Unrewarded Labor

A man persuaded Weary Willie, with some difficulty, to try to work on a job for thirty dollars at eight dollars a day, on the condition that he would forfeit ten dollars a day for every day that he idled. [1967 US dollars. Ed.] At the end of the month neither owed the other anything, which entirely convinced Willie of the folly of labor. Can you tell just how many days' work he put in and on how many days he idled?

(Source: 536 Curious Problems & Puzzles by Henry Ernest Dudeney, edited by Martin Gardner)

There were two unstated assumptions: it's a 30-day month and fractions of a day's labor or slacking are possible. Anyway this one is pretty easy: solve for $x$ where $x$ is the amount of days he worked during the month:

\begin{align*}8x - 10(30 - x) &= 0 \\
8x - 300 + 10x &= 0 \\
18x &= 300
\end{align*}

Final answer: Willie worked 16⅔ days and idled 13⅓ days during the month.