Percentage change is the most quoted calculation in everyday life and the most misquoted with it, because its formula is simple and its behaviour is not. A rise and the matching fall do not have the same percentage. A rate moving from 20 to 25 percent has risen 5 percentage points and 25 percent, and both figures are correct about different things. And two successive percent changes cannot be averaged into an overall change. This guide works the formula, the asymmetry, the percentage-point distinction, and successive changes, each on numbers that can be checked by hand.

The formula

Percentage change equals the new value minus the old value, divided by the old value, times 100. From 80 to 100: 20 divided by 80 is 0.25, times 100 is positive 25 percent, an increase. From 100 to 80: negative 20 divided by 100 is negative 0.20, times 100 is negative 20 percent, a decrease. The old value is always the denominator, and the sign of the result is the direction.

The denominator is the whole discipline. Dividing by the new value instead gives a different figure that is sometimes called percent difference in disguise, and dividing by the average gives percent difference outright. If the question is how much something changed from where it was, the old value is the only honest denominator.

Worked examples: prices, salaries, and populations

A price rises from $40 to $50: the change is $10, divided by $40, which is 25 percent, an increase. A salary rises from $50,000 to $57,500: the change is $7,500, divided by $50,000, which is 15 percent. A town’s population falls from 20,000 to 18,500: the change is negative 1,500, divided by 20,000, which is negative 7.5 percent, a decrease. Same formula, same three steps, whatever the subject.

The old value is the denominator in every row. Change the denominator and the answer changes with it.

The asymmetry: up and down are not mirror images

A rise from 100 to 150 is 50 percent. The fall back from 150 to 100 is 33.33 percent, because the divisor changed from 100 to 150. The value returned to where it started, but the percentages do not cancel, and anyone who adds them — 50 minus 33.33 is not zero — has discovered the asymmetry the hard way. The general rule: a fall of x percent needs a rise of x divided by (100 minus x), times 100, to recover. A 20 percent fall needs a 25 percent rise. A 50 percent fall needs a 100 percent rise.

The asymmetry is the reason investment losses hurt more than equal gains help. A portfolio that falls 50 percent must double to recover, and a portfolio that falls 20 percent must rise 25 percent. The percentages look symmetric; the money is not.

Successive changes multiply, they do not add

Two changes of 10 percent and then negative 10 percent do not cancel: 100 becomes 110, then 99. The second change applies to the new value, not the original, so the base moved between the two calculations. The correct combination multiplies the factors: 1.10 times 0.90 is 0.99, a net change of negative 1 percent. Three changes combine the same way: multiply all the factors, subtract one, times 100.

Adding percentages is the error behind most wrong compound claims, from sale prices to salary histories. A price cut 20 percent and then cut a further 20 percent is not cut 40 percent: it is cut 36 percent, because 0.80 times 0.80 is 0.64. Whenever changes follow one another, multiply; addition is only valid when the changes share a base, which successive changes never do.

Percentage points: the distinction that stops misreporting

A rate moving from 20 percent to 25 percent rises by 5 percentage points and by 25 percent. Percentage points are the arithmetic difference between two percentages. Percent is that difference relative to the starting percentage. News reports that say a rate rose 25 percent when it moved from 20 to 25 percent are using percent where percentage points belong, and the honest figure is the 5-point move.

The rule for writing is to name the unit every time a percentage of a percentage is in play. Interest rates, tax rates, unemployment rates, and election shares move in percentage points. Prices, salaries, revenues, and populations move in percent. A sentence that mixes them without naming them will mislead someone, usually in the direction of the bigger number.

Mistakes: the wrong base, the averaged change, the missing sign

First, the wrong base: dividing by the new value, or by the average, and calling the result percent change. Second, averaging successive percent changes instead of multiplying their factors. Third, dropping the sign: a change of 20 percent without a sign is neither a rise nor a fall, and a report that omits it has omitted the answer. Fourth, quoting a percent change above 100 percent as if it were impossible: a value that triples has risen 200 percent, because the change is twice the starting value, and increases have no 100 percent ceiling. Only decreases are capped, at negative 100 percent, where the value reaches zero.

Frequently asked questions

What is the formula for percentage change?

(New – old) / old x 100. From 80 to 100 that is +25%; from 100 to 80 it is -20%. The old value is always the denominator.

Why does a 50% fall need a 100% rise to recover?

Because the base changes. Falling from 100 to 50 divides by 100, but recovering from 50 to 100 divides by 50, so the recovery percentage is twice the fall.

Do successive percentage changes add up?

No, they multiply. A 10% rise followed by a 10% fall gives 100 x 1.10 x 0.90 = 99, a net change of -1%, not 0%.

What is the difference between percent and percentage points?

Percentage points are the arithmetic difference between two percentages: 20% to 25% is 5 points. Percent is that difference relative to the starting value: the same move is a 25% increase.

Can percentage increase exceed 100%?

Yes. A value that triples has risen 200%, because the change is twice the starting value. Only decreases are capped, at -100%, where the value reaches zero.

What is the most common percentage change mistake?

Dividing by the wrong base: the new value or the average instead of the old value. Dividing by the average gives percent difference, a different calculation answering a different question.