Direct and inverse proportion
Direct proportion
Two quantities are in direct proportion if, when one doubles, the other doubles — they increase together at a constant rate.
Written as: y ∝ x, which means y = kx for some constant k.
Finding k: given one pair of values, substitute to find k. Then use k for all other calculations.
Example: y is directly proportional to x. When x = 4, y = 20. Find y when x = 7. k = y/x = 20/4 = 5. So y = 5x. When x = 7: y = 35.
Other forms of direct proportion:
- y ∝ x² → y = kx² (directly proportional to the square)
- y ∝ √x → y = k√x (directly proportional to the square root)
- y ∝ x³ → y = kx³ (directly proportional to the cube)
Inverse proportion
Two quantities are in inverse proportion if, when one doubles, the other halves — as one increases, the other decreases at a rate that keeps their product constant.
Written as: y ∝ 1/x, which means y = k/x or xy = k.
Example: y is inversely proportional to x. When x = 3, y = 12. Find y when x = 9. k = xy = 3 × 12 = 36. y = 36/x. When x = 9: y = 4.
Other forms of inverse proportion:
- y ∝ 1/x² → y = k/x²
- y ∝ 1/√x → y = k/√x
Recognising proportion from a table
| Direct proportion | Inverse proportion |
|---|---|
| y/x = constant | xy = constant |
| Graph: straight line through origin | Graph: reciprocal curve |
| As x increases, y increases | As x increases, y decreases |
CCEA context
CCEA Paper 2 often presents a proportion problem in context: speed and time (inverse), cost and quantity (direct), pressure and volume (inverse — Boyle's Law). You must:
- Write the proportionality statement (∝).
- Convert to an equation with k.
- Find k from given values.
- Use the equation for required calculations.
⚠Common mistakes
- Forgetting "through the origin": direct proportion always gives a straight line through the origin; y = 3x + 5 is NOT direct proportion.
- Confusing direct and inverse: more speed → less time (inverse), not more time.
- Not finding k first: trying to scale directly without establishing the constant.
- y ∝ x²: students sometimes write y = kx instead of y = kx².
- Units: in context problems, check units are consistent before substituting.
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