Two pipes $A$ and $B$ can separately fill a cistern in $60\ minutes$ and $75\ minutes$ respectively. There is a third pipe at the bottom of the cistern to empty it , If all the three pipes are simultancously opened, then the cistern is full in $50\ minutes$, In how much time, the third pipe alone can empty the cistern ?

Using the least common multiple of 60, 75, and 50, assume the cistern's total capacity is 300 units. Pipe A fills at 5 units per minute, pipe B fills at 4 units per minute, and the combined filling time of 50 minutes means all three pipes have a net rate of 300 / 50 = 6 units per minute. Replacing the third pipe's emptying rate with x, the equation is 5 + 4 - x = 6, which means x = 3 units per minute. Therefore, the third pipe alone takes 300 / 3 = 100 minutes to empty the cistern.

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