Increase and decrease do not cancel out

A very widespread mistake is to believe that an increase of $p\%$ followed by a decrease of $p\%$ brings you back to the starting value. This is not so, and the reason is purely algebraic: the two changes are applied to different bases.

Caution

An increase of $p\%$ followed by a decrease of $p\%$ (or vice versa) does not bring you back to the initial value! $\frac{100+p}{100}\cdot\frac{100-p}{100} = \frac{10000-p^2}{10000} < 1 \quad\text{for every }p>0.$

The product of the two factors equals $\frac{10000-p^2}{10000}$, which is strictly less than $1$ for every $p>0$: the final value is therefore always lower than the initial one. It is the same asymmetry that we shall find again when comparing VAT added and VAT extracted.

Links

Topics: Percentages
Concepts: Percentage increase · Percentage decrease · Percentage change
Skills: Calculating · Proving