Pipe A can fill an empty tank in 10 h, Pipe B can fill the same tank, when empty, in 15 h, and Pipe C can empty the same tank, when completely full, in 12 h. Find the time taken to fill half the tank if all three pipes are opened simultaneously when the tank is empty.

LCM of 10, 15, 12 is 60 units which can be assumed as the total capacity of the tank.

Pipe A can fill an empty tank in 10 h.

The efficiency of Pipe A = $$\frac{60}{10}$$ = 6 units/h

Pipe B can fill the same tank, when empty, in 15 h.

The efficiency of Pipe B = $$\frac{60}{15}$$ = 4 units/h

Pipe C can empty the same tank, when completely full, in 12 h.

The efficiency of Pipe C = -$$\frac{60}{12}$$ = -5 units/h [Here negative sign represents that the pipe is used to empty the tank.]

Time taken to fill half the tank if all three pipes are opened simultaneously when the tank is empty = $$\frac{\frac{60}{2}}{6+4-5}$$

= $$\frac{30}{5}$$

= 6 h