Percentage calculator

Which box for which question

Almost every percentage question in ordinary life is one of these three wearing different clothes. A discount is the first one: 30% off a $80 jacket means working out 30% of 80, which is 24, and paying 56. A tip is the same shape — 18% of a 64 bill is 11.52. So is tax, commission and any "add 20%" instruction, because all of them take a percentage of a number you already have.

Marks, shares and hit rates are the second box. Scoring 37 out of 45 is 37 ÷ 45 × 100, or 82.2%. So is "we closed 12 of 80 leads" (15%) and "what percentage of my salary is rent". Whenever you have a part and a whole and want the rate between them, that is the one to use.

Anything described as growth, a rise, a fall, a markup or a variance against last year is the third box. It needs two numbers in time order — where you started and where you ended — and it reports the move as a percentage of where you started. Sales going from 200 to 250 is +25%; going from 250 back to 200 is −20%, and the asymmetry is not a mistake.

Where people get percentages wrong

The most common error is adding percentages that are taken from different bases. Two consecutive 10% rises are not a 20% rise, they are 21%, because the second one is charged on the already-larger number. The same trap runs backwards through discounts: an extra 20% off a sale price that is already 30% off gives 44% off the original, not 50%.

The second is confusing a percentage with a percentage point. If an interest rate moves from 4% to 5%, that is one percentage point, but it is a 25% increase in the rate. Both statements are true and they describe very different things, which is why the phrasing in a news story or a lender's letter is worth reading twice.

Questions people ask

How do I work out a discount or a price after tax?

Use the first box. For a discount, put the discount rate in X and the price in Y to get the amount you save, then subtract it from the price — 30% of 80 is 24, so the jacket costs 56. For tax or a markup, do the same and add it instead. If you want the final figure in one step, work out 70% of the price for a 30% discount, or 120% for a 20% markup.

What are the three formulas?

X% of Y = X ÷ 100 × Y, so 15% of 200 is 30. X is what percent of Y = X ÷ Y × 100, so 30 out of 200 is 15%. change = (new − old) ÷ |old| × 100, so 200 to 250 is +25%.

Why does a 20% fall then a 20% rise not get me back?

Because each percentage is taken from a different starting number. 100 falls 20% to 80, then 80 rises 20% of 80, which is 16, giving 96. To undo a 20% fall you need a 25% rise. The bigger the fall, the bigger the rise needed.

What happens if I start from zero or a negative number?

Percentage change from zero has no meaning, since there is nothing to compare against, so the answer is left blank. Starting from a negative number, the change is measured against the size of that number ignoring its sign: −100 to −50 is a +50% change.