List price with a discount

Statement

A shop buys a lamp for $12$ euros. It sets a list price $P$ so that, after a $20\%$ discount, it still makes a $25\%$ profit on the cost. What is $P$?

Solution

Discounted price. With a $20\%$ discount the customer pays $80\%$ of the list price, i.e. $0.8\,P$.

Profit condition. A $25\%$ profit on the cost means the discounted price must equal $1.25 \cdot 12 = 15$ euros: $0.8\,P = 15 \implies P = \frac{15}{0.8} = 18.75.$ Check: $80\%$ of $18.75$ is $15$ euros, i.e. the cost $12$ plus $25\%$ ($3$ euros). Correct. $\boxed{P = 18.75 \text{ euro}}$