Increase and discount of the same value

Statement

A pair of shoes costs $80\ \text{€}$. In January the price rises by $25\%$; in March it is discounted by $25\%$. What is the final price? Does it return to $80\ \text{€}$?

Solution

The two multiplicative factors are applied in sequence: $80\cdot 1{.}25 = 100, \qquad 100\cdot 0{.}75 = \boxed{75\ \text{€}}.$ The price does not return to $80\ \text{€}$: the overall factor is $1{.}25\cdot 0{.}75 = 0{.}9375 < 1$, i.e. a $6{.}25\%$ reduction.