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Two pipes $P$ and $Q$ can fill a tank in $6$ hours and $9$ hours respectively, while a third pipe $R$ can empty the tank in $12$ hours. Initially, $P$ and $R$ are open for $4$ hours, Then $P$ is closed and $Q$ is opened. After $6$ more hours $R$ is closed. The total time taken to fill the tank (in hours) is ____

- $13.50$
- $14.50$
- $15.50$
- $16.50$

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P can fill the tank in $6$ hours

$\Rightarrow$ In $1$ hr P can fill $\frac{1}{6}$ of the tank.

Q can fill the tank in $9$ hours

$\Rightarrow$ In $1$ hr Q can fill $\frac{1}{9}$ of the tank.

R can empty the tank in $12$ hours

$\Rightarrow$ In $1$ hr R can empty $\frac{1}{12}$ of the tank.

P and R are opened for $4$ hours

$\Rightarrow$ They fill $4*\left ( \frac{1}{6}-\frac{1}{12} \right ) = 4*\frac{1}{12}=\frac{1}{3}$ of the tank.

$\Rightarrow$ $1-\frac{1}{3}=\frac{2}{3}$ of the tank is still empty.

Then P is closed and Q is opened. After $6$ more hours R is closed.

$\Rightarrow$ Q and R are opened together for $6$ hours.

$\Rightarrow$ They fill $6*\left ( \frac{1}{9}-\frac{1}{12} \right ) = 6*\frac{1}{36}=\frac{1}{6}$ of the tank.

$\Rightarrow$ $\frac{2}{3}-\frac{1}{6}=\frac{4-1}{6}=\frac{1}{2}$ of the tank is still empty.

Now only Q is opened

$\because$ Q can fill a tank in $6$ hr

$\Rightarrow$ Q can fill $\frac{1}{2}$ of the tank in $6*\frac{1}{2}$ hours = $3$ hours = $3$ hours.

$\therefore$ Total Time to fill the tank = $4$ hours+$6$ hours+$3$ hours= $13$ hours

So Option A. $13.50$ is the correct answer.

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