Direct and Inverse Proportion

Grade 5

Two ways quantities can be linked

When two quantities are connected, they're either in direct proportion or inverse proportion.

In direct proportion, both quantities increase or decrease together — double one and you double the other. Think: more workers, more bricks laid (same time). In inverse proportion, one goes up while the other goes down — double one and the other halves. Think: more workers, less time to finish the job.

The unitary method

The cleanest way to handle both types is the unitary method: find the value for one unit first, then scale to whatever the question asks for.

The unitary method works for both direct and inverse proportion — the only difference is whether you multiply or divide at the final step.

Direct proportion — worked example

EXAMPLE

5 pens cost £3.75. How much do 8 pens cost?

Find the cost of 1 pen: £3.75 ÷ 5 = £0.75. Scale up to 8 pens: 8 × £0.75 =£6.00.

Inverse proportion — worked example

EXAMPLE

4 workers take 15 days to complete a project. How long would 6 workers take, assuming they all work at the same rate?

More workers means fewer days — this is inverse proportion. Find the total work: 4 × 15 = 60 worker-days. Divide by the new number of workers: 60 ÷ 6 =10 days.

The common mistake is to treat an inverse proportion question like a direct one and multiply instead of divide at the end. Always ask yourself first: if one quantity goes up, does the other go up (direct) or down (inverse)?

Try this one

A recipe for 6 people requires 450 g of flour.

How much flour is needed for 10 people? [2]

Nova's hint:

This is a direct proportion problem.

Ask yourself: 'What is the amount per person, and how do I scale that up?' For example, if 4 people need 200 g, one person needs 200 ÷ 4 = 50 g.

Try to Solve it with Nova?