Question

Suppose three friends Rama, Rahim and John can paint a fence in $2$ hours. If Rama does it alone, he takes $5$ hours while Rahim alone takes $6$ hours. How long John takes to finish the job alone?

Answer 1

Here there are three people involved. The reasoning we apply: if Rama, Rahim and John each alone can to the job in $t_1,t_2$ and $t_3$ units of time respectively, then the time $t$ taken to finish the job by three of them working together is given by
$\frac{1}{t}=\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}$.
We know that $t=2,t_1=5$ and $t_2=6$. Hence,
$\frac{1}{t_3}=\frac{1}{t}-\frac{1}{t_1}-\frac{1}{t_2}=\frac{1}{2}-\frac{1}{5}-\frac{1}{6}=\frac{15-6-5}{30}=\frac{4}{30}=\frac{2}{15}$.
Hence, $t_3=\frac{15}{2}=7\frac{1}{2}$. Thus John alone takes $7\frac{1}{2}$ hours to finish the job.
These ideas work in problems involving pipes and tank. When a tank is filled, we have to take positive sign and when being emptied one has to take negative sign.