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Increasing & decreasing by a percentage
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Express a quantity as a percentage of another

Increasing & decreasing by a percentage

These two techniques are the basis for everything else we will do with percentages. They are essential, in that you will not able to progress beyond this point in mathematics, until you master it, so take your time, work through the examples and then practise.

To change a quantity (up or down) by a percentage, we have two possible approaches: the slow and stupid way (SAS) and the fast and clever way (FAC). TBH, the only reason I will show you the SAS method is that it helps in explaining the FAC method.

Because we already know how to find a percentage of a quantity, we can do that and then either add or subtract, depending on whether the problem is an increase or a decrease. Now many students see this method as being The one I prefer because it is fairly obvious. However, what they don't know is just how big a disadvantage they will be at later on.

Here is a simple example: Increase £200 by 15%

We can work this out using the SAS method by finding the increase and then adding this to the original £200. \[\begin{aligned}[t] &15\%  \text{ of } 200 = 0.15 \times 200 = £30\\ &200 + 30 = £230 \end{aligned}\] The method works but the problem is this: it takes two separate steps. Every time we use the method, we have to perform two distinct tasks. And when we decide to increase the new value by $15\%$, we have to do both steps again and again and so on. It becomes very laborious, particularly if you need to perform the increase several hundred times, which, we have to do, especially if we are dealing with investments with interest calculated on a daily basis. It would be so much easier if we can find a way to do this which would enable us to work out an introductory step and then apply a single multiplier each time subsequently.

So let us begin over again. Increase £200 by 15%.

What we are going to do, is to combined the two steps above, into a single calculation which we can then apply repeatedly as required.
\[\begin{aligned}[t] 230 &= 100\%  \text{ of } 200 + 15\%  \text{ of } 200 \\ 230 &= 115\%  \text{ of } 200\\ 230 &= 1.15 \times 200\\ \hline \end{aligned}\]

Now, if we had to do what I did in this example each time, we would have a method which would be laborious in the extreme. But the formulas below make things much quicker and once you get the hang of them, you will find multipliers very quickly indeed. To this end, we must find the multiplier (in this case $1.15$), after which we can multiply by it as often as we like.

The Multiplier formulas

Increase

To increase by $p\%$ we apply the following formula to get the multiplier

\[ \text{Multiplier } = \frac{100+p}{100} \]

Decrease

To decrease by $p\%$ we apply the following formula to get the multiplier

\[ \text{Multiplier }  = \frac{100-p}{100} \]