Percentages can be a little trickier if the figure you are starting with is not 100% of the quantity in question. If the quantity in question has already been changed (i.e. increased or decreased), then you are not dealing with the 100% figure.

For example, you go into a shop and see that the latest iPhone is on sale at 30% off. If the price has been reduced by 30%, then the price you see in the shop is clearly not the original price of the iPhone. Since the quantity in question is the price of the iPhone, we should consider the original price of the iPhone to represent 100% of this quantity. Therefore, the price you see in the shop must represent only 70% of the original price (100% minus the 30% for the sale).

The key point here is that the price you see in the shop does not represent 100%; instead, it represents only 70%.

Why is this important? Let’s add some figures to this example to make things clearer.

Say the original price of the iPhone, before any sale, was £1,000. This represents 100% of the original price (i.e. full price).

In the sale, the price is reduced by 30%. The sale price is therefore calculated as the original price (100%) minus 30% for the sale, for a total of 70% of the original price (i.e. we need to multiply by 0.70):

\begin{aligned}\text{Sale price}&=\text{Original price}\times 0.70 \\[12pt]&=£1,000\times 0.70 \\[12pt]&=£700\end{aligned}

Say you were asked to calculate 1% of the price of the iPhone. Well normally, to calculate 1% you simply divide by 100 don’t you? But do we divide the original price of £1,000 by 100 to calculate 1%, or do we divide the sale price of £700 by 100 to calculate 1%? If we do each in turn:

\begin{aligned}1\%=\frac{£1000}{100}=£10\end{aligned}

Or,

\begin{aligned}1\%=\frac{£700}{100}=£7\end{aligned}

Which 1% figure is correct?

The answer is of course £10, the 1% figure we get if we divide the original price by 100. Dividing by 100 only gives you 1% if the figure you are dividing represents 100%, i.e.:

\begin{aligned}\frac{100\%}{100}=1\%\end{aligned}

Since the original price represents 100%, we can be confident that dividing this by 100 does indeed give 1%.

The mistake we made when dividing the sale price of £700 by 100 was that the sale price did not represent 100%. It represented only 70%, so dividing this by 100 actually gave:

\begin{aligned}\frac{\text{Sale price}}{100}=\frac{£700}{100}=\frac{70\%}{100}=0.7\%\end{aligned}

Which is not the percentage we were asked to find!

If we only knew the sale price and we still had to calculate 1%, we should not divide by 100 but instead we should divide by 70:

\begin{aligned}\frac{\text{Sale price}}{70}=\frac{£700}{70}=\frac{70\%}{70}=1\%=£10\end{aligned}

And this gives the same 1% = £10 result as before.

These sorts of considerations are important if the quantity you are dealing with has been changed.

Where this is the case, there are set steps you should follow:

1) Establish the quantity in question – in our example, this was the price of the iPhone.
2) Establish the 100% figure – in our example, this was the original price of the iPhone before the sale.
3) Establish your current percentage in relation to the 100% figure – in our example, this was 70% (i.e. the percentage the sale price represented after having decreased the original price by 30%).
4) Calculate 1% – in our example, this was achieved by dividing the sale price (70%) by 70.
5) Calculate the percentage you need using 1% – in our example, we could have taken the 1% figure (£10) and multiplied it by 100 to get back to the original price of £1,000.

Try the following examples to get to grips with this framework for handling reverse percentages:

A games console is on sale and has been reduced by 40%. The new price of the console is £360. What was the original price?

1) Quantity in question = console price​
2) 100% figure = the original price of the console
3) Current percentage = 60% (100% of the original price minus 40% for the sale)
4) Calculate 1%:

\begin{aligned}1\%\ \text{of original price}&=\frac{\text{Sale price (i.e. 60\%)}}{60} \\[12pt]&=\frac{£360}{60} \\[12pt]&=£6\end{aligned}

5) Calculate the percentage you need using 1%:

\begin{aligned}\text{Original price}&=100\% \\[12pt]&=1\%\times 100 \\[12pt]&=£6\times 100 \\[12pt]&=£600\end{aligned}

A basketball team scored 102 points in their game today, which was 20% more than they scored in their last game. How many points did the team score in their last game?

1) Quantity in question = points scored​
2) 100% figure = the points scored in the original game
3) Current percentage = 120% (100% of points scored in the original game, plus an additional 20% scored in today’s game)
4) Calculate 1%:

\begin{aligned}1\%\ \text{of original points scored}&=\frac{\text{New points scored (i.e. 120\%)}}{120} \\[12pt]&=\frac{102}{120} \\[12pt]&=0.85\end{aligned}

5) Calculate the percentage you need using 1%:

\begin{aligned}\text{Original points scored}&=100\% \\[12pt]&=1\%\times 100 \\[12pt]&=0.85\times 100 \\[12pt]&=85\end{aligned}