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

Four questions, not one

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

1. A percentage of an amount. What is 20% of 250?

amount × 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. Percentage change. How much did it rise from 250 to 300?

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

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

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

If you are unsure which you need, look at what is missing: a missing part is case 1; a missing percentage is case 2 or 3; a missing final figure is case 4.

Change is always measured against the starting value

This is where most people slip. Going from 250 to 300 is a 20% rise, because the €50 difference is compared against the 250 you started from. But falling from 300 back to 250 is not a 20% drop; it is 16.7%, because the reference point is now 300.

The same absolute difference produces two different percentages depending on which end you look from. That is why a news story can truthfully say a price "rose 20% and then fell 17%" while describing an exact return to where it began.

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

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, which is 37% off.

Undoing a percentage means dividing, not subtracting

If 21% VAT has been added to a price and you want the net amount, do not subtract 21%; divide by 1.21.

• €100 + 21% = €121
• €121 − 21% = €95.59 ← wrong
• €121 ÷ 1.21 = €100 ← correct

It is the same idea seen from the other side: the 21% was calculated on 100, not on 121, so subtracting it from the total takes off too much.

Percentage points

When the thing rising is itself a percentage, there are two correct ways to describe it, and they are worth distinguishing.

If an interest rate goes from 2% to 3%, it has risen one percentage point. It has also increased 50% in relative terms, since 1 out of 2 is half. Both statements are true and describe exactly the same event; the second sounds far more dramatic, which is precisely why it pays to notice which one you are being told.

When the problem comes with units instead of percentages

A percentage is a direct rule of three in which the first quantity is 100. When the question arrives not in percentage terms but with kilos, euros, days or hours, the natural set-up is the other one: how to do a rule of three sets out the four-step method, the difference between direct and inverse, and the four cases where there is no proportion and cross-multiplying returns a wrong number.