Algebra › Proportions
Proportions
A proportion says two ratios are equal. Recipes, maps, scale models — they all live on this idea.
Scale a proportion
Direct — both grow together. Inverse — one up, the other down.
Recipe
cups flour
2
eggs
3
ratio cups flour:eggs = 2:3 (scaled ×1)
Direct: y = kx. Double one, double the other. The ratio stays constant.
Direct proportion
When two quantities grow at the same rate, they're directly proportional. Double one, double the other. Half one, half the other. We write it as y = k · x where k is a constant called the constant of proportionality.
- Cups of flour to pancakes baked.
- Hours worked to wages earned.
- Distance on a map to actual distance on the ground.
Inverse proportion
When one quantity grows as the other shrinks — and their product stays the same — they're inversely proportional: y = k / x.
- Speed and time on the same trip.
- Workers on a job and days to finish (more workers → fewer days).
- Pressure and volume of a gas at constant temperature.
Cross-multiply to solve
a/b = c/d then a · d = b · c. This turns a fraction equation into a clean linear one — a tiny trick you'll use forever.Worked example
A recipe uses 200 g flour for 3 cookies. How much flour for 12 cookies?
Set up the proportion: 200 / 3 = x / 12. Cross-multiply: 200 × 12 = 3x, so x = 800 g.
Quick check
- 5 apples cost £2. What do 15 apples cost? (Direct.)
- It takes 4 painters 6 hours to paint a house. How long for 12 painters? (Inverse.)
- If
3/4 = x/20, find x.
Answers: £6, 2 hours, and 15.
If 3/4 = x/20, what is x?
4 painters take 6 hours. How long for 12 painters? (Inverse proportion.)