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

The ratio of red to blue to yellow sweets in a bag is 4 : 5 : 1.

There are 60 sweets in total.

(a) How many blue sweets are there? [2]

(b) What fraction of the sweets are yellow? Give your answer in its simplest form. [1]

Nova's hint:

This is a three-part ratio problem combined with fractions of an amount.

Ask yourself: 'How many total parts are there, and what is one part worth?' Once you have one part, multiply by the relevant number of parts for each colour.

Try to Solve it with Nova?