Direct and inverse proportion equations
WJEC Higher candidates must move beyond simple "if A then B" arithmetic and construct algebraic models of proportion.
Direct proportion
y is directly proportional to x: y ∝ x → y = kx.
Variants:
- y ∝ x² → y = kx².
- y ∝ √x → y = k√x.
- y ∝ x³ → y = kx³.
The constant k is found from a given pair of values, then used to answer further parts.
Inverse proportion
y is inversely proportional to x: y ∝ 1/x → y = k/x.
Variants:
- y ∝ 1/x² → y = k/x².
- y ∝ 1/√x → y = k/√x.
As x increases, y decreases (and vice versa).
Method
- Write the proportionality statement M1.
- Replace ∝ with "= k" and the appropriate function (M1 or A1).
- Substitute the given pair to find k M1.
- Re-write the formula with k filled in.
- Substitute the new value to answer the question A1.
✦Worked example— Worked example — direct
y is directly proportional to the square of x. When x = 4, y = 80. Find y when x = 6.
- y = kx² M1
- 80 = k × 16 → k = 5 A1
- y = 5x²
- y = 5 × 36 = 180 A1
✦Worked example— Worked example — inverse
y is inversely proportional to the cube root of x. When x = 8, y = 6. Find y when x = 27.
- y = k / ³√x M1
- 6 = k / 2 → k = 12 A1
- y = 12 / ³√27 = 12 / 3 = 4 A1
WJEC exam tip
The first M1 in every proportion question is awarded for setting up the algebraic statement (y = kxⁿ or y = k/xⁿ). Candidates who jump to ratio arithmetic skip this mark and often arrive at the wrong answer when the relationship is not linear.
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