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The three questions percentages answer

What is X% of Y: multiply Y by X divided by 100. Fifteen per cent of 240 is 240 times 0.15, which is 36.

X is what percentage of Y: divide X by Y and multiply by 100. Thirty-six out of 240 is 15%.

Percentage change from X to Y: subtract, divide by the original, multiply by 100. From 240 to 276 is 36 divided by 240, which is a 15% increase. Nearly every percentage problem is one of these three wearing different clothes.

Percentage points against percentages

This distinction causes real confusion in reporting and it is worth being precise about. A rate moving from 4% to 6% has risen by two percentage points, and by 50%.

Both statements are true and they sound wildly different, which is why the choice between them is frequently made for effect rather than for clarity.

The convention is that percentage points describe the arithmetic difference between two percentages, and percentage change describes the relative change. Anyone reading a claim about a percentage rising by a percentage should check which is meant.

Why increases and decreases do not cancel

A 20% increase followed by a 20% decrease does not return to the starting value. It returns to 96% of it, because the decrease applies to the larger amount.

The same asymmetry means a 50% loss requires a 100% gain to recover. An investment falling from £1,000 to £500 needs to double to get back.

This is the arithmetic behind why volatility is costly, and it applies to prices, to portfolios and to any sequence of proportional changes. The order does not matter; the direction of the asymmetry always does.

Compounding percentages

Successive percentage changes multiply rather than add. Three consecutive 10% increases give a total increase of 33.1%, not 30%, because each applies to the result of the last.

The general form is to multiply the factors: 1.1 times 1.1 times 1.1 is 1.331.

Mixed changes work the same way. A 20% increase followed by a 15% decrease is 1.2 times 0.85, which is 1.02, a net 2% increase. Adding the percentages would have given 5%, which is wrong.

Reversing a percentage

Recovering an original amount from a result requires division. A price of £120 after a 20% increase was £100, found by dividing by 1.2.

Subtracting 20% from £120 gives £96, which is wrong, and the error grows with the size of the change.

The same applies to tax. Extracting the net amount from a VAT-inclusive price of £120 at 20% means dividing by 1.2 to get £100, not taking 20% off. That specific error appears in a great many spreadsheets.

Percentages of percentages are the case that catches most people, and it is worth working one through. If a survey finds that 40% of respondents own a car, and 25% of those car owners drive to work, then the proportion of all respondents who drive to work is 25% of 40%, which is 10%, not 25% and not 65%. The multiplication is straightforward and the mistake is in reading the second percentage as applying to the whole rather than to the subgroup. Any time a percentage is quoted about a group that was itself defined by a percentage, the two multiply.

The same care applies to percentages above 100, which are perfectly valid and are frequently misread. An increase of 150% means the result is two and a half times the original, not one and a half times, because the increase is added to the whole. Something that has risen by 100% has doubled. A figure described as 300% of the original is three times it, where a figure described as a 300% increase is four times it. Those two phrasings differ by an entire multiple and they appear interchangeably in ordinary reporting, which is worth pausing over whenever the number matters.