Why doesn't a 20% rise cancel a 20% fall?

Because the two percentages are taken from different bases. Rising 20% from 100 gives 120; falling 20% from 120 removes 24, not 20, leaving 96. The shortfall is 4%, which is 20% of 20%.

What is the difference between percent and percentage points?

Percentage points measure the arithmetic gap between two percentages; percent measures the relative change. Going from 4% to 5% is a rise of one percentage point and a rise of 25% at the same time.

How much do I need to gain to recover a loss?

More than you lost, always. After a 20% fall you need 25%; after 30% you need 42.9%; after 50% you need 100%. The formula is 1 ÷ (1 − loss) − 1.

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Percentage calculator

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Up 20% then down 20% leaves you 4% short

Percentages are the arithmetic people are most confident about and most often wrong about, because they quietly change what they are a percentage of. A 20% rise followed by a 20% fall does not return you to where you started — it leaves you 4% down, and the second 20% is simply a percentage of a bigger number than the first.

How it works

Works out what X% of a number is, what percentage one number is of another, and the percentage change between two values.
Keeps the base explicit, which is where almost every percentage mistake actually starts.
Handles successive changes correctly, by multiplying rather than adding them.

X% of Y = Y × X ÷ 100

A as a percentage of B = A ÷ B × 100

percentage change = (new − old) ÷ old × 100

successive changes multiply: (1 + a) × (1 + b), they never add

Worked example

The same percentage applied up and then down, and the gain needed to undo a loss.

100 up 20% → 120, then down 20% → 96
the shortfall is 4%, which is exactly 20% squared
up 50% then down 50% → 75, a 25% shortfall
to recover from −20% you need +25%, not +20%

Equal percentages up and down never cancel, and the gap grows with the square of the percentage: 10% leaves you 1% short, 30% leaves you 9% short, 50% leaves you 25% short.

Reading the result

Recovering a loss always takes a larger percentage than the loss itself, because the gain is measured against the reduced amount. Down 50% needs +100% to get back; down 80% needs +400%. This is why a large drawdown is far worse than it looks.
The order makes no difference. Down 20% then up 20% also gives 96, because multiplication commutes — which means the sequence of a rise and a fall never explains the shortfall, only the shifting base does.
Percent and percentage points are different quantities and are constantly confused. A rate moving from 4% to 5% has risen by one percentage point but by 25% in relative terms, and both numbers are correct descriptions of the same change.
X% of Y always equals Y% of X, which is occasionally a shortcut worth having. 18% of 50 is awkward; 50% of 18 is 9, and they are the same number.
Stacked discounts do not add up. 30% off then a further 20% off is 44% off in total, not 50%, because the second discount applies to the already reduced price.

Common questions

Why doesn't a 20% rise cancel a 20% fall?: Because the two percentages are taken from different bases. Rising 20% from 100 gives 120; falling 20% from 120 removes 24, not 20, leaving 96. The shortfall is 4%, which is 20% of 20%.
What is the difference between percent and percentage points?: Percentage points measure the arithmetic gap between two percentages; percent measures the relative change. Going from 4% to 5% is a rise of one percentage point and a rise of 25% at the same time.
How much do I need to gain to recover a loss?: More than you lost, always. After a 20% fall you need 25%; after 30% you need 42.9%; after 50% you need 100%. The formula is 1 ÷ (1 − loss) − 1.