Inverse Proportion
There are many proportion-style questions. Here we are focusing specifically on inverse proportion, or the questions similar to the below:
It takes 10 workers 30 days to build a house. How long would it take 5 workers?
You might intuitively figure out that since we have halved the number of workers, it’s going to take them double the time. Now imagine the question is the following:
It takes 15 workers 12 days to build a bridge. How long would it take 9 workers?
A quick way to answer all questions of this style is to figure out the number of ‘worker-days’; i.e. how long would it take 1 worker to build the bridge? We figure this out by multiplying 15 by 12 = 180 days.
You can think of this as since 1 worker is doing the work of 15, it will take them 15x the amount of time.
We then distribute the days to the new number of workers. In this case, 180 days distributed to 9 workers = 20 days.
You can also think of this as since 9 workers are doing the work of 1, it will take them 1/9th the amount of time.
This can easily be applied to other similar questions, for example:
5 pipes can fill a bathtub in 23 hours and 30 minutes. How long would it take 3 pipes?
- 23 hours and 30 minutes = 1410 minutes
- 5 * 1410 = 7050 minutes
- 7050 / 3 = 2350 minutes
- 2350 minutes = 39 hours and 10 minutes
Principles
The above technique should be able to get you through all of these types of questions, but if you are more interested in the GCSE maths concept behind it here’s a brief writeout below.
Direct proportion occurs when one value increases as the other increases. Inverse proportion occurs when one value increases while the other decreases. Let’s think of them as value X and value Y.
Direct proportion => X ∝ Y (X is directly proportional to Y)
Inverse proportion => X ∝ 1/Y (X is inversely proportional Y, or X is proportional to the inverse of Y)
X and Y are related by some unknown value we can call ‘K’. We’ll come back to what K represents at the end, but for now we can rewrite this proportion as equations:
Direct proportion => X = KY (every increase in Y increases X by this K value)
Inverse proportion => X = K/Y
If we go back to our initial example, it takes 12 days for 15 workers to build a bridge. The value of X is what we are trying to find out, so in this case we will put X as the number of days and Y as the number of workers.
12 = K/15
It follows then that 12 * 15 = 180, and K = 180. Therefore the general equation for the relation between the number of days and workers is X = 180/Y. If we wanted to find how long it would take 9 workers we substitute 9 for Y. X = 180 / 9 = 20 days.
‘K’ is the initial value, in context being how long it would take 1 worker to build the bridge. It can be applied also to how long it would take 1 pipe to fill a bathtub, 1 programmer to build a piece of software etc.