GCSE Maths · Ratio, proportion and rates of change

Inverse proportion GCSE Questions and Worked Answers

Two positive quantities are inversely proportional when multiplying one by a factor divides the other by that factor. Their product stays constant: xy = k, or y = k/x for x ≠ 0.

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Start with the meaning

What you need to know about inverse proportion

A job needs 24 hours of one worker's time. Two equally productive workers could share it and finish in 12 hours; four could finish in 6 hours. Each time the number of workers doubles, the time halves. This is inverse proportion. The amount of work stays fixed: workers multiplied by hours is always 24 worker-hours in this model.

See the idea first

The product stays the same

Let x be the number of workers and y the time in hours. Then xy = 24, so y = 24/x. More generally we write y = k/x, where k is the constant product. This assumes equal working rates and a job that can be divided without extra delays.

One job, equal working rates
Workers x	Hours y	Product xy
1	24	24
2	12	24
4	6	24
8	3	24

The positive part of y = 24/x bends down towards the axes without touching them. For the worker model, x counts whole workers and y is hours; the dots show the table values. With zero workers, the job has no finite completion time in this model.

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.

Share a fixed job

What the problem asks: Six identical machines finish a job in 10 hours. How long would four take?

How to solve it: The job needs 6 × 10 = 60 machine-hours. Four machines take 60 ÷ 4 = 15 hours, assuming the same rates and no delays.

Change speed over a fixed distance

What the problem asks: A journey takes 3 hours at a constant 60 km/h. How long at 90 km/h?

How to solve it: The fixed distance is 60 × 3 = 180 km. At 90 km/h it takes 180 ÷ 90 = 2 hours. This models travel time without stops.

Construct an inverse equation (Higher)

What the problem asks: y is inversely proportional to x, and y = 8 when x = 3. Find the equation.

How to solve it: Use y = k/x. The known pair gives k = xy = 24, so y = 24/x. The product, not the ratio, is constant.

Use an inverse-square model (Higher)

What the problem asks: y is inversely proportional to x². What happens to y when x doubles?

How to solve it: The denominator becomes four times as large, so y becomes one quarter as large. Use y = k/x², not y = k/x.

A reliable routine

For inverse proportion with a fixed product

The product method applies when one shared total, such as a fixed amount of work or distance, remains unchanged. First identify that fixed total; a vague statement that one quantity decreases is not enough.

Check what stays fixed and whether the assumptions justify inverse proportion.
Multiply the known pair to find the constant product.
Divide that product by the new input to find its matching output.
Check the direction and scale: doubling the input should halve the output in this model.

Check: Do not use y = k/x at x = 0. For a positive constant k, the graph is a curve, not a straight downward line, and it does not cross either axis. A context such as worker numbers uses only positive inputs.

Fully worked

Inverse proportion GCSE worked examples

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

Example 1

Workers and time

3 marks

Question

Five equally productive workers finish a job in 12 days. How long would eight workers take, assuming the job can be shared evenly?

The fixed work is

5\times12=60\text{ worker-days}

For eight workers,

t=60\div8=7.5

The time is 7.5 working days. More workers produce a shorter time.

Example 2

A fixed distance

3 marks

Question

A journey takes 4 hours at 50 km/h. Find its travel time at a constant 80 km/h, with no stops.

First find the fixed distance.

d=50\times4=200\text{ km}

Then divide by the new speed.

t=200\div80=2.5\text{ hours}

This is 2 hours 30 minutes, not 2 hours 50 minutes.

Example 3

Test products

2 marks

Question

For x = 2, 3, 6, the corresponding y-values are 12, 8, 4. Do these pairs fit inverse proportion?

2\times12=24

3\times8=24

6\times4=24

The product is constant, so these pairs fit y = 24/x. The table alone does not prove a rule beyond its values.

Example 4

Construct an equation

Higher only3 marks

Question

y is inversely proportional to x. When x = 5, y = 18. Find a formula for y in terms of x, then use it to find y when x = 12.

Use y = k/x.

k=5\times18=90

Hence y = 90/x.

y=90/12=7.5

The product remains 90.

Example 5

Inverse square

Higher only4 marks

Question

y is inversely proportional to x². When x = 3, y = 20. Find y when x = 6.

Use y = k/x².

20=\frac{k}{9}

k=180

At x = 6,

y=\frac{180}{36}=5

Doubling x quarters y, which checks the result.

