Question : A tap can fill a tank in $5 \frac{1}{2}$ hours. Because of a leak, it took $8 \frac{1}{4}$ hours to fill the tank. In how much time (in hours) will the leak alone empty 30% of the tank?
Option 1: $\frac{99}{20}$
Option 2: $\frac{5}{2}$
Option 3: $\frac{9}{2}$
Option 4: $\frac{17}{2}$
Correct Answer: $\frac{99}{20}$
Solution :
Time taken by A to fill the tank = $5\frac{1}{2}\ \text{hours} = \frac{11}{2}\ \text{hours}$
Because of leak, the tank filled in = $8\frac{1}{4}\ \text{hours} = \frac{33}{4}\ \text{hours}$
LCM of 11 and 33 = 33
Capacity of the tank = 33
Efficiency of A to fill the tank = $(33×\frac{2}{11}) = 6$
Combined Efficiency of leak + A = $\frac{33}{\frac{33}{4}}=4$
Efficiency of Leak = 4 – 6 = – 2
30% of the tank = $33×\frac{30}{100}=\frac{99}{10}$
Required time to empty = $\frac{\frac{99}{10}}{2}=\frac{99}{20}\ \text{hours}$
Hence, the correct answer is $\frac{99}{20}\ \text{hours}$.
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