Inverse Proportion

Grade 5

The opposite of direct proportion

With inverse proportion, as one quantity goes up, the other comes down. The product of the two values stays the same — that's what makes it inverse.

Imagine sharing a fixed amount of work between people: the more workers you have, the less time each person needs to spend. The total work doesn't change — it's just spread differently.

The equation — y = k/x

Every inverse proportion relationship fits the equation y = k/x. Rearranged, that's xy = k — which shows that the product of any pair of values is always the constant k.

Your method is the same as direct proportion: find k first using a known pair, then use it to find the unknown.

y = k ÷ x, so k = x × y

To find k for inverse proportion, multiply the two values (not divide). Multiply the x and y values from any known pair and you've got k.

Worked example

EXAMPLE

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

Find k: 4 × 15 = 60. Now use y = k ÷ x: 60 ÷ 6 =10 days.

Direct or inverse — how to tell

Always check which type you have before you start. Ask: if x doubles, does y double (direct) or halve (inverse)? Getting this wrong means solving a completely different equation. A quick common-sense check on your answer goes a long way.

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?