How does the rule of three work with inverse proportionality?

The rule of three is a very practical process to solve problems which require four numbers of which three are known.

But before explaining how the rule works, it is important to stress the fact that it can only be used if the required numbers are directly or indirectly proportional. A simple example will help us to understand the difference between these two concepts.

Let´s see the following example: It takes 8 hours for a group of 10 workers to gather grapes. If the number of workers is raised to 12, and the pace of work is the same. How long does it take them to gather grapes? First of all let's build a table and we will represent through `x` the value we want to find.

Then let´s think and ask: If the number of workers is raised, what will happen to the time required to gather the grapes? It would be reduced since there are more workers and so it would take them less time to perform the same job. Whenever one of the quantities is increased the other one diminishes and so we are faced with inverse proportionality.

We cannot forget about inverting one of the terms and so we will get the following relation: `12/10 = 8/x`

In order to find the wanted value, it's enough to make a crossed multiplication and solve the equation:

`12 xx x = 10 xx 8 hArr x = (10 xx 8)/12 hArr x = 80/12 hArr x ~~ 6.67`

It will take the workers about 6.67 hours, which means about 6 hours and 40 minutes.

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