# Puzzles

## Dirty work

Timothy, Urban, and Vincent are digging identical holes in a field.

When Timothy and Urban work together, they dig 1 hole in 4 days.

When Timothy and Vincent work together, they dig 1 hole in 3 days.

When Urban and Vincent work together, they dig 1 hole in 2 days.

Working alone, how long does it take Timothy to dig one hole?

#### Show answer & extension

#### Hide answer & extension

Let \(t\), \(u\) and \(v\) be the prortion of a hole dug in one day by Timothy, Urban and Vincent respectively. Then the following equations hold:

$$4t+4u=1$$$$3t+3v=1$$$$2u+2v=1$$

These can be solved to find that \(t=\frac{1}{24}\) so it will take Timothy 24 days to dig a hole.

#### Extension

Working alone, how long does it take Urban to dig one hole?

Working alone, how long does it take Vincent to dig one hole?