Example 6

Find an input

Higher only4 marks

Question

y is inversely proportional to x². When x = 2, y = 27. Find the positive x when y = 3.

k=yx^2=27(2^2)=108

Thus

3=\frac{108}{x^2}

3x^2=108

x^2=36

The question asks for the positive input, so x = 6. Without that restriction, −6 would also satisfy the equation.

10 original questions · total 26 marks

Inverse proportion GCSE exam-style questions

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

Before you startAllow about 31 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Double the workers

2 marks

Three equal-rate workers take 16 hours for a divisible job. How long would six take under the same conditions?

Show worked answer

3\times16=48

48\div6=8

Six workers take 8 hours: double the workers, half the time.

Fewer machines

3 marks

Eight identical machines complete a job in 9 hours. How long would six machines take at the same rate?

Show worked answer

8\times9=72\text{ machine-hours}

72\div6=12\text{ hours}

Fewer machines need longer.

Fixed-distance travel

3 marks

A journey takes 2 hours at 90 km/h. How long at 60 km/h, without stops?

Show worked answer

d=90\times2=180\text{ km}

t=180\div60=3\text{ hours}

The distance, not the time, stays fixed.

Check a table

2 marks

Do pairs (2, 15), (3, 10), (5, 6) fit inverse proportion?

Show worked answer

The products are 2 × 15 = 30, 3 × 10 = 30 and 5 × 6 = 30. They fit the inverse model y = 30/x.

Decreasing is not enough

2 marks

For x = 1, 2, 3, y = 9, 8, 7. Are the pairs inversely proportional?

Show worked answer

The products are 9, 16 and 21, which differ. The values decrease, but they are not inversely proportional.

Model assumptions

2 marks

A manager says 20 painters will paint a tiny room ten times faster than two. Give two reasons the inverse model may fail.

Show worked answer

There may not be enough space for everyone to work at once, and some jobs such as drying cannot be shared to reduce their duration. Equal productivity and fully divisible work are assumptions, not guarantees.

Given formula

2 marks

The model is y = 48/x. Find y when x = 8.

Show worked answer

y=48/8=6

The pair has product 8 × 6 = 48.

Find a constant

Higher only3 marks

y is inversely proportional to x. When x = 4, y = 15. Find a formula for y in terms of x, then use it to find y when x = 10.

Show worked answer

$k=4(15)=60$ The product xy stays 60, so the formula is y = 60/x. $y=\frac{60}{10}=6$

Find x

Higher only3 marks

y is inversely proportional to x. When x = 3, y = 14. Find a formula for y in terms of x, then find x when y = 7.

Show worked answer

$k=3(14)=42$ The product xy stays 42, so y = 42/x. $7=42/x$

7x=42

x=6

Halving y doubles x.

Inverse square

Higher only4 marks

y is inversely proportional to x². When x = 2, y = 18. Find y when x = 6.

Show worked answer

k=18(2^2)=72

y=\frac{72}{6^2}=2

Tripling x divides y by nine.

Examiner-style feedback

Common inverse proportion mistakes

Dividing to find k

For y = k/x, multiply the known x and y. The ratio y/x is the constant in direct proportion, not inverse proportion.

Scaling both in the same direction

For a fixed positive product, multiplying x by a factor divides y by that factor.

Forgetting model assumptions

Worker-time calculations need fixed work, equal rates and no extra coordination or space constraints.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Ordinary inverse proportion preserves a product.
Identify what stays fixed.
More input means less output for positive quantities.
A power in the denominator changes the scale factor.

Quick answers

Inverse proportion FAQ

Is every downward straight line inverse proportion?

No. For y = k/x, the graph is curved. A falling straight line generally has neither a constant product nor an inverse-proportion relationship.

How does inverse square differ from ordinary inverse proportion?

In y = k/x², yx² stays constant. Doubling x quarters y. In y = k/x, xy stays constant and doubling x halves y.

Build connected skills

What to revise next

Compare constant ratios

Direct proportion

See when both quantities share the same scale factor.

Revise direct proportion

Find a different unknown

Rearranging formulae

Work safely with denominators and powers.

Revise rearranging formulae

Content standards

Curriculum and rights review

R10 and R13: inverse numerical proportion and supplied models across tiers; constructing equations and inverse-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