Percentage calculator: discounts, increases and reverse VAT

How percentages are calculated

A percentage is simply a fraction with 100 as its denominator: 20% is 20/100, that is 0.2. Everything else follows from there. 20% of 150 is 150 multiplied by 0.2, which is 30. To find what percentage 30 is of 150, take the ratio, 30/150 = 0.2, and multiply by 100.

Increases and discounts are applied by multiplying by a factor rather than in two steps: a 20% increase is × 1.20, a 20% discount is × 0.80. This form is far more convenient when changes stack up: two successive 20% discounts do not make 40%, but × 0.80 × 0.80 = × 0.64, an overall discount of 36%.

The reverse calculation is the one people get wrong most often. If a €122 price includes VAT at 22%, the net amount is not 122 minus 22% of 122 (which would give €95.16) but 122 divided by 1.22, that is €100. The reason is that the percentage was taken on the starting value, not on the final one: to go back you divide by the factor, you do not subtract the same share.

Percentage change between two values is always measured against the starting value. From 80 to 100 is a 25% increase (20/80); from 100 to 80 is a 20% fall (−20/100). They are not symmetric, which is why a stock that loses 50% then has to gain 100% to get back to where it started.

Common mistakes

Subtracting the percentage from the final price to strip out VAT or a discount: the result is always too low, and the error grows with the rate.
Adding two successive percentage changes instead of multiplying the factors: +10% followed by +10% gives +21%, not +20%.
Measuring the change against the final value rather than the starting one: it flips the denominator and returns a different percentage.
Confusing percentage points with percentages: going from 4% to 6% is a rise of 2 percentage points, but of 50% in relative terms.
Applying a discount to the VAT-inclusive price when it was agreed on the net amount: the order of the two operations changes the total due.

Frequently asked questions

How do you calculate a percentage of a number?

Divide the percentage by 100 and multiply by the number. 15% of 200 is 200 × 0.15 = 30. Alternatively, divide the number by 100 and multiply by the percentage: the result is identical.

How do you remove VAT from a price?

Divide the gross price by 1 plus the rate. With VAT at 22%, a price of €122 corresponds to a net amount of 122 / 1.22 = €100, and the tax is the difference, €22. Subtracting 22% from the €122 would give the wrong answer.

How do you calculate the percentage change between two values?

Change = (final value − starting value) / starting value × 100. From 80 to 100 the change is (100 − 80) / 80 = +25%. The sign tells you whether it is a rise or a fall.

Do two 10% discounts add up to a 20% discount?

No. The second discount applies to the already reduced price, so the overall factor is 0.90 × 0.90 = 0.81: the effective discount is 19%, not 20%. The same holds for successive increases, in the opposite direction.

What is the difference between percentage points and percent?

Percentage points measure the difference between two percentages; percent measures their relative change. A rate going from 2% to 3% rises by 1 percentage point, but by 50%.

How this calculation works

Percentage of a total: result = total × percentage / 100. What percentage: percentage = part / total × 100. Increase or discount: result = amount × (1 ± percentage / 100). Change between two values: change = (final − starting) / |starting| × 100. Reverse: starting value = final amount / (1 ± percentage / 100). That last formula divides by the factor precisely because the percentage had been taken on the starting value.

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