Percentage Calculator

The three formulas

X% of Y = Y × X ÷ 100  ·  A of B = A ÷ B × 100%  ·  change = (B − A) ÷ A × 100%

Worked examples: 15% of 80 is 12. 30 out of 120 is 25%. Going from 50 to 65 is a 30% increase, while 65 back to 50 is only a 23.1% decrease — the base changed, which is the whole trick of percentages and the reason the third tab exists.

The three questions, and telling them apart

Nearly every percentage error is a right answer to the wrong question. There are only three, and naming the one you are asking removes most mistakes before they happen.

Finding a part: "what is 15% of 240?" — multiply, 240 × 0.15 = 36. This is tips, VAT on an exclusive price, commission, a discount amount. Finding a rate: "36 is what percent of 240?" — divide, 36 ÷ 240 = 15%. This is exam marks, market share, conversion rates. Finding the whole: "36 is 15% of what?" — divide by the rate, 36 ÷ 0.15 = 240. This is working back from a deposit to a purchase price, or from VAT paid to the amount it was charged on, and it is the one people most often attempt by adding a percentage, which does not work.

Why a 20% rise then a 20% fall does not return you to the start

Take 100, add 20% to reach 120, then take 20% off: 120 × 0.8 = 96. You are 4% down, not level. The second percentage was taken from a bigger number, so it removed more than the first added. The general result is that a rise of p% followed by a fall of p% always lands below where you began, by p²/100 percent.

This is not a curiosity. It is the arithmetic behind "was £200, now £150, was £200" pricing cycles, and it is why investment losses are harder to recover than they look: a 50% loss needs a 100% gain to get back to even, and a 20% loss needs 25%. Symmetric-sounding percentages are not symmetric.

Percentage points are not percent

If an interest rate moves from 5% to 7%, it has risen by two percentage points and by 40 percent. Both are correct and they mean very different things, which is why the distinction is a favourite of anyone trying to make a change sound bigger or smaller than it is. A vaccine that cuts risk from 2% to 1% has reduced it by one percentage point and by half; a headline can honestly choose either. When you read a percentage change, ask what it is a percentage of.

Reversing a percentage correctly

Removing 15% VAT from an inclusive price of R1,150 is not subtracting 15%. Subtracting gives R977.50, which is wrong. The inclusive price is 115% of the original, so you divide: 1,150 ÷ 1.15 = R1,000. The error is small on one item and adds up quickly across a book of invoices — and it always runs the same direction, understating the pre-tax amount.

The same trap applies to any "after the increase" figure. If a salary is £52,000 after a 4% rise, the previous salary was 52,000 ÷ 1.04 = £50,000, not 52,000 × 0.96 = £49,920.

Mental shortcuts worth knowing

Percentages commute: 8% of 25 is the same as 25% of 8, which is 2 — and one of those is trivial. Use it whenever one side is a friendly number. 10% is a decimal-point shift, and 5% is half of that, so 15% is "move the point, then add half again" — the standard way to work out a tip without a phone. 1% is two decimal shifts, which makes any percentage buildable from tens and ones: 37% is three tens and seven ones.