GCSE Maths · Ratio, proportion and rates of change

Direct proportion GCSE Questions and Worked Answers

Two quantities are directly proportional when multiplying one by a factor multiplies the other by the same factor. Their ratio stays constant: y = kx, where k is the amount of y per unit of x.

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Start with the meaning

What you need to know about direct proportion

One pen costs £1.50, two cost £3 and four cost £6. There is no fixed fee or discount: every pen has the same price. Doubling the number of pens doubles the total cost. This same-factor relationship is direct proportion. The cost per pen stays £1.50, regardless of how many you buy.

See the idea first

One constant rate connects every pair

Let x be the number of pens and y their total cost in pounds. Dividing y by x gives 1.50 whenever x is non-zero. We call this constant k, so y = kx here means y = 1.5x. The symbol ∝ means ‘is proportional to’; it does not mean ‘equals’.

Same price, no fixed fee
Pens x	Cost y (£)	Cost per pen y/x (£)
1	1.5	1.5
2	3	1.5
4	6	1.5
6	9	1.5

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Scale a cost or recipe

What the problem asks: A recipe for 4 people uses 300 g of flour. How much for 10 people?

How to solve it: Each person needs 300 ÷ 4 = 75 g. For 10 people use 750 g, assuming the same portion size. Alternatively multiply by 10/4.

Test a proposed relationship

What the problem asks: For x = 2, 4, 6, the y-values are 5, 10, 15. Is y directly proportional to x?

How to solve it: Each y/x is 2.5, so the given pairs fit y = 2.5x. A finite table supports that model for these pairs; it does not by itself prove the rule outside them.

Construct a proportional equation (Higher)

What the problem asks: y is directly proportional to x, and y = 12 when x = 3. Find the equation.

How to solve it: The statement y ∝ x becomes the equation y = kx; k is the unknown constant. The known pair gives 12 = 3k, so k = 4 and y = 4x. The proportionality statement justifies using one constant for every pair.

Use proportion to a power (Higher)

What the problem asks: y is proportional to x², and y = 18 when x = 3. Find y when x = 5.

How to solve it: Now the constant ratio is y/x², not y/x. Use y = kx²: 18 = 9k gives k = 2, so the required value is 2 × 25 = 50.

A reliable routine

For a direct-proportion quantity problem

The unitary method finds the amount for one unit, then scales it to the required number. It works because the amount per unit is constant. Do not use it when a fixed charge, discount or changing rate breaks that assumption.

Identify the matched pair of quantities and the constant-rate assumption.
Divide to find the amount for one unit.
Multiply by the new number of units, keeping units consistent.
Check that the output changed by the same scale factor as the input.

Check: Two quantities both increasing is not enough. A taxi fare with a fixed starting charge may increase with distance but is not directly proportional to distance. The graph of y = kx is a straight line through the origin.

Fully worked

Direct proportion GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Find one, then many

2 marks

Question

Six identical notebooks cost £15. Find the cost of ten at the same unit price.

One notebook costs

15\div6=2.50

Ten cost

10\times2.50=25

The total is £25. The unit cost stays £2.50.

Example 2

Scale a recipe

3 marks

Question

A recipe for 6 people uses 450 g of rice. Find the amount for 14 people, keeping portions equal.

One portion uses

450\div6=75\text{ g}

Fourteen portions use

75\times14=1050\text{ g}

That is 1.05 kg. Multiply by the scale factor 14 ÷ 6. Do not add 8 g just because there are 8 more people.

Example 3

Check ratios

2 marks

Question

For x = 2, 4, 6, the corresponding y-values are 7, 13, 19. Are these pairs in direct proportion?

Compare the ratios.

7/2=3.5

13/4=3.25

They differ, so the pairs are not directly proportional. The y-values increase regularly, but fit y = 3x + 1 rather than y = kx.

Example 4

Construct the rule

Higher only3 marks

Question

y is directly proportional to x. When x = 8, y = 20. Find a formula for y in terms of x, then use it to find y when x = 14.

Write y = kx and use the known pair.

20=8k

k=2.5

Therefore y = 2.5x. At x = 14,

y=2.5(14)=35

Example 5

Proportional to a square

Higher only4 marks

Question

y is directly proportional to x². When x = 4, y = 48. Find y when x = 7.

