# 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.

Next page: Summary of rate problems

1. Thank you

teejayash on 27 Oct 2007
2. You have an error in the first formula - the second addition should be time taken B.

lbacon on 30 Dec 2008
3. Thanks for the correction. Fixed.

joel on 3 Jan 2009
4. God bless you Joel.

don_eze on 31 Aug 2011