Direct and inverse proportion

Two quantities are in direct proportion if, when one doubles, so does the other. Use the unitary method: find the value for one, then multiply.

1. Divide to find the value for 1 item (or 1 person).
2. Multiply by the number you need.

Example: 5 pens cost £3.40. How much do 8 pens cost?

1. 1 pen: £3.40 ÷ 5 = £0.68
2. 8 pens: 8 × £0.68 = £5.44

Example: a recipe for 4 people uses 300 g of flour. How much is needed for 10 people?

1. 1 person: 300 ÷ 4 = 75 g
2. 10 people: 10 × 75 = 750 g

Two quantities are in inverse proportion if, when one doubles, the other halves: more workers means less time. Their product stays the same.

1. Multiply to find the total (for example, worker-days).
2. Divide by the new number.

Example: 6 workers take 10 days to build a wall. How long would 4 workers take?

1. Total work: 6 × 10 = 60 worker-days
2. 4 workers: 60 ÷ 4 = 15 days

"y is proportional to x" is written y ∝ x, which means y = kx for a number k, the constant of proportionality.

In the same way, y ∝ x² means y = kx², y ∝ 1/x means y = k/x, and y ∝ 1/x² means y = k/x².

1. Write the equation with k, and substitute the given values to find k.
2. Rewrite it with that k, then use it.

Example: y ∝ x², and y = 45 when x = 3. Find y when x = 4.

1. y = kx², so 45 = k × 9 and k = 5
2. y = 5x², so y = 5 × 16 = 80

Example: y ∝ 1/x², and y = 2 when x = 6. Find y when x = 3.

1. 2 = k/36, so k = 72
2. y = 72/x² = 72/9 = 8

• y = kx (direct proportion) is a straight line through the origin: line A, y = 2x.
• y = k/x (inverse proportion) is a curve that falls more and more slowly and never touches the axes: curve B, y = 12/x.
• y = kx² is a curve through the origin that gets steeper.

Example: a graph of y against x is a straight line through (0, 0) and (4, 10). Write its equation.

1. A straight line through the origin means direct proportion: y = kx
2. 10 = k × 4, so k = 2.5
3. y = 2.5x