11. Andrew, Bob, and Chris are working together to finish a group project. If Andrew doesn't help, it would take 2 hours. If Bob doesn't help, it would take 3 hours. If Chris doesn't help, it would take 4 hours. Now they just learned that Dave can also help them out. If Dave works on the project alone, it would take him half a day. What is the least amount of time it would take for them to finish the group project?
Answer: $\frac{8}{5}$ hours or 96 minutes
Solution: Let $A, B, C$, and $D$ be the number of hours Andrew, Bob, Chris, and Dave can finish the project when working alone. Then we have
$$
\begin{aligned}
\frac{1}{B}+\frac{1}{C} & =\frac{1}{2} \\
\frac{1}{A}+\frac{1}{C} & =\frac{1}{3} \\
\frac{1}{A}+\frac{1}{B} & =\frac{1}{4} \\
\frac{1}{D} & =\frac{1}{12}
\end{aligned}
$$

The least amount of time it would take to finish the group project would be achieved when all four work together on the project. This would be $x$ hours, where
$$
\begin{aligned}
\frac{1}{x} & =\frac{1}{A}+\frac{1}{B}+\frac{1}{C}+\frac{1}{D} \\
& =\frac{1}{2}\left[\left(\frac{1}{B}+\frac{1}{C}\right)+\left(\frac{1}{A}+\frac{1}{C}\right)+\left(\frac{1}{A}+\frac{1}{B}\right)\right]+\frac{1}{D} \\
& =\frac{15}{24}
\end{aligned}
$$

This means that $\frac{24}{15}=\frac{8}{5}$ hours or 96 minutes would be the least amount of time needed for the project.