--- title: "Proportion" description: "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 g" canonical: https://lightmysky.com/learn/mathematics/proportion-mt_5mIcmKRCgA source: https://lightmysky.com/learn/mathematics/proportion-mt_5mIcmKRCgA.md retrieved: 2026-10-19 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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 Subject: Mathematics · Area: Ratio & Proportion · Ages 12 to 14 Page: https://lightmysky.com/learn/mathematics/proportion-mt_5mIcmKRCgA ## Ready when they can - 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) ## Lesson: Direct and Inverse Proportion 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. **Example.** 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. *(drawing: This job always takes 24 worker-days. Two workers take 12 days, four workers take 6 days, and eight workers take just 3 days.)* **Example.** 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. **Tip.** 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. **Recap.** 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](https://creativecommons.org/licenses/by/4.0/) ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Proportional Reasoning Vocabulary](https://lightmysky.com/learn/mathematics/proportional-reasoning-vocabulary-mt_ATYLKt0je-) - [Proportion Graphs](https://lightmysky.com/learn/mathematics/proportion-graphs-mt_FspV_imUGK) - [Dividing Quantities by Ratio](https://lightmysky.com/learn/mathematics/dividing-quantities-by-ratio-mt_nOCPx5qw0Z) - [Linear Function Graphs](https://lightmysky.com/learn/mathematics/linear-function-graphs-mt_WBdHkc2HTf) - [Ratio Notation and Relationships](https://lightmysky.com/learn/mathematics/ratio-notation-and-relationships-mt_yK51ZnKA8m) ## Opens up - [Reacting Masses, Limiting Reactants and Yield](https://lightmysky.com/learn/science/reacting-masses-limiting-reactants-and-yield-mt_C2K6kp5Chv) - [Isotopes and Relative Atomic Mass](https://lightmysky.com/learn/science/isotopes-and-relative-atomic-mass-mt_NFoRZd6UN_) - [Direct Proportion as a Formula](https://lightmysky.com/learn/mathematics/direct-proportion-as-a-formula-mt_nhSf9I7Bjz)