Proportion
Recognise and solve problems involving direct proportion (as one quantity increases, the other increases at a constant rate) and inverse proportion (as one increases, the other decreases), including graphical and algebraic representations
What a learner can do afterwards
- Identify whether a real-world relationship is direct or inverse proportion and justify the choice
- Set up and solve a direct-proportion equation (e.g. if 4 pens cost £6, find the cost of 10)
- Sketch graphs showing direct proportion (straight line through origin) and inverse proportion (curve)
1 · Read
Direct proportion means two quantities grow at exactly the same rate. Double one and the other doubles too, so their ratio never changes: one is always the same multiple of the other. You have met this on a graph already, because direct proportion is a straight line through the origin. A gym charging £8 per class is direct proportion: 2 classes cost £16, 5 classes cost £40, always £8 each. A car at a steady speed works the same way, since distance divided by time always gives the same speed.
Here is the find-one-first method. 3 notebooks cost £4.50, and you want 8 notebooks. One notebook costs £4.50 ÷ 3 = £1.50, so 8 notebooks cost £1.50 × 8 = £12. Because the price per notebook never changes, this works for any number you like, even 100.
Inverse proportion is the mirror image. One quantity goes up while the other goes down, and their product always stays the same fixed number. A job might need exactly 24 worker-days: more workers means fewer days, because workers times days must stay 24. On a graph, inverse proportion makes a curve that falls as you move right, never a straight line.
5 printers need 12 hours to finish a batch of flyers. How long would 10 printers take? First find the fixed amount of work: 5 × 12 = 60 printer-hours. Then share it over the new team: 60 ÷ 10 = 6 hours. Multiply first to find the total, then divide by the new amount.
To tell the two apart, ask what stays fixed. If dividing one quantity by the other gives the same number every time, it is direct proportion. If multiplying them together gives the same number every time, it is inverse proportion. Try it on the printers: 5 × 12 and 10 × 6 both give 60, so that pairing must be inverse.
In direct proportion two quantities grow at the same rate, their ratio stays fixed, and the graph is a straight line through the origin. In inverse proportion one goes up as the other goes down, their product stays fixed, and the graph is a curve.
Adapted from OpenStax Algebra and Trigonometry (CC-BY 4.0), openstax.org · license
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.