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

Two pipes A and B can separately fill a cistern in 60 minutes and 75 minutes respectively. There is a third pipe in the bottom of the cistern to empty it. If all the three pipes are simultaneously opened, then the cistern is full in 50 minutes. In how much time the third pipe alone can empty the cistern?
[A]100 minutes
[B]120 minutes
[C]90 minutes
[D]110 minutes

100 minutes
Let the third pipe empty the cistern in x minutes.
Part of cistren filled in 1 minute when all three pipes are opened simultaneously
= \frac{1}{60} + \frac{1}{75} - \frac{1}{x}
According to the question,
= \frac{1}{60} + \frac{1}{75} - \frac{1}{x} = \frac{1}{50}
=> \frac{1}{x} = \frac{1}{60} + \frac{1}{75} - \frac{1}{50}
=>\frac{5+4-6}{300}= \frac{3}{300}
=> \frac{1}{x} = \frac{3}{300}
\therefore x = \frac{300}{3} = 100 minutes
Hence option [A] is the right answer.

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