Direct and inverse proportion

Proportion is tested on OCR J560 Papers 2 and 3. Higher-tier extends beyond y ∝ x to y ∝ x², y ∝ √x and y ∝ 1/x etc. The constant of proportionality k is always found from a given value pair.

Direct proportion

y is directly proportional to x means: y = kx (for some constant k > 0).

As x doubles, y doubles. The graph is a straight line through the origin.

Notation: y ∝ x.

Method:

1. Write y = kx.
2. Substitute a known pair (x, y) to find k.
3. Use the equation to find any other value.

Example: y ∝ x and y = 15 when x = 5. Find y when x = 8.

1. 15 = k × 5 → k = 3.
2. y = 3x. When x = 8: y = 24.

Higher direct proportion

y ∝ x²: y = kx² (e.g. kinetic energy ∝ speed²). y ∝ √x: y = k√x. y ∝ x³: y = kx³.

Example: y ∝ x² and y = 48 when x = 4. Find y when x = 5.

• 48 = k × 16 → k = 3.
• y = 3x². When x = 5: y = 3 × 25 = 75.

Inverse proportion

y is inversely proportional to x means: y = k/x.

As x doubles, y halves. The graph is a hyperbola.

Notation: y ∝ 1/x.

Example: y ∝ 1/x and y = 12 when x = 4. Find y when x = 6.

• 12 = k/4 → k = 48.
• y = 48/x. When x = 6: y = 48/6 = 8.

Inverse proportion variants

y ∝ 1/x²: y = k/x². y ∝ 1/√x: y = k/√x.

Example: y ∝ 1/x² and y = 4 when x = 3. Find y when x = 6.

• 4 = k/9 → k = 36.
• y = 36/x². When x = 6: y = 36/36 = 1.

Comparing effects

"If x is multiplied by 4, what happens to y?"

• y ∝ x: y × 4.
• y ∝ x²: y × 16.
• y ∝ 1/x: y ÷ 4.
• y ∝ 1/x²: y ÷ 16.

Common OCR exam mistakes

1. Confusing direct and inverse proportion — direct means y increases with x; inverse means y decreases as x increases.
2. Not finding k before using the formula.
3. For y ∝ x²: when x doubles, y quadruples (× 4), not doubles (× 2).
4. Writing y = k + x instead of y = kx for direct proportion.