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

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?