The model is y = kx².

48=k(4^2)

k=3

So y = 3x².

y=3(7^2)=147

A change in x is squared before scaling y.

Example 6

Find the input

Higher only4 marks

Question

y is directly proportional to the square root of x. When x = 9, y = 12. Find x when y = 20.

Use

y=k\sqrt x

12=k\sqrt9=3k

Hence k = 4. For y = 20,

20=4\sqrt x

\sqrt x=5

x=25

Check: 4√25 = 20.

10 original questions · total 27 marks

Direct proportion GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 32 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Cost

2 marks

Four identical pencils cost £3. What do twelve cost at the same price per pencil?

Show worked answer

Twelve is three times four, so the total is three times £3.

3\times3=9

They cost £9.

Unit rate

2 marks

Five identical bottles hold 3.5 litres altogether. How much do eight hold?

Show worked answer

3.5\div5=0.7

Each holds 0.7 litres.

8\times0.7=5.6

Eight hold 5.6 litres.

Recipe

3 marks

A recipe for 8 people uses 600 g of potatoes. How much for 5 people?

Show worked answer

600\div8=75

75\times5=375

Use 375 g, assuming equal portions.

Ratio test

2 marks

Do pairs (2, 6), (5, 15), (8, 24) fit direct proportion? Explain.

Show worked answer

6/2=3

15/5=3

24/8=3

All have the same ratio 3, so the pairs fit y = 3x.

Fixed charge

2 marks

Delivery costs £4 plus £2 per parcel. Is total cost directly proportional to the number of parcels?

Show worked answer

No. One parcel costs £6 and two cost £8, not £12. The fixed £4 charge means doubling the parcel count does not double the total.

Use a given model

2 marks

Two quantities follow y = 7x. Find y when x = 3.5.

Show worked answer

Substitute into the supplied rule.

y=7(3.5)=24.5

The constant rate is 7.

Find k

Higher only3 marks

y is directly proportional to x. When x = 6, y = 27. Find the equation and y when x = 10.

Show worked answer

27=6k

k=4.5

The equation is y = 4.5x, so at x = 10, y = 45.

Find x

Higher only3 marks

y is directly proportional to x. When x = 4, y = 18. Find a formula for y in terms of x, then use it to find x when y = 54.

Show worked answer

$18=4k$ $k=4.5$ The formula is y = 4.5x. $54=4.5x$

x=12

Equivalently, tripling y triples x.

A squared input

Higher only4 marks

y is directly proportional to x². When x = 2, y = 20. Find y when x = 5.

Show worked answer

20=k(2^2)

k=5

y=5(5^2)=125

Use the square, not the input itself.

Square-root model

Higher only4 marks

y is directly proportional to √x. When x = 16, y = 10. Find y when x = 36.

Show worked answer

10=k\sqrt{16}=4k

k=2.5

y=2.5\sqrt{36}=15

Examiner-style feedback

Common direct proportion mistakes

Assuming any increase is proportional

Check a constant ratio or the same-factor rule, not just whether both numbers get larger.

Confusing ∝ and =

Replace a proportionality statement with an equality containing an unknown constant k.

Ignoring the power

If y ∝ x², doubling x multiplies y by four, not two.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Direct proportion preserves the ratio.
Find one unit, then scale.
A fixed fee breaks direct proportion to quantity.
For a power model, apply the power before multiplying by k.

Quick answers

Direct proportion FAQ

Must the graph go through the origin?

For y directly proportional to x, the mathematical model y = kx passes through (0, 0). A physical context may restrict which part of the graph is meaningful.

Is direct proportion the same as y proportional to x²?

Not as a relationship between y and x. In y = kx², y is directly proportional to x²; its graph against x is generally curved.

Build connected skills

What to revise next

Compare a fixed total

Inverse proportion

Understand why one quantity can shrink as the other grows.

Revise inverse proportion

See the graph

Straight-line graphs

Connect a constant rate to gradient.

Revise straight-line graphs

Content standards

Curriculum and rights review

R10 and R13: numerical direct proportion and supplied models across tiers; constructing proportional equations and power models are labelled Higher. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content