How to calculate a percentage: the formula and the four cases

Four questions, not one

"Calculating a percentage" sounds like a single operation, but in practice it is four different calculations, and almost every mistake comes from applying the formula of one to the question of another.

1. The percentage of a number. What is 20% of 250?

number × percentage ÷ 100 → 250 × 20 ÷ 100 = 50

2. What percentage one number is of another. What percentage is 30 of 250?

part ÷ total × 100 → 30 ÷ 250 × 100 = 12%

3. The percentage change. How much did it rise from 250 to 300?

(final − initial) ÷ initial × 100 → (300 − 250) ÷ 250 × 100 = 20%

4. Applying an increase or a reduction. What is 250 with 20% off?

number × (1 ± percentage ÷ 100) → 250 × 0.8 = 200

If you are not sure which one you need, look at which figure is missing: if a part is missing it is case 1; if the percentage is missing, case 2 or 3; if the final result is missing, case 4.

Change is always measured against the starting value

This is where most people go wrong. Going from 250 to 300 is a 20% rise, because the €50 difference is compared with the 250 you started from. But falling from 300 to 250 is not a 20% drop: it is 16.7%, because the reference is now 300.

The same difference in euros gives two different percentages depending on where you stand. That is how a news story can truthfully say a price "rose 20% and then fell 17%" and be describing an exact return to the starting point.

Why percentages do not add up

If a €100 price rises 20% it becomes €120. If it then falls 20% it does not return to €100 but to €96: the second percentage applies to a larger base.

The same goes for chained discounts. 30% in the window plus 10% at the till is not 40%: €100 less 30% is €70, and less another 10% is €63, a total discount of 37%.

The general rule: to chain percentages you multiply the factors, you do not add the percentages. A 30% and a 10% discount are 0.7 × 0.9 = 0.63, that is 37% off. The discount calculator does the sum in two steps for exactly this reason.

Undoing a percentage means dividing, not subtracting

If 22% VAT has been added to a price and you want the net amount, do not subtract 22%: divide by 1.22.

• €100 + 22% = €122
• €122 − 22% = €95.16 ← wrong
• €122 ÷ 1.22 = €100 ← correct

It is the same idea as before seen from the other side: the 22% was calculated on 100, not on 122, so subtracting it from the total takes off too much. In Italy this operation has a name, the scorporo dell'IVA, and it works for any rate.

Percentage points

When the thing that rises is already a percentage, there are two correct ways to say it and it pays to keep them apart.

If an interest rate goes from 2% to 3% it has risen by one percentage point. It has also increased by 50% in relative terms, because 1 out of 2 is a half. Both statements are true and describe exactly the same thing; the second sounds far more dramatic, which is precisely why you should notice which one the person telling you is using.

When the problem comes with units rather than percentages

A percentage is a direct proportion in which the first quantity is 100. When the question does not arrive in per cent but in kilos, euros, days or hours, the natural setup is the other one: the rule of three calculator applies the method, explains the difference between direct and inverse proportion, and flags the cases where the proportion does not exist and cross-multiplying gives a wrong number.