A pipe can fill a cistern in 12 minutes and another pipe can fill it in 15 minutes, but the third pipe can empty it in 6 minutes. The first two pipes are kept open for 5 minutes in the beginning and then the third pipe is also opened. The number of minutes taken to empty the cistern is

Using the one-unit method, the rates of the two filling pipes are 1/12 and 1/15, and the emptying pipe's rate is 1/6. In the first 5 minutes, the amount of water filled is 5 * (1/12 + 1/15) = 5 * (9/60) = 3/4 of the cistern. When the third pipe is opened, the net emptying rate becomes 1/6 - (1/12 + 1/15) = 10/60 - 9/60 = 1/60 of the cistern per minute. To empty the 3/4 of the cistern currently filled, the time required is (3/4) / (1/60) = 45 minutes.

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