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 $9$ hr
$\Rightarrow$ Q can fill $\frac{1}{2}$ of the tank in $9*\frac{1}{2}$ hours $=4.5$ hours.
$\therefore$ Total Time to fill the tank $=4$ hours $+6$ hours $+ 4.5$ hours $=14.5$ hours
So, Option B. $14.50$ is the correct answer.