Percentage Calculator

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A rise of 50% followed by a fall of 50% does not return you to the start — it leaves you 25% down. Percentages are multiplicative, not additive.

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Percentage change is not percentage points

If an interest rate moves from 4% to 5%, that is a rise of one percentage point but a 25% increase. Both statements are true and they describe the same event, which is exactly why the distinction gets abused in headlines and sales decks. When someone reports a large percentage change in something that is itself a percentage, check which one they mean.

Why increases and decreases do not cancel

A stock that falls 50% needs to rise 100% to get back to where it started. The reason is that each percentage applies to a different base: the fall is measured against the original price, the recovery against the reduced one. This is the single most common percentage error in personal finance, and it works against you in both directions — a 20% discount followed by a 20% price rise leaves the item 4% cheaper than it began.

Reverse percentages and the VAT problem

If a price including 20% tax is 120, the pre-tax price is not 120 minus 20% — that gives 96, which is wrong. It is 120 divided by 1.2, which gives 100. Subtracting the percentage uses the wrong base. The reverse mode above does the division correctly, which matters whenever you need to extract a tax-exclusive price, an original price before a discount, or a pre-commission amount.

Percentage difference versus percentage change

Percentage change has a direction: from A to B. Percentage difference is symmetric and divides by the average of the two values, so comparing A to B gives the same answer as comparing B to A. Scientific and engineering work usually wants the symmetric version; business reporting almost always wants directional change.

Frequently Asked Questions

What is the difference between a percentage point and a percent?

A move from 4% to 5% is one percentage point, but a 25% increase. Both are correct descriptions of the same change, which is why the wording matters whenever the quantity being measured is itself a percentage.

Why does a 50% loss need a 100% gain to recover?

Each percentage applies to a different base. The 50% loss is measured against the original value; the recovery is measured against the reduced value, which is only half as big, so it needs to double.

How do I find the original price before tax or a discount?

Divide rather than subtract. A price of 120 including 20% tax has a pre-tax value of 120 ÷ 1.2 = 100. Subtracting 20% from 120 gives 96, which is the classic mistake.

What is percentage difference?

It divides the gap between two values by their average, so it is symmetric — comparing A to B gives the same figure as B to A. Percentage change, by contrast, has a direction and uses the starting value as its base.