Direct and inverse proportion
This topic deepens R7 by formalising the two main types of proportional relationship and includes algebraic forms used at higher tier.
Direct proportion
y is directly proportional to x (written y ∝ x) means y = kx for some constant k. Doubling x doubles y; halving x halves y. Graph: straight line through origin.
Worked example: y ∝ x, and y = 21 when x = 6. Find y when x = 10.
- k = 21/6 = 3.5.
- y = 3.5 × 10 = 35.
Inverse proportion
y is inversely proportional to x (y ∝ 1/x) means y = k/x, equivalently xy = k. Doubling x halves y. Graph: a hyperbola (curved).
Worked example: y ∝ 1/x, and y = 8 when x = 5. Find y when x = 20.
- k = 8 × 5 = 40.
- y = 40/20 = 2.
Power forms (Higher tier)
- y ∝ x² → y = kx²; doubling x quadruples y.
- y ∝ √x → y = k√x.
- y ∝ 1/x² → y = k/x².
Worked example: y ∝ x². When x = 3, y = 18. Find y when x = 5.
- k = 18/9 = 2.
- y = 2 × 25 = 50.
Real-world problems
Workers and time — more workers, less time → inverse. Petrol used and distance — direct. Light intensity and distance from source — inverse square (y ∝ 1/d²).
Worked example: 4 workers can paint a wall in 9 hours. How long for 6 workers?
- "Workers × time = constant": 4 × 9 = 36.
- 6 × T = 36 → T = 6 hours.
Solving in three steps
- Identify the relationship (direct, inverse, power form).
- Use given values to find k.
- Substitute new values to find unknown.
⚠Common mistakes
- Treating "more X means less Y" as direct — it's inverse.
- Forgetting to square or square-root for power forms.
- Not using the same units — convert before solving.
- Mixing up "y when x = …" and "x when y = …" — re-read.
- Assuming linear when it's quadratic — sketch a quick table to check.
➜Try this— Quick check
If P ∝ 1/V² and P = 50 when V = 4, find P when V = 5.
- k = 50 × 16 = 800.
- P = 800/25 = 32.
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