How does the rule of three work with direct proportionality?

short answers for big questions

The rule of three is a very practical process to solve problems, which requires 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 and indirectly proportional. A simple example will help us to understand the difference between these two concepts.

So, we have the following problem: a bicycle covers 80 km in 2 hours keeping the same pace. How long does it take to cover 20 km? First of all let's build a table in which `x` will represent the value we want to find:

Route (km)Time (h)
bicycle distance

Then let´s think and ask: if the route reduces, what will happen to time? As the bike will cover a shorter distance, less time will b required. So, the time will be reduced as well. Whenever both quantities are reduced or increased, we are faced with direct proportionality.

Therefore we have the following relation: `80/20 = 2/x`

In order to find the wanted value, it is enough to proceed with a crossed multiplication and solve the equation:

`80 xx x = 20 xx 2 hArr x = (20 xx 2)/80 hArr x = 40/80 hArr x = 0.5`

It will take the bike 0.5 hours, which means it needs 30 minutes to cover 20 km.

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