# Combined work rate problems

Last updated: 22 Nov 2008

Combined work rate problems are a bit more complex than standard rate problems.

These problems involve have two people (or machines) working at different rates.

### Example

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?

The first thing to remember is that GMAT combined work rate problems are all idealized, i.e. if people are working together they are unaffected by one another so they don't do less work because they are arguing or taking extra tea breaks or more because they inspire one another.

With this in mind it should be obvious that it will take them less than 2 days working together since Alex can do it on his own in 2 days, but how much less?

### Solve the example combined work rate problem

There is a general formula for solving combined work rate problems but it will be useful for you to understand where it comes from and to do this we will work through this example from first principles.

To work out how long they will take to build a house when they are working together we need to start by finding out that the rate at which each person works. Then we can add these rates together to find out how fast they work together. Finally we can work out how long it will take them to do the job.

We know that

$\text{Work rate} = \frac{1}{\text{Time taken}}$

So looking at our example, we can see that

$\text{Alex's work rate} = \frac{1}{\text{time taken by Alex}} = \frac{1}{2}$

i.e. Alex can build $$\frac{1}{2}$$ a house each day.

And

$\text{Bob's work rate} = \frac{1}{\text{time taken by Bob}} = \frac{1}{3}$

i.e. Bob can build $$\frac{1}{3}$$ of a house each day.

To find their work rate when they are working together we simply add the rates at which they work individually, so

$\begin{split} \text{Work rate (together)} &= \text{Alex's work rate} + \text{Bob's work rate} \\ &= \frac{1}{2} + \frac{1}{3} \\ &= \frac{3}{6} + \frac{2}{6} \\ &= 5/6 \end{split}$

Therefore, together, they can build $$\frac{5}{6}$$ of a house each day.

We also know that

$\text{Work rate (together)} = \frac{1}{\text{Time taken together}}}$

Taking the inverse of both sides of the equation

$\begin{split} \text{Time taken together} &= \frac{1}{\text{Work rate (together)}} \\ &= \frac{1}{\frac{5}{6}} \\ &= \frac{6}{5} \\ &= 1 \frac{1}{5} \end{split}$

We can see that it will take Alex and Bob $$1 \frac{1}{5}$$ days to build a house working together.

Next page: General formula for combined work rate