Bill and Jill are hired to paint a line on a road. If Bill works by himself, he could paint the line in $B$ hours. If Jill works by herself, she could paint the line in $J$ hours. Bill starts painting the line from one end, and Jill begins painting the line from the other end one hour later. They both work until the line is painted. Which is the following is an expression for the number of hours that Bill works?
(A) $\frac{B(J+1)}{B+J}$
(B) $J+1$
(C) $\frac{B J}{B+J}+1$
(D) $\frac{B+J-1}{2}$
(E) $\frac{B(J-1)}{B+J}$
Solution 1

In one hour, Bill paints $\frac{1}{B}$ of the line and Jill paints $\frac{1}{J}$ of the line.

Let $t$ be number of hours during which both Bill and Jill paint the line. Since Bill paints for one hour before Jill starts to paint and since the painting of the line is completely finished after the $t$ hours that they both work, then $\frac{1}{B}+t\left(\frac{1}{B}+\frac{1}{J}\right)=1$ or

$$
t=\frac{1-\frac{1}{B}}{\frac{1}{B}+\frac{1}{J}}=\frac{\left[\frac{B-1}{B}\right]}{\left[\frac{B+J}{B J}\right]}=\frac{J(B-1)}{B+J}
$$

So Bill works for $t+1=\frac{J(B-1)}{B+J}+1=\frac{B J-J}{B+J}+\frac{B+J}{B+J}=\frac{B J+B}{B+J}=\frac{B(J+1)}{B+J}$ hours.

## Solution 2

Suppose that Bill could paint the line in 1 hour only, ie. $B=1$.

Then when Jill joins the painting after 1 hour, Bill would actually be finished the painting. In other words, if $B=1$, then the total time that Bill spends painting the line, regardless of the value of $J$, is 1 hour. If we substitute $B=1$ into the five choices, we obtain
(A) $\frac{J+1}{J+1}=1$
(B) $J+1$
(D) $\frac{J}{2}$
(E) $\frac{J-1}{J+1}$
(C) $\frac{J}{J+1}+1$

The only choice which is equal to 1 , regardless of the value of $J$, is the first one.

Answer: (A)