Two pipes P and Q can fill an empty tank in 20 min and 10 min respectively. R can empty a full tank in 15 minutes. If all three pipes are opened together, how much time (in minutes) will they take to fill the tank?

Let's assume the capacity of the tank is 60 units.(It is the LCM of 20, 10 and 15.)

Two pipes P and Q can fill an empty tank in 20 min and 10 min respectively.

The efficiency of pipe P = $$\frac{60}{20}$$ = 3 units/min

The efficiency of pipe Q = $$\frac{60}{10}$$ = 6 units/min

R can empty a full tank in 15 minutes.

The efficiency of pipe R = -$$\frac{60}{15}$$ = - 4 units/min [Here negative sign represents the pipe is used for emptying the tank.]

Time taken by all the three pipes together to fill the tank = $$\frac{60}{\left(3+6-4\right)}$$

= $$\frac{60}{5}$$

= 12 minutes