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

A printing machine prints pages at a constant rate. It prints 360 pages in 6 minutes.

(a) How many pages does it print in 10 minutes? [2]

(b) A document has 480 pages. How many minutes does it take to print this document? [2]

(c) Two identical machines work together. How long does it take them to print 480 pages? [2]

Nova's hint:

Parts (a) and (b) use direct proportion; part (c) uses inverse proportion (more machines, less time).

Ask yourself: 'How many pages per minute does one machine print, and what happens to the time when two machines share the work?'

For example, if 1 machine takes 10 minutes, then 2 machines take 10 ÷ 2 = 5 minutes.

Try to Solve it with Nova?