PRACTICAL GUIDE
Percentage Calculations Explained With Simple Examples
The three calculations people actually need, the two that reliably go wrong, and why a rise and a fall of the same percentage never cancel out.
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A percentage of a number
Convert the percentage to a decimal and multiply. Twenty per cent of 150 is 0.20 multiplied by 150, which is 30. The same works in reverse and catches people out less often than it should: because multiplication commutes, four per cent of 75 is the same as 75 per cent of 4, which is 3. That trick turns some awkward mental arithmetic into easy arithmetic.
What percentage one number is of another
Divide the part by the whole and multiply by 100. Thirty out of 150 is 30 divided by 150, times 100, which is 20 per cent. The only real difficulty is deciding which number is the whole, and that is a question about the situation rather than about the maths: the whole is whatever the answer is meant to be a share of.
Percentage change
Subtract the old value from the new one, divide by the old value, and multiply by 100. From 80 to 100 is 20 divided by 80, which is a 25 per cent increase. From 100 to 80 is minus 20 divided by 100, which is a 20 per cent decrease.
Those are different numbers for the same pair of values, and that is not a mistake. The denominator is always where you started, so the direction of travel changes the answer. This is why an increase and a decrease of the same percentage never return you to where you began: a price that rises 50 per cent and then falls 50 per cent ends up a quarter lower than it started.
Percentages and percentage points
A rate moving from 20 per cent to 25 per cent has risen five percentage points, and has also risen 25 per cent. Both statements are true and they measure different things: one is the difference between the rates, the other is that difference relative to the starting rate. Reports that mix them up are usually reaching for the larger number, so it is worth checking which one is being quoted.
Working backwards from a total
If a price includes 23 per cent tax and you want the amount before tax, do not subtract 23 per cent from the total. Divide by 1.23. Subtracting takes 23 per cent of the larger number and leaves you short; dividing undoes the multiplication that produced the total. The same applies to any figure that has already had a percentage added to it.
Frequently asked questions
Why do a 50% rise and a 50% fall not cancel out?
Because each is measured against a different starting point. A rise takes 50 per cent of the original; the fall then takes 50 per cent of the larger, increased figure, which is a bigger amount. 100 becomes 150 becomes 75. To return exactly to 100 from 150 you would need to fall by a third.
How do I remove a tax already included in a price?
Divide the total by one plus the rate as a decimal. For 23 per cent, divide by 1.23. Subtracting 23 per cent instead gives a number that is too low, because it takes the percentage of the total rather than of the base you are trying to recover.
What is the difference between percentage and percentage points?
Percentage points are the plain difference between two rates; a percentage change is that difference relative to the starting rate. Going from 20 per cent to 25 per cent is five percentage points and a 25 per cent increase. Both describe the same movement and imply very different things.
How do I apply two percentage changes in a row?
Multiply them rather than adding them. A 10 per cent discount followed by a further 10 per cent is 0.9 times 0.9, which is 0.81 — a 19 per cent discount in total, not 20. The order does not matter, but the fact that they compound does.