Percentage of a Percentage Calculator

“Of” is the multiplication sign, and per cent is just “over 100”

Every figure on this page comes from those two lines. The first is the whole method: per cent means “per hundred”, so 20% is the number 0.20 and 30% is 0.30, and the word of between two numbers has always meant multiply. Multiply them and you get 0.06 — six hundredths, which written as a percentage is 6%. Nothing in that depends on the figures being percentages; it is the same multiplication you would do for 20% of 30 apples, and it lands on 6 for the same reason.

The step that trips people is what the answer is a percentage of. It is 6% of the original whole, not 6% of the 30%. If 30% of a school’s pupils cycle to school and 20% of those cyclists are in Year 7, then Year 7 cyclists are 6% of the school — not 6% of the cyclists, who are a much larger 20% of them. Both statements are true about the same children and they are not interchangeable, so name the base out loud before you quote the number. Multiplication also does not care about order: 20% of 30% and 30% of 20% both give 6%, which is a quick self-check when a result looks wrong.

Two inputs that look like errors are not. A percentage above 100% is ordinary — 150% of 30% is 45%, because 1.5 × 0.30 = 0.45 — and a negative first percentage expresses a reduction, giving a negative share. Only a genuinely empty or non-numeric entry is refused. No authority is cited for any of this because none exists to cite: multiplying two proportions is elementary arithmetic, set out in full in the cards above. What a particular contract, syllabus, or tax rule means by its two percentages comes from that document, not from mathematics.

Half of a half is a quarter; 10% of 10% is 1%

The results people look up most often are worth recognising on sight, because each one is a single multiplication. 50% of 50% is 25% — half of a half is a quarter, which is the version everyone already knows in words. 10% of 10% is 1%, and that pattern holds: taking a tenth of a tenth always moves the answer one decimal place. 25% of 80% is 20%. Every cell below is (row ÷ 100) × column, computed by the same engine as the calculator above rather than typed out by hand:

Read a row as the slice you are taking and a column as the percentage you are taking it from. The whole grid falls below its own column heading, which is the visual form of the rule that a percentage of a percentage is always smaller than either — as long as both are under 100%. Push the first figure over 100% and the result climbs above the column instead, which is exactly what taking more than all of something should do.

Up 10%, then down 10%, leaves you down 1% — never back where you started

The second mode answers a different question: not one percentage of another, but several percentage changes applied one after another, reduced to the single change they really come to. The instinct is to add the rates — +10% and −10% cancelling to nothing — and it is wrong, because the second change is applied to a total the first change already moved. A £1,000 balance up 10% is £1,100; 10% off that is £110, not £100, so you land on £990. Overall: −1%.

What the calculator multiplies is growth factors rather than rates. A +10% change is a factor of 1.10, a −10% change is 0.90, and 1.10 × 0.90 = 0.99, which is a 1% loss. Once you are multiplying, order stops mattering — down 10% then up 10% also lands on 0.99 — and the gap between the honest answer and the naive sum grows with every step you add. The tool reports both figures side by side for that reason. Some common stacks:

The stacked-discount rows are the ones with money on them. A coupon taking 20% off a sale item already reduced by 10% is not 30% off: it is 28% off, because the coupon applies to the already-reduced price. That is a real difference of two percentage points on the sticker, and it always favours the seller. The last row is the harshest version of the same arithmetic: a value that doubles and then halves is exactly back where it began, so a +100% gain is undone by a −50% fall — which is why a 50% loss needs a 100% gain to recover, not another 50%.

A rate moving from 5% to 6% rose by one point and by 20% at once

There is a third question that sounds like this page’s and is not, and mixing it up is how a headline ends up overstating a change by twenty times. When a rate that is itself a percentage moves — an interest rate, an unemployment rate, a tax rate, a conversion rate — there are two correct ways to describe the move, and they give very different numbers. From 5% to 6% is a rise of one percentage point, which is plain subtraction. It is also a rise of 20% in relative terms, because 1 ÷ 5 = 0.20. Both are right. Only one is what most readers hear.

