# General formula for combined work rate

Last updated: 3 Jan 2009

The formula for solving combined work rate problems is

\[
\frac{1}{\text{Time taken by A}} + \frac{1}{\text{Time taken by B}} = \frac{1}{\text{Time taken by A and B together}}
\]

This general formula can be extended if more than 2 people (or machines are working together

\[
\frac{1}{T_{A}} + \frac{1}{T_{B}} + \frac{1}{T_{C}} + ... = \frac{1}{T_{Together}}
\]

Where \(T_{A}\), \(T_{B}\) and \(T_{C}\) are the times taken by A, B and C respectively to complete the task alone and \(T_{Together}\) is the time taken by them to complete the task when they are all working together.

### Example

We can use this formula to solve the same problem we faced on the previous page

If Alex can build a house in 2 days and his apprentice Bob can build a house in 3 days, then how long will it take Alex and Bob to build a house when they are working together?

Putting the information from the question into the formula gives us

\[
\begin{split}
\frac{1}{2 \text{ days}} + \frac{1}{3 \text{ days}} &= \frac{1}{\text{Time working together}} \\
\frac{5}{6} &= \frac{1}{\text{Time working together}}
\end{split}
\]

Invert both sides of the equation

\[
\text{Time working together} = \frac{6}{5} = 1 \frac{1}{5} \text{ days}
\]

So Alex and Bob will take \(1 \frac{1}{5}\) days to build a house when they are working together.

Thank you

You have an error in the first formula - the second addition should be time taken B.

Thanks for the correction. Fixed.

God bless you Joel.