Question : An inlet pipe can fill an empty tank in $4 \frac{1}{2}$ hours, while an outlet pipe drains a filled tank in $7 \frac{1}{5}$ hours. The tank is initially empty, and the two pipes are alternately opened for an hour each, till the tank is filled, starting with the inlet pipe. In how many hours will the tank be filled?
Option 1: 24 hours
Option 2: $20 \frac{1}{4}$ hours
Option 3: $20 \frac{3}{4}$ hours
Option 4: $22 \frac{3}{8}$ hours
Correct Answer: $22 \frac{3}{8}$ hours
Solution :
An inlet pipe can fill an empty tank in $4\frac{1}{2}$ hours while an outlet pipe drains a filled tank in $7\frac{1}{5}$ hours.
In 1 hour the inlet pipe can fill $\frac{2}{9}$th of the tank.
In 1 hour the outlet pipe can empty $\frac{5}{36}$th of the tank.
In 2 hours the pipes fill $\frac{2}{9} - \frac{5}{36} = \frac{8 - 5}{36}$ of the tank $=\frac{1}{12}$th of the tank
Tank filled in 22 hours $=\frac{1}{12}\times 11 = \frac{11}{12}$
Remaining work $=1-\frac{11}{12}=\frac{1}{12}$
After 22 hours it is the inlet pipe's turn, so the time taken by the inlet pipe to fill the $\frac{1}{12}$ part = $\frac{1}{12}\times \frac{9}{2} = \frac{3}{8}$ hours
So, the tank will be filled in $=22+\frac{3}{8} = 22\frac{3}{8}$ hours
Hence, the correct answer is $22\frac{3}{8}$ hours.
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