Percentage Calculator

How to use this calculator

1. Choose the mode that matches your questionThe five modes cover genuinely different calculations. Using the wrong one is the main way people get a plausible but wrong answer.
2. For percentage change, A is the originalOrder matters here and nowhere else. A is where you started, B is where you ended up. Reversing them gives a different number, not the same one with a sign flipped.
3. Use reverse mode for VAT and discountsIf you have the price after a discount or including tax and want the figure before, this is the mode. Subtracting the percentage from the final figure gives the wrong answer.
4. Read the workingEach mode shows the substitution and a note on the error people usually make with it.

Why a rise and the matching fall are different numbers

Percentage change is always measured against the starting value, and that single fact explains most of the confusion percentages cause.

Go from 100 to 150 and that is a 50% rise: the increase of 50 is measured against the original 100. Now go back from 150 to 100. The fall of 50 is measured against 150, so it is a 33.3% decrease.

The same absolute movement, two different percentages, because the base changed. A 50% rise is undone by a 33.3% fall — never by another 50%.

This is why a share price that falls 50% needs to double, a 100% rise, just to get back to where it started. And why a shop offering 20% off, then adding 20% back on, does not return to the original price: 100 becomes 80, and 20% of 80 is 16, so you end at 96.

The same trap appears in reverse percentages. If a jacket costs £48 in a 20% off sale, the original price is not £48 plus 20%. The £48 is 80% of the original, so the original is £48 ÷ 0.8 = £60. Adding 20% to £48 gives £57.60, which is wrong by £2.40 — because the 20% was taken off the larger original, not the smaller sale price.

The rule that resolves all of these: to undo a percentage change, divide by the multiplier rather than subtracting the percentage.

Worked example: the five modes on the same numbers

Take A = 15 and B = 80 and run each mode.

<b>What is 15% of 80?</b> Convert to a decimal and multiply: 0.15 × 80 = 12. Worth noting that percentages are commutative — 15% of 80 and 80% of 15 both give 12, and the second is easier to do in your head.

<b>15 is what percentage of 80?</b> Divide the part by the whole: 15 ÷ 80 = 0.1875, so 18.75%.

<b>Percentage change from 15 to 80?</b> The difference is 65. Divide by the original: 65 ÷ 15 = 4.333, so a 433.3% increase. Going the other way, from 80 to 15, is an 81.25% decrease — the same movement, a very different percentage.

<b>Increase 15 by 80%?</b> The change is 15 × 0.8 = 12, so the result is 27. Decreasing 15 by 80% gives 3. And note that increasing by 80% then decreasing by 80% gives 5.4, not back to 15.

<b>15 is the figure after an 80% change — what was it before?</b> The original was multiplied by 1.8 to reach 15, so divide: 15 ÷ 1.8 = 8.33. The common error is taking 80% off 15, which gives 3 — wrong by a factor of nearly three.

Percentages that come up constantly

The removing-VAT row is the one that catches people out most often in business. To take 20% VAT off a gross figure you divide by 1.2, which is equivalent to multiplying by 0.8333 — not by subtracting 20%. Subtracting 20% from a gross £60 gives £48 rather than £50, understating the net by £2 on every invoice.

Percentage points, compounding and other traps

A percentage of a percentage is where reporting most often misleads. If an interest rate moves from 4% to 5%, that is a rise of one percentage point — but it is also a 25% increase in the rate. Both statements are true, and they sound wildly different. Headlines tend to choose whichever is more dramatic.

The rule is that percentage points measure the arithmetic gap between two percentages, while percent measures the proportional change. A poll moving from 40% to 44% has risen four percentage points, or 10%.

Compounding is the second trap. Three successive 10% rises do not make 30%. Each applies to the result of the last, so 100 becomes 110, then 121, then 133.1 — a 33.1% rise overall. Over longer periods the gap widens considerably, which is the whole basis of compound interest.

Averaging percentages is usually wrong. If one shop converts 10% of 1,000 visitors and another converts 30% of 100, the average conversion rate is not 20%. It is 130 conversions out of 1,100 visitors, which is 11.8%. Percentages can only be averaged directly when the bases are equal, and they rarely are.

Percentage of a negative number behaves oddly and is worth avoiding. Percentage change from a negative starting value — a loss becoming a smaller loss — produces figures that are technically correct and practically meaningless. Report the absolute movement instead.

And very large percentage changes usually signal a small base. A 400% increase sounds enormous until you learn it was from 1 to 5. Whenever a percentage change looks dramatic, the useful question is what the starting number was.

Assumptions and limitations

• Percentage change from zero is undefined — there is no base to measure against
• Reverse mode cannot work back from a −100% change, which leaves nothing
• Does not compound across periods — three 10% rises are not one 30% rise
• Does not distinguish percentage points from percent, which is a wording question rather than an arithmetic one
• Averaging percentages with different bases is invalid and this does not warn you
• Percentage change from a negative starting value is arithmetically valid but usually meaningless
• Results are rounded for display; the underlying calculation is at full precision

Questions people ask about this in United Kingdom