An empty tank has three inlet and two outlet pipes connected to it. An inlet pipe can fill an empty tank in 12 hours, while an outlet pipe can drain a filled tank in 10 hours. All inlet pipes have the same efficiency and both outlet pipes have the same efficiency. If all inlet pipes and outlet pipes connected to the empty tank are opened together, find the time in which the tank is completely filled.

Assume that the capacity of the tank is $$x$$ litres, the efficiency of an inlet pipe is $$y$$ litres per hour, and that of an outlet pipe is $$z$$ litres per hour.

So, given that an inlet pipe can fill an empty tank in 12 hours or, 

$$\dfrac{x}{y}=12$$

$$y=\dfrac{x}{12}$$ . . . (1)

Also, given that an outlet pipe can drain a filled tank in 10 hours or,

$$\dfrac{x}{z}=10$$

$$z=\dfrac{x}{10}$$ . . . (2)

Now, all the pipes are opened together to an empty tank,

so the time taken to fill the tank is $$\dfrac{x}{3y-2z}$$

Putting the values of $$y$$ and $$z$$ from equations (1) and (2) we get

Time taken = $$\dfrac{x}{3y-2z}=\dfrac{x}{\dfrac{3x}{12}-\dfrac{2x}{10}}=20$$ hours

Hence, the answer is 20 hours.