Direct proportion
- Two quantities are in direct proportion if they increase at the same rate: double one and the other doubles.
- The unitary method finds the value of one first: 5 pens cost £3.50, so 1 pen costs 70p and 8 pens cost £5.60.
- The graph of direct proportion is a straight line through the origin.
Inverse proportion
- Two quantities are in inverse proportion if one goes up as the other goes down at the same rate: double one and the other halves.
- 4 builders take 6 days to build a wall. 1 builder would take 24 days, so 3 builders take 8 days.
- The product stays the same: 4 × 6 = 3 × 8 = 24.
- The graph of inverse proportion is a curve that gets closer and closer to the axes.
Proportion equations (Higher)
- ∝ means 'is proportional to'. y ∝ x means y = kx; y ∝ x² means y = kx²; y ∝ 1/x means y = k/x.
- Find k from the values you are given, then write the equation. If y ∝ x² and y = 36 when x = 3, then 36 = 9k, so k = 4 and y = 4x². When x = 5, y = 100.
- If y is inversely proportional to x and y = 8 when x = 3, then k = 24 and y = 24/x. When x = 6, y = 4.
Key terms
- Direct proportion
- When one quantity is multiplied, the other is multiplied by the same number.
- Inverse proportion
- When one quantity is multiplied, the other is divided by the same number.
- Unitary method
- Finding the value of one unit first, then scaling up.
- Constant of proportionality
- The number k in equations such as y = kx.
- Proportional to (∝)
- Linked by direct proportion, such as y ∝ x.