Percentage calculator

Pick the question you actually want answered, type two numbers, and read the answer together with the arithmetic that produced it. Nothing is sent anywhere — the calculation happens in this browser tab, which is the only sensible place for arithmetic this small to happen.

The five questions hiding behind one word

When somebody says "work out the percentage", they could mean any of five different sums, and the answers are not interchangeable. The first is the one most people picture: a share of a quantity. Fifteen percent of two hundred is thirty. The second runs the other way — you have two quantities and want to know how one relates to the other. Twenty-five out of two hundred is twelve and a half percent.

The third is change over time. Something was forty, now it is fifty, and the interesting number is the twenty-five percent it climbed. The fourth is applying a change you already know about: take twelve hundred and add twenty percent to it. The fifth is the one almost nobody names out loud — the gap between two figures where neither is the original. Thirty and fifty differ by fifty percent when measured that way, which is not the same as saying thirty grew by sixty-seven percent to reach fifty.

Most calculators put three of these on one page as three unlabelled boxes and leave you to work out which one matches your situation. That first step is where the mistake usually happens, not in the arithmetic. So this page names each question in full before you type anything, and shows the completed sentence next to the result so you can check that the answer belongs to the question you meant to ask.

A rise and a fall of the same size do not cancel out

Take a hundred, add fifty percent, and you have a hundred and fifty. Take fifty percent off that and you land on seventy-five, not back where you started. The two operations look symmetrical and are not, because the second one is measured against a bigger number than the first. In general, a rise of p percent followed by a fall of p percent leaves you at 1 − (p/100)² of the original, which is always less than one.

The same asymmetry shows up in reporting. If a figure falls from a hundred to eighty, that is a twenty percent drop; getting back to a hundred is a twenty-five percent rise. Both statements describe the same twenty units of movement, and both are correct, because percentage change divides by wherever you started. Quoting one of those numbers and implying the other is the most common way percentages are used to mislead without anybody actually lying.

Every change worked out on this page prints the return journey beside it, so the lopsidedness sits on screen instead of being a fact you are expected to carry around. A fifty percent loss needs a hundred percent gain to recover. A ninety percent loss needs nine hundred.

Percentage points are a different unit from percentages

If an interest rate goes from two percent to three percent, it has risen by one percentage point — and by fifty percent. Those are two true statements about the same event, using two different units, and swapping one for the other changes the size of the claim by a factor of fifty.

The rule is simple once it is stated: when you subtract one percentage from another, the result is measured in percentage points. When you divide one by the other, the result is a percentage. A survey moving from forty percent support to fifty percent support has gained ten percentage points, which is a twenty-five percent increase in support. Both numbers are worth knowing and they answer different questions.

Mixing them up is not a rare pedantic error. It appears in news coverage of interest rates, unemployment, election polling and tax bands almost every week, and it usually inflates the story. If you are ever unsure which one a source means, subtract the two figures and see which of the quoted numbers you get.

Discounts and mark-ups do not add together

A shop offers thirty percent off, then another twenty percent off at the till. That is not fifty percent off. The second discount applies to the already-reduced price, so a hundred becomes seventy, and seventy becomes fifty-six — a combined discount of forty-four percent. Stacked reductions always come out smaller than their sum, and the gap grows as the individual discounts get bigger.

The same arithmetic applies in the other direction with mark-ups, surcharges and compound growth. Two consecutive ten percent increases produce twenty-one percent, not twenty. Over enough periods the difference stops being a rounding detail: ten percent a year for ten years is not a hundred percent, it is a hundred and fifty-nine.

The useful mental model is that percentages compose by multiplication, not addition. Thirty percent off means multiplying by 0.7. Twenty percent off means multiplying by 0.8. Doing both means multiplying by 0.56, and the order does not matter. Once you think in multipliers, the "add them up" mistake becomes hard to make, and the "apply a percentage" mode on this page shows the single-multiplier form of every calculation for exactly that reason.

Working backwards from a price that already includes tax

Suppose a price of one hundred and five already includes a five percent tax. The tax component is not five percent of a hundred and five. It is a hundred and five divided by 1.05, giving the pre-tax figure of a hundred, and the tax is the five that is left over. Multiplying the gross figure by the rate overstates the tax every time, and the error grows with the rate: at twenty percent, the naive method gives twenty instead of the correct sixteen and two thirds.

Expense claims and bookkeeping go wrong here more often than anywhere else in percentage arithmetic, and the cause is always the same: which figure the rate was applied to before anyone wrote the total down. Tax-exclusive pricing, common in the United States, adds the percentage on top: a fifty dollar item with an eight percent tax costs fifty-four. Tax-inclusive pricing, which is the norm across the European Union, Taiwan, Japan and most of Asia, means the percentage is already baked into the figure on the label and has to be divided back out.

The "A is what percent of B" mode handles this directly if you already know both figures, and the reverse of any mark-up is shown under "to undo it you need" — which for a twenty percent addition is not twenty percent but sixteen point six seven.

Where the arithmetic itself quietly goes wrong

Computers store decimals in binary, and most decimal fractions have no exact binary representation, in the same way that a third has no exact representation in decimal. That is why a naive calculation of ten percent of a price sometimes produces a long tail of digits ending in a stray four. The value is off by less than a quadrillionth, which never matters for the answer but looks alarming on a page that claims to show its working, so results here are settled to twelve significant figures before they are displayed — far more precision than any real question needs, and far less than the point at which the noise appears.

Rounding is a separate trap. Rounding each line of an invoice and then totalling gives a different figure from totalling and then rounding once, which is why tax authorities specify which of the two they require rather than leaving it to the software. Neither is more correct in the abstract; they are different conventions, and disagreements between two systems are usually this and not a bug.

The last one is genuine precision loss. Numbers above roughly nine quadrillion cannot be held exactly by a browser, so beyond that limit the final digits become an approximation. That threshold is far outside any percentage question a person actually asks, but if you do type a number that large this page says so rather than presenting an approximation as a fact.

A calculator has no reason to talk to a server

Every tool on this site runs on your own device, and for file conversion that is a meaningful privacy claim: your contract or your passport scan genuinely never leaves the machine. For a percentage calculator the honest version of the claim is smaller, and worth stating plainly rather than dressing up. Two numbers and a percentage are not sensitive on their own.

What is worth stating is the other half. Sending "fifteen" and "two hundred" to a server, waiting for a round trip, and getting "thirty" back is an absurd amount of machinery for one multiplication — and it is what a surprising number of calculator sites still do, because a page load that touches the server is a page load that can be measured, retargeted and monetised. The arithmetic here is a few lines of JavaScript that your browser already downloaded with the page. It runs offline after your first visit, it returns an answer before you finish typing, and there is no queue.

Verifying that takes about ten seconds and requires trusting nobody. Put the network inspector on screen, then edit one of the figures. The list of requests stays exactly as long as it already was, because there is no endpoint on the far side for it to grow towards.