Working Together (Solution)

The trick here is to understand that completion times do not add, but rates do. For example:

If Alice finishes a job in 2 hours and Bob finishes it in 3 hours, when working together they do not finish it in 5 hours (2+3). But if Alice finishes 1/2 the job per hour and Bob finishes 1/3 of it per hour, when working together they do finish 5/6 of it per hour (1/2 + 1/3)..

So now we can set up equations. Let’s call \(A\) Alice’s per hour rate, \(B\) Bob’s per hour rate, and \(C\) Charlie’s per hour rate.

\[\begin{align*} A + B &= \frac{1}{2} \\ A + C &= \frac{1}{3} \\ B + C &= \frac{1}{4} \\ \end{align*}\]

What is \(A + B + C\)?

Now it seems pretty simple, you add all three equations and divide by two:

\[\begin{align*} 2A + 2B + 2C &= \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \\ A + B + C &= \frac{13}{24} \\ \end{align*}\]

So, when all three work together, their per hour rate (\(A + B + C\)) is \(\frac{13}{24}\), so the time it takes them to complete the job is just 1 over the rate, so \(\frac{24}{13}\) hours.