Direct & Inverse Proportion
Direct & Inverse Proportion
Proportion describes the relationship between two quantities that change together.
Direct Proportion
Two quantities are in direct proportion if when one doubles, the other doubles. When one is multiplied by a factor, the other is multiplied by the same factor.
The relationship is written as: y ∝ x which means y = kx where k is the constant of proportionality.
Worked Example: y is directly proportional to x. When x = 4, y = 10. Find y when x = 6.
- y = kx → 10 = k × 4 → k = 2.5
- y = 2.5x
- When x = 6: y = 2.5 × 6 = 15
Direct Proportion to Powers and Roots (Higher)
y can be proportional to x², x³, √x etc.
y ∝ x² means y = kx²
Worked Example: y is directly proportional to x². When x = 3, y = 36. Find y when x = 5.
- y = kx² → 36 = k × 9 → k = 4
- y = 4x²
- When x = 5: y = 4 × 25 = 100
Worked Example: y is directly proportional to √x. When x = 16, y = 12. Find x when y = 15.
- y = k√x → 12 = k × 4 → k = 3
- y = 3√x
- 15 = 3√x → √x = 5 → x = 25
Inverse Proportion
Two quantities are in inverse proportion if when one doubles, the other halves. As one increases, the other decreases.
The relationship is: y ∝ 1/x which means y = k/x
Worked Example: y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.
- y = k/x → 8 = k/3 → k = 24
- y = 24/x
- When x = 6: y = 24/6 = 4
Inverse Proportion to Powers (Higher)
y ∝ 1/x² means y = k/x²
Worked Example: The intensity of light (I) is inversely proportional to the square of the distance (d). At d = 2m, I = 50. Find I at d = 5m.
- I = k/d² → 50 = k/4 → k = 200
- I = 200/d²
- At d = 5: I = 200/25 = 8
Recognising Proportion from Tables and Graphs
| Type | Test | Graph Shape |
|---|---|---|
| y ∝ x | y/x is constant | Straight line through origin |
| y ∝ x² | y/x² is constant | Parabola through origin |
| y ∝ 1/x | xy is constant | Reciprocal curve (hyperbola) |
Three-Step Method for All Proportion Problems
1. Write the proportionality statement and convert to an equation with k
2. Find k by substituting the given values
3. Use the equation to answer the question
Exam Tips
- Always start by writing the proportionality statement (e.g. y ∝ x²) and the equation (y = kx²)
- Show the working to find k — this earns method marks
- Direct proportion graphs pass through the origin; inverse proportion graphs never touch either axis
- Read carefully whether it says "proportional to x" or "proportional to x²" — they are different relationships