Part-Time Carpenter

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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:

\(\ds 2 x\) \(=\) \(\ds 3 \paren {30 - x}\)
\(\ds \leadsto \ \ \) \(\ds 5 x\) \(=\) \(\ds 90\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds 18\)


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 $\pounds 3 \times 12 = \pounds 36$.

$\blacksquare$


Sources