Direct and inverse proportion problems - Edexcel GCSE Maths (Non-calculator) Exam Questions

Practise 29 exam-style questions on Direct and inverse proportion problems for Edexcel GCSE Maths (Non-calculator). Write your answer as you would in the exam and get it marked in seconds against the mark scheme, point by point, with an AI tutor on hand if you get stuck.

29 questions
1–5 marks each
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20 questions, marked as you go

Sample questions with model answers

Here are 4 of the questions in this topic, each with a model answer, an explanation of why it earns the marks and the mark scheme an examiner would use.

Question 1

2 marks

Here are the graphs of four relationships between \(x\) and \(y\), labelled A, B, C and D.

Write down the letter of the graph that shows \(y\) is directly proportional to \(x\), and the letter of the graph that shows \(y\) is inversely proportional to \(x\).

A coordinate grid with x from 0 to 8 and y from 0 to 10; each grid square is 1 unit across, and the axes cross at the origin O. Line A is a straight line sloping upwards from left to right crossing the y-axis at (0, 3) and passing through (2, 5). Curve B is a curve: it gets closer and closer to the line y = 0 (the x-axis) at both ends without reaching it. Line C is a straight line sloping downwards from left to right crossing the y-axis at (0, 7) and crossing the x-axis at (7, 0). Line D is a straight line sloping upwards from left to right passing through the origin and (2, 4). Line A and curve B cross at about (1.7, 4.7). Line A and line C cross at (2, 5). Line A and line D cross at (3, 6). Curve B and line C cross at about (1.4, 5.6) and about (5.6, 1.4). Curve B and line D cross at (2, 4). Line C and line D cross at (7/3, 14/3).

Model answer

Directly proportional: D (a straight line through the origin)
Inversely proportional: B (a curve that gets closer to both axes)

Explanation

When \(y\) is directly proportional to \(x\), the graph is a straight line through the origin, so line A (which does not pass through the origin) is not direct proportion. When \(y\) is inversely proportional to \(x\), \(y = \frac{k}{x}\): as \(x\) increases \(y\) decreases, and the curve never meets the axes. Line C goes down but is a straight line, so it is not inverse proportion.

Mark scheme

1 mark
B1 for D (directly proportional)
correct answer with no working scores full marks
1 mark
B1 for B (inversely proportional)

Question 2

2 marks

The graph shows the cost, in pounds, of buying different lengths of ribbon.
The cost is directly proportional to the length of ribbon.

Work out the cost of 45 m of ribbon.

A graph of Cost (£) (vertical axis, 0 to 12, marked every 1) against Length of ribbon (m) (horizontal axis, 0 to 20, marked every 2), on a grid. The graph is a straight line; its values can be read from the grid.

Model answer

From the graph, 10 m costs £6.
45 m = 4.5 × 10 m
Cost = 4.5 × 6 = £27

Explanation

The graph only goes up to 20 m, so read off an easy pair of values and use direct proportion to scale up. Any accurate pair works: 20 m for £12 gives 40 m for £24, and 5 m for £3, so 45 m costs £27.

Mark scheme

1 mark
M1 for reading a correct pair of values from the graph, e.g. 10 m costs £6, or 60p per metre, and scaling up to 45 m
e.g. 4.5 × 6 or 20 m = £12 used with 5 m = £3; correct answer with no working scores full marks
1 mark
A1 for £27
cao

Question 3

3 marks

\(y\) is inversely proportional to \(x\).
When \(x = a\), \(y = 12\).
When \(x = a + 3\), \(y = 8\).

Find the value of \(a\).

Model answer

\(xy = k\), so \(12a = 8(a + 3)\)
\(12a = 8a + 24\)
\(4a = 24\)
\(a = 6\)
Check: \(6 \times 12 = 72\) and \(9 \times 8 = 72\)

Explanation

For inverse proportion the product \(xy\) is the same for every pair of values. Writing the product for both pairs and setting them equal gives a linear equation in \(a\).

Mark scheme

1 mark
M1 for \(12a = 8(a + 3)\) oe
e.g. \(k = 12a\) and \(k = 8(a + 3)\); correct answer with no working scores full marks
1 mark
M1 for \(4a = 24\) oe
1 mark
A1 for \(a = 6\)
cao

Question 4

4 marks

Sam changes £500 into euros. The exchange rate is £1 = €1.15
On holiday, he spends €460.
When he gets home, he changes the euros he has left back into pounds. The exchange rate is now £1 = €1.25

How many pounds does Sam get?

Model answer

£500 = 500 × 1.15 = €575
Euros left = 575 − 460 = €115
€115 = 115 ÷ 1.25 = £92

Explanation

Multiply by the rate to change pounds into euros, and divide by the rate to change euros back into pounds. A common error is to multiply by 1.25 at the end, which gives more euros, not pounds.

Mark scheme

1 mark
P1 for 500 × 1.15 (= €575)
correct answer with no working scores full marks
1 mark
P1 for 575 − 460 (= €115)
1 mark
P1 for 115 ÷ 1.25
ft their 115
1 mark
A1 for £92
cao

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