# Ask Uncle Colin: Multiple workers

*Ask Uncle Colin* is a chance to ask your burning, possibly embarrassing, maths questions – and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I have a problem I just can’t get to the bottom of:

Georg, Bertrand and Kurt are building a wall. Working alone, Georg would take one day to complete the wall; Bertrand would take two days; and Kurt would take three. How long would it take them, working together?

Am I stupid?

Weary, I Learn Especially Slowly?

Hello, WILES – no, you’re not stupid, it’s trickier than it looks! It’s tempting to add the three numbers together and say the wall would take six days, but that doesn’t make any sense – the three working together ought to be quicker than any one of them working alone!

The way to do it is to think about how many walls each builder would build in a given time period – you might pick a day, if you like fractions, or (e.g.) six days if you don’t.

- Georg would build one wall in one day, or six walls in six days
- Bertrand would build half a wall in one day, or three walls in six days
- Kurt would build a third of a wall in one day, or two walls in six days

Altogether, in one day, the team would build $1 + \frac 12 + \frac 13 = \frac{11}{6}$ of a wall in one day, which works out to $\frac{6}{11}$ of a day to build a wall. Alternatively, they’d build 11 walls in six days – which is one wall every $\frac{6}{11}$ of a day.

In general, the time to build a wall works out to be $\frac{1}{\frac{1}{G} + \frac{1}{B} + \frac{1}{K}}$, or $\frac{GBK}{BK + KG + GB}$.

-- Uncle Colin