# 2.3 Inverse proportion

In Section 2.2 you saw that direct proportion described relationships between two quantities, where as one increased, so did the other. Sometimes as one quantity increases the other decreases instead of increasing. This is called indirect proportion. Team tasks are often an example of this. The time taken to do a job is indirectly proportional to the number of people in the team.

A difficulty with the real-life context of such problems is that, in many cases, it is hard to believe that people working in a team will work at the same rate regardless of the size of the team, unless the team work independently, i.e. ‘in parallel’. The main idea behind this type of problem is that increasing the number of people working decreases the time taken to complete the task. (An obvious exception to this is decision-making in a committee: if two people can reach a decision in an hour, four people are liable to take twice as long!)

Such problems can be compared with certain problems involving speed: doubling the number of people working is the same as doubling the speed at which the team work. In either case the time is halved. It is useful to find out how long it would take one person to do the whole job, then divide by the number of people sharing the work. This is a good approach to most indirect proportion problems.

## Example 9

A team of five people can deliver leaflets to every house on a housing estate in three hours. How long will it take a team of just two people?

### Answer

Take the same approach as in Examples 7 and 8, and first work out how long it would take *one* person to deliver leaflets to the estate.

It will take one person five times as long as a team of five people. (If you find this hard to accept, imagine that the estate consists of five streets, and that each person delivers leaflets to one of these streets in the three hours.) So each street takes 3 hours to leaflet. It would take one person 5 × 3 hours to leaflet all five.

So it takes one person 15 hours to deliver leaflets to the whole estate. Two people will take half this time, so two people take hours.

(As a check: you would expect two people to take longer than five.)