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

Comments (4):

  1. Thank you

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

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

    joel on 3 Jan 2009 (permalink)
  4. God bless you Joel.

    don_eze on 31 Aug 2011 (permalink)

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