Two pipes $A$ and $B$ can fill a cistern is $12$ minutes and $15$ minutes respectively, but a third pipe $C$ can empty the full tank in $6$ minutes. $A$ and $B$ are kept open for $5$ minutes in the beginning and then $C$ is also opened. In what time will the cistern be emptied ?

In the first 5 minutes, pipes A and B fill (1/12 + 1/15) * 5 = 3/4 of the cistern. When pipe C is opened, the net rate becomes an emptying rate of 1/6 - 1/12 - 1/15 = 1/60 per minute. To empty the 3/4 filled portion at this rate, the time required is (3/4) / (1/60) = 45 minutes. The result is 45 min.

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