proportion equation?
Write y = kx, where k is the constant of proportionality. Find k by substituting a known pair of values and dividing: k = y ÷ x. Once you have k, you can find y for any x (or x for any y) by substituting into y = kx.
What does the gradient of a direct proportion graph tell you?
The gradient of the straight-line graph is the constant of proportionality k. A steeper gradient means a larger k — one unit of x corresponds to more units of y. For example, if a currency exchange gives €1.30 per £1, the gradient of the euros-against-pounds graph is 1.30.
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