Direct and inverse proportion problems – how to solve them

Direct proportion

Say you work in a garden centre and it takes you an hour to plant 6 shrubs. You want to work out how long it would take you to plant 18 shrubs.

The answer to this question is probably obvious to many people – it would take you 3 hours, because there are 3 lots of 6 in 18. The amount of time it takes you to plant shrubs is in direct proportion to the number of shrubs, assuming you’re not accounting for things like coffee breaks or customers asking you to sell stuff to them.

In other words, direct proportion means that you have two quantities (in this case time and number of shrubs) such that when one quantity increases the other one does too, by the same factor (in this example, the factor is 3).

Another example: suppose you’re at your local supermarket/convenience store and you’re buying five tins of cat food costing £0.59 each. You want to work out how much the five tins will cost altogether.

Believe it or not, this calculation is an example of direct proportion in action. As the number of tins goes up, the total cost also goes up by the same factor. Five tins will cost five times as much as one tin, ten tins will cost ten times as much as one tin. Etc. Etc. And if you’re given the price of two tins to start with and want to work out the cost of – say – seven tins, you just divide by two to find the price of one tin, and multiply by seven.

Another way to think of direct proportion is visually – if you plot a graph of one quantity versus another and it goes up in a straight line, then they go up in direct proportion:


Inverse proportion

Sometimes one quantity goes down proportionately as the other one goes up. This is known as inverse or indirect proportion. To explain this, it’s probably better to give an example rather than getting bogged down in mathsspeak straight away.

Suppose 3 men can repaint a house in 4 days. You want to work out how long it would take (a) one of the men on his own, (b) 6 men and (c) 2 men to paint the house.

You can see straight away that one man is going to take longer than three men – it’s common sense, innit! Again, assuming he doesn’t have any extra breaks and that all the men are equally quick at painting, he is going to take three times as long as long as the 3 men did, i.e. 12 days.

Six men will be able to get the work done in half as much time as 3 men, so it would take them 2 days.

Two men is a harder one to work out. But if you know that one man takes 12 days, then two men will take half of that, i.e. 6 days.

If you want to lay it out to see if there’s a pattern, you could write it like this:

A third (1/3) of the original number of men take 3 times as much time
Two thirds (2/3) of the original number of men take 3/2 as much time (because 6 = 4 x 3/2)
2 x the original number of men take half (1/2) the time

You will see from looking at this that there is indeed a pattern. The number on the right is the reciprocal of the number on the left, i.e.

1/3 is the reciprocal of 3
2/3 is the reciprocal of 3/2
2 is the reciprocal of 1/2

If you did it as a graph, the line would start high and go down from there. But it wouldn’t go in a straight line this time – it would form what’s known as a hyperbola:


What kinds of quantities are inversely proportional?

We’ve seen that the more people you have working on a job the less time it takes. So that’s one example of inverse proportion. Another example is the relationship between speed and time taken to reach a destination – the higher the speed, the less time taken. Another example from the world of physics is the relationship between pressure and volume – as the volume increases, the pressure decreases and vice versa.

So there you have it. Hope that’s been useful!

© Empress Felicity September 2011