Two pipes, when working one at a time can fill a cistern in 2 hours and 3 hours, respectively while a third pipe can drain the cistern empty in 6 hours. All the three pipes were opened together when the cistern was $$\frac{1}{6}$$ full. How long will it take for the cistern to be completely full?

Cistern filled by first pipe in =2 hour

Cistern filled by second pipe in =3 hour

third pipe can drain the cistern empty in 6 hours

Part of the cistern filled in $$1 hour by first pipe = \frac{1}{2}

Part of the cistern filled in $$1 hour by second pipe = \frac{1}{3}

Part of the cistern empty in $$1 hour by third pipe = \frac{1}{6}

So

Part of the cistern filled in $$1 hour = \frac{1}{2}+\frac{1}{3}-\frac{1}{6}$$

$$1 hour = \frac{3+2-1}{6}$$

$$1 hour = \frac{4}{6}$$

$$1 hour = \frac{2}{3}$$

istern was $$\frac{1}{6}$$ full 

pending  $$\frac{5}{6}$$

$$\frac{2}{3}$$ part filled in 1 hour

$$\frac{5}{6}$$ part filled =$$\frac{1}{\frac{2}{3}}\times\frac{5}{6}$$

$$\frac{5}{4}=1 hour 15 minutes$$