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Percentages don’t always work in exactly the way that you might think. The example in the previous section of the erroneous calculation of Nett from Gross price should have put you on your guard.
For another example consider the following question - is taking P% of something and then Q% of the result the same as taking P%+Q% of the original value?
Putting this in more practical terms if you are offered a discount of 10% on a price and then offered a further discount of 5% is this the same as a 15% discount?
Taking P% and then Q% of a value is just
and this is quite clearly not the same as
To see that this is the case just try a few examples using a spreadsheet and you will quickly discover that the results that the two formulas produce are very different. For example, if you take 10% of 100, i.e. 10 and then take 20% of this the result is 2 but (10+20)% i.e. 30% of 100 is 30.
In other words if you take successive percentages of something they don’t add they multiply. Taking 10% followed by taking 20% of the result is the same as taking 2% of the original (i.e.0.1*0.2).
However when reasoning or bargaining with percentages it is usual to speak of “another 1%” in the sense of increasing the percentage offered by adding one percent. In this case it does look as if percentages add but notice that all the percentages refer to a proportion of the same whole.
The best way to think of this is by considering slices of a pie. If you are negotiating for an additional percentage of the whole pie then certainly percentages add but if you are arguing for a percentage of the slice you have already been allocated then they multiply.
A 10% gain and a 10% loss
So in many cases you can be mislead by adding percentages. This is also true if you increase or decrease a value by P% and then Q%. In general
- Increasing or decreasing a quantity by P% and then Q% is not the same as increasing or decreasing it by (P%+Q%)
This gives us the answer to the discount on a discount problem posed earlier. Unless you know if the sales person intended to give you a total discount of 15% or alternatively a further discount on the discounted price the actual price you will have to pay isn’t clear.
In most cases an additional discount is usually meant in the literal sense of adding to the initial discount - i.e. the intended discount is 10%+5% and not 5% more off the discounted price - what is important is that you realise that there are two possible interpretations.
As another example of how percentages do not add up in the way that you might suppose consider the following simple problem. A trader makes a 10% profit on an investment in the first part of the day but before the day is out has made a 10% loss - is the total profit zero?
The commonsense answer to this question is that the 10% gain is wiped out by the 10% loss, but this reasoning doesn’t take into account the fact that the percentages are calculated on different values.
Suppose the initial investment was $100, then a 10% gain is $10 and this makes the total investment worth $110. The following 10% loss reduces the investment by $110*0.1 or $11, making the final investment worth only $99. Hence a 10% gain followed by a 10% loss doesn’t take you back to square one.
Also notice that it doesn’t matter in which order the lost or gain occurred - the answer is exactly the same, a 1% loss on the initial value.
Of course if both the percentages were quoted relative to the initial investment then the percentages would add up because 10% of $100 is always $10. Again what is important here is that you realise that there are two possible meanings and you should always discover what value a percentage change is using as its starting point.
The actual percentage
If you are offered a discount of P% and then a further discount of Q% on the discounted price, it is worthwhile asking what actual discount you are receiving on the original price? This sounds like a difficult question but it is very easy. As the twice discounted price is given by:
if the total actual discount percentage is D% then the final discounted price is:
cancelling price from both sides gives the relationship between Q, P and D:
Finally solving for D gives:
The same is true of percentage increase but in this case the formula for the actual percentage increase is (1+P)*(1+Q)-1. That is, if a value is increased by P% and then the result further increased by Q% the total percentage increase is:
For example, if a sales person offers you a further 5% discount on a price that has already been discounted by 10% the total discount is 1-(1-0.1)*(1-0.05) which works out to 14.5% rather than the naive guess of 15%.
If you are initially charged 8% more for a holiday booking and then just before you leave a 15% fuel surcharge is added then the total percentage surcharge is (1+0.08)*(1+0.15)-1 or 24.2% increase on the original price.
This is the first and simplest example of an actual or effective percentage. Later when we look at percentages as interest rates we will meet the best known example of actual percentages - the APR or Actual Percentage Rate. For now let’s look at another simpler, but practical, question concerning percentages and interest rates.