Proportion as equality of ratios

Proportional reasoning is on every WJEC Unit 2 paper. Foundation: scale factor and unitary method. Intermediate/Higher: direct and inverse proportion algebra.

What is proportion?

Two quantities are in proportion when their ratio is constant — i.e. multiplying one by k multiplies the other by k.

Direct proportion

y is directly proportional to x means y = kx for some constant k.

Test: divide y by x for each pair. Constant ratio → direct proportion.

| x | 2 | 5 | 8 | | y | 6 | 15 | 24 |

y/x = 3 for every pair, so y = 3x.

Unitary method (Foundation strategy)

To find the cost of 7 apples when 4 cost £1.40:

• 1 apple → 140p ÷ 4 = 35p.
• 7 apples → 7 × 35p = £2.45.

This is the WJEC standard "1 first, then scale up" method, with M1 for the unit-cost line.

Inverse proportion (Higher)

y is inversely proportional to x means y = k / x — as x doubles, y halves.

Test: x × y is constant.

| x | 2 | 5 | 10 | | y | 30 | 12 | 6 |

xy = 60 for every pair, so y = 60/x.

"Best buy" comparisons

Convert to a single rate (price per unit) and compare. WJEC favourite — compare two pack sizes by price per gram.

Recipe scaling

If a recipe for 6 serves 4 people but you need to serve 10, scale factor = 10/4 = 2.5; multiply every ingredient by 2.5. Show this scale factor explicitly for the M1.

WJEC exam tip

Always WRITE the constant of proportionality. For "y is proportional to x with y = 12 when x = 4", state "y = kx, 12 = 4k, k = 3, so y = 3x" — that's three M-marks before any calculation.