A tap can fill a bath in $20$ minutes and another tap can fill it in $30$ minutes. Amit opens both the taps simultaneously. When the bath should have been full, he finds that the waste pipe was open. He then closes the waste pipe and in another $4$ minutes the bath is full. In what time would the waste pipe empty it ?

Let the waste pipe empty the tank in x minutes; the net rate of all three operating is 1/20 + 1/30 - 1/x = 1/12 - 1/x. Amit finds the tank unfilled at the expected time, and closing the waste pipe allows the two filling taps to finish the remaining work in 4 minutes at their combined rate of 1/12, meaning 4/12 = 1/3 of the tank was left. Since 2/3 was filled before, (1/12 - 1/x) * t = 2/3 where t is the expected time, but realizing the net rate caused a delay means the remaining 1/3 corresponds to the waste pipe working against the taps for those 4 extra minutes, establishing 1/x = 1/12 - 1/12 + 1/12 * 1/3, yielding x = 36. The result is 36 min.

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