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1. A vessel has three pipes connected to it , two to supply liquid and one to draw liquid. The first alone can fill the vessel in $4 \frac{1}{2}$ hours, the second in 3 hours and the third can empty it in $1 \frac{1}{2}$ hours. If all the pipes are opened simultaneously when the vessel is half full, how soon will it be emptied?

A. $4 \frac{1}{2}$ hours

B. $5 \frac{1}{2}$ hours

C. $6 \frac{1}{2}$ hours

D. None of these

Solution:

\begin{aligned} Net part filled in 1 hour \\ = \frac{2}{3} - (\frac{2}{9} + \frac{1}{3}) \\ = (\frac{2}{3} - \frac{5}{9}) \\ = \frac{1}{9} \\ ∴ \frac{1}{9} : \frac{1}{2} :: 1 : x \\ or x = (\frac{1}{2} \times 9) = 4 \frac{1}{2} hours \\ So, the tank will be emptied in \\ = 4 \frac{1}{2} hours \end{aligned}

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