MAT Exam 2023 Paper

Concept:The leak reduces the net filling rate. Its emptying time is found by dividing the cistern capacity by the leak rate.Explanation:Let the capacity of the cistern be $1$ unit.Time to fill without leak $= 2.5$ hours.Filling rate without leak $= \frac{1}{2.5} = \frac{2}{5}$ per hour.Time to fill with leak $= 2.5 + \frac{1}{4} = 2.75$ hours.Net filling rate with leak $= \frac{1}{2.75} = \frac{4}{11}$ per hour.The leak rate is the difference between the filling rate and the net rate.Leak rate $= \frac{2}{5} - \frac{4}{11} = \frac{2}{55}$ per hour.Time for the leak to empty the full cistern $= \frac{1}{\text{leak rate}} = \frac{1}{\frac{2}{55}} = \frac{55}{2} = 27.5$ hours.Answer:$27.5$ hours, i.e. Option D.