Percentage Calculator

The four questions people actually ask

What is X percent of Y, which is simple multiplication. X is what percent of Y, which is division. What is the percentage change from X to Y, which is the difference divided by the original. And the reverse: if the result after a percentage change is X, what was the original? Each is a different operation and mixing them up is where most errors come from.

Percentage change is always relative to the original

Going from 40 to 50 is a 25 percent increase, because the change of 10 is measured against the starting value of 40. Going from 50 back to 40 is a 20 percent decrease, because the same change of 10 is now measured against 50. The two are not symmetric, which surprises people constantly. A 50 percent fall requires a 100 percent rise to recover.

Successive percentages do not add

A 20 percent increase followed by a 20 percent decrease does not return you to the start. Multiplying by 1.2 and then by 0.8 gives 0.96, a 4 percent net loss. Compounding always applies: to combine percentage changes you multiply the factors, never add the percentages. This is the arithmetic behind why a portfolio down 50 percent needs to double to break even.

Percentage points versus percent

These are routinely confused in reporting, and the difference is large. If an interest rate rises from 2 percent to 3 percent, that is a rise of one percentage point but a 50 percent increase in the rate. Both statements are true and they describe very different magnitudes, which is why the distinction is worth being precise about.

Reverse percentages and VAT

To remove a percentage that has already been added, divide rather than subtract. A price of 120 including 20 percent VAT has a net value of 120 divided by 1.2, which is 100, not 120 minus 20 percent, which would give 96. Subtracting the percentage applies it to the wrong base. The same logic applies to reversing any markup or discount.