Maths · KS3 and GCSE · Interactive lesson
Direct and inverse proportion
Find a unit price and scale quantities together. Then explore why more equally effective workers can take less time on an ideal fixed job.
Choose your view
Explore an interactive example, then try the questions at your own pace.
KS3 / GCSE · start with the idea
By the end of this lesson: Use the amount for one to scale a quantity.
The interactive diagram and controls are in Stage 2, “Explore one example”. Read the idea here, then try it next.
- Find the price of one item.
- Each extra identical item adds that same cost.
- Twice as many items cost twice as much when the unit price stays fixed.
Useful starting points
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What do the words mean?
- Direct proportion
- Two quantities scale by the same factor; their ratio is constant.
- Unit rate
- The amount for one unit, such as pounds per item.
- Inverse proportion
- One quantity increases by a factor as the other decreases by that factor, keeping their product constant.
- Equally effective workers
- Workers who each do the same amount of work in the same time.
- Worker-hour
- The amount of work one worker does in one hour in this simplified model.
Change the example
This is the lesson example. Your practice question uses its own numbers.
Something to try
Keep the price of one item the same. Double the number of items. What happens to the total cost?
Notice what changes
How particle motion works +
Starts paused. Press Play, then add energy to see the warmer particles move faster.
Particles keep the same size. Liquid particles move around; solid particles vibrate around fixed positions. This is a simplified model, with the change in motion exaggerated to make it easier to see.
About the playback +
About this model
Identical items have a fixed price per item, with no delivery charge or bulk discount. Zero items would cost £0. A fixed extra charge would stop the total being directly proportional to item count.
More about this diagram
Still picture. Play particle motion in Stage 2.
Total cost = £12
Still picture of the example from Stage 2. Use the back arrow to change it.
Work through it
Starting example: read the steps below
- One item costs £3. This price stays fixed.
- There are 4 items.
- Cost = 4 × £3 = £12.
- Check: £12 ÷ 4 = £3 per item.
Try on your own
Your turn. No rush.
Question 1
8 identical notebooks cost £56. The price per notebook stays the same, with no extra fees or discounts. What do 1 notebooks cost?
About these questions
Questions appear in a mixed order. Work at your own pace and stop whenever you like.
These questions use their own numbers, separate from the lesson example. You can stop whenever you like. Answers last for this visit to the page.
A good place to pause
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Can you find the unit price and explain why both quantities scale by the same factor?
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Explain it in your own words and add a small example. This can help you remember and spot what to ask Maurice.
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Useful starting points
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Notes for parents and tutors
These lessons cover selected KS3 and GCSE ideas for England; they are not a complete course. The level notes explain where GCSE methods or optional extensions begin. Maurice can help you choose material for your course and exam board.
The content draws on the DfE Maths curriculum, DfE Science curriculum and AQA GCSE Physics specification. Level labels are guidance, not exam-board endorsements. For AQA Radioactivity, calculating net decline as a ratio is Higher-tier work. Net means the overall change; ask Maurice whether this extension is relevant to your course. Explanations and practice were reviewed during development on 21 October 2026; Maurice should check their fit before assigning work.
Topic labels and your lesson-view preference are saved in this browser. This tab also remembers the last lesson stage, so you can choose to continue there. If you choose revision, selected activities and suggested revisit dates are saved here too. Recent question codes are also kept to reduce repeats. Reflection notes are saved automatically per topic when you type them; you can delete a note in the recap. They stay in this browser and are visible to other people using it. Practice answers and enquiry details are not saved. Clearing browser data removes the labels.