Two pipes can fill a cistern in $14$ hours and $16$ hours respectively, The pipes are opened simultaneously and it is found that due to leakage in the bottom, $32$ minutes extra are taken for the cistern to be filled up. If the cistern is full, in what time would the leak empty it ?

The first two pipes have a combined rate of 1/14 + 1/16 = 15/112 of the cistern per hour, meaning they would normally fill it in 112/15 hours (7 hours and 28 minutes). Accounting for the 32 extra minutes, the actual filling time becomes 8 hours, making the combined rate with the leak 1/8 of the cistern per hour. The leak's rate is the difference between these rates, which is 15/112 - 1/8 = 1/112 of the cistern per hour, so the leak alone would empty the full cistern in 112 hours.

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