Pipes P and Q can fill a tank in 18 hours and 27 hours respectively. Pipe R is an emptying pipe. When the tank is empty, P and Q are opened together for 6 hours and then both are closed and R is opened. The tank is now emptied in 30 hours. In how many hours can P and R together fill half the tank?

Correct option is C

​Given :

Pipe P fills the tank in 18 hours

rate =$$\frac{1}{18}$$​ tank/hr

Pipe Q fills the tank in 27 hours 

rate =$$\frac{1}{27}$$​ tank/hr

Pipe R empties the tank (rate to be found)

Formula Used :
​$$\text{Work} = \text{Rate} \times \text{Time}$$​​
Solution :

Work done by P and Q in 6 hours

​$$\text{Rate of P + Q} = \frac{1}{18} + \frac{1}{27}$$​​

​$$= \frac{3 + 2}{54} = \frac{5}{54}$$​​

Work done in 6 hours:
6$$\times \frac{5}{54} = \frac{30}{54} = \frac{5}{9}$$​​

Emptying by pipe R

Pipe R empties $$\frac{5}{9}$$​ tank in 30 hours.

​$$\text{Rate of R} = \frac{\frac59}{30} = \frac{5}{270} = \frac{1}{54}$$​​

So, pipe R empties the tank in 54 hours.

$$\text{Rate of P and R} = \frac{1}{18} - \frac{1}{54} \\ \ \\= \frac{3 - 1}{54} = \frac{2}{54} = \frac{1}{27} \\ \ \\\text{Time} = \frac{1/2}{1/27} = \frac{27}{2} \\ \ \\= 13\frac12 \text{ hours}$$​