Part-Time Carpenter

Classic Problem
A carpenter agrees to work, on the condition that:
 * he is paid $\pounds 2$ for every day he works
 * he forfeits $\pounds 3$ for every day he does not work.

At the end of $30$ days he finds he has paid out exactly as much as he has received.

Solution
$18$ days.

Proof
Let $x$ be the number of days he worked.

Then:

Checking our work:

For $18$ days and at $\pounds 2$ per day, he earns $\pounds 2 \times 18 = \pounds 36$.

For $12$ days, forfeiting $\pounds 3$ per day, he forfeits $\pounds 3 \times 12 = \pounds 36$.