Direct Proportion

Grade 4

What direct proportion means

Two quantities are in direct proportion when they increase and decrease together at the same rate. Double one, and the other doubles too. Halve one, and the other halves.

Classic examples: the cost of buying multiple identical items, the distance you travel at a constant speed, the amount of ingredients you need when scaling a recipe up.

The key test: if you divide one quantity by the other and always get the same number, they're in direct proportion. That constant is called k.

The equation — y = kx

Every direct proportion relationship can be written as y = kx, where k is the constant of proportionality. On a graph, this is a straight line through the origin.

Finding k is always your first move: pick a known pair of values, divide y by x, and you've got it. Then you can find any other value.

y = kx, so k = y ÷ x

Worked example

EXAMPLE

5 identical pens cost £3.25. How much do 8 pens cost?

Find k (cost per pen): 3.25 ÷ 5 = £0.65. Now use y = kx: 0.65 × 8 =£5.20.

Watch out for the graph

A direct proportion graph is always a straight line that passes through (0, 0). If the line doesn't go through the origin, the quantities might still be linear — but they're not in direct proportion.

Try this one

The cost, C pence, of producing a batch of sweets is directly proportional to the cube root of the number of sweets, n, in the batch.

When n = 8, C = 90.

(a) Find a formula for C in terms of n. [2]

(b) Find the cost when n = 27. [1]

Nova's hint:

This is direct proportion to a fractional power — specifically the cube root.

Ask yourself: 'How do I write cube root as a power, and what form does my equation take?' Try \( C = k \times n^{1/3} \) and substitute the known values to find k before doing any other part.

Try to Solve it with Nova?