That distinction is not part of this calculator, and leaving it out is deliberate: our Percentage Change Calculator has a Percentage Points mode built for exactly it, and duplicating the tool here would only split one question across two of our pages. Use this page when you are taking a slice of a percentage or stacking changes; use that one when a single rate has moved and you need both readings of the move. The rule for choosing between them when you write it up: quote percentage points when the base is itself a percentage and your reader could otherwise mistake a relative change for an absolute one, and say “percentage points” in full rather than trusting the % sign to carry the difference.

A 10% commission on a 30% margin is 3% of the sale, not 10%

Most of the time this arithmetic arrives attached to money, and the base amount field is what turns the share back into currency. Enter both percentages and the whole, and the calculator shows the intermediate subtotal as well as the final figure — the subtotal being the number people forget exists. On a $5,000 sale at a 30% gross margin, the margin is $1,500; a 10% commission paid on the margin rather than the sale is $150, which is 3% of the sale. Quote “a 10% commission” without naming its base and you have described a payment three times larger than the one being made.

Marks work the same way and are the other common arrival. When a syllabus says coursework is 40% of the module and the practical is 25% of the coursework, the practical is 10% of the module — 0.25 × 0.40 = 0.10 — and that is the weight to use when you work out what a score is worth overall. Tax and fee rules stack identically: a surcharge levied as a percentage of a tax that is itself a percentage of a price ends up as a small percentage of the price, and the only way to get it right is to multiply along the chain in the order the rule sets out. Where the document is ambiguous about whether two percentages compound or both apply to the original amount, the arithmetic cannot settle it — that is a reading of the document, and it is worth confirming before the figure is used.

In a spreadsheet, the trap is a stray ×100

Both modes are one short formula, and the mistake to avoid in each is the same one — multiplying by 100 in a cell that is already formatted as a percentage, which inflates the answer a hundredfold:

• A percentage of a percentage, with 20% in A1 and 30% in B1 stored as real percentages: =A1*B1 returns 6% once the result cell is formatted as Percentage. If instead the cells hold the plain numbers 20 and 30, use =A1*B1/100 and read the answer as a percentage.
• Successive changes, with the rates in A1:A3 as real percentages: =PRODUCT(1+A1:A3)-1, entered normally in current Excel and Google Sheets, gives the combined change. Older Excel versions need it confirmed with Ctrl+Shift+Enter as an array formula.

There is nothing to download for any of this and no account to create: the calculator above is a free web page that runs entirely in your browser, on desktop or mobile, and the numbers you type are never sent anywhere.

Related calculators

Other percentage tools:

More in Math, or browse all calculators.

Read the guide

The combine mode above has a guide of its own: Why a Percentage Loss and an Equal Percentage Gain Don’t Cancel Out works the same compounding through in prose, with the recovery arithmetic this calculator does not model — what a given loss needs as a gain to get back to level, and why that figure grows so fast.

Sources and methodology

The arithmetic on this page cites no authority because none exists to cite — multiplying two proportions is elementary mathematics, and it is written out in full above rather than asserted. What is cited below is the part that is somebody else’s behaviour rather than mathematics: what a spreadsheet actually does with the formulas in the section above, which is the one place a correct method still returns a wrong number.

• Microsoft Support — PRODUCT function (Excel)
• Microsoft Support — format numbers as percentages in Excel
• Google Docs Editors Help — PRODUCT (Sheets)
• How we calculate — methodology

Method. Every figure here — the calculator, the reference grid, and the stacked-change table — is produced in your browser by one engine, src/lib/percentage-of-percentage.ts, so the tables and the tool cannot drift apart. That engine is checked on every change against 55 hand-written assertions, including the cases most likely to be got wrong: that +10% then −10% is −1% rather than 0%, that two stacked discounts do not add, that a −100% step cannot be recovered, and that the order of the two percentages does not matter. The count and the per-case breakdown are published on the formula verification page.