Jacob agrees to work after school for Mr. Daniels, his school’s head custodian, who says he'll pay Jacob $240 and the front bumper of a 1958 Ford Thunderbird for one year of work, including the summer. Jacob quits after seven months. Mr. Daniels does some calculation and pays Jacob $100 plus the bumper. What did Mr. Daniels decide was the value of the bumper?

## Solution

Let $b$ be the value of the bumper. We are given that the value of a whole years work is $240 + b$. Therefore,

$\text{value of one month's work } = \frac{1}{12}(240 + b) = 20+\frac{b}{12}.$So, the value of seven month's work is

$7(20 + \frac{b}{12}) = 140 + \frac{7b}{12} = 100 + b,$where the last part of this equality is what Mr. Daniels actually paid Jacob. Solving for $b$ we get,

$\begin{aligned} b &=& 140 + \frac{7b}{12} - 100 \\ \frac{5b}{12} &=& 40, \\ b &=& \$96. \end{aligned}$