A tap can fill a tank in \(5\frac{1}{2}\) hours. Because of a leak, it takes \(8\frac{1}{4}\) hours to fill the tank. In how many hours can the leak alone empty 30% of the full tank?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\frac{99}{20}\)
With the leak open the tank fills more slowly. The leak's rate is the tap's rate minus the net filling rate.
Tap fills \(\displaystyle \frac{1}{5.5}=\frac{2}{11}\) of the tank per hour; with the leak, the net filling is \(\displaystyle \frac{1}{8.25}=\frac{4}{33}\) per hour
Leak empties \(\displaystyle \frac{2}{11}-\frac{4}{33}=\frac{6-4}{33}=\frac{2}{33}\) of the tank per hour, so it empties the full tank in \(\frac{33}{2}\) hours
Time to empty 30% \(\displaystyle =\frac{33}{2}\times\frac{3}{10}=\frac{99}{20}\) hours