Two pipes A and B can fill a tank in 5 hours and 6 hours respectively. Pipe C can empty it in 12 hours. If all three pipes are opened together, then the time taken to fill the tank is:

$3\frac{9}{17}$ hours

The correct answer is Option (4) → $3\frac{9}{17}$ hours

Rate of filling for pipe A = $\frac{1}{5}$ tank/hour

Rate of filling for pipe B = $\frac{1}{6}$ tank/hour

Rate of emptying for pipe C = $\frac{1}{12}$ tank/hour

Net rate when all pipes are open: $\frac{1}{5} + \frac{1}{6} - \frac{1}{12}$

Compute LCM of denominators: LCM(5,6,12) = 60

Net rate = $\frac{12}{60} + \frac{10}{60} - \frac{5}{60} = \frac{17}{60}$ tank/hour

Time to fill the tank = $\frac{1}{\text{net rate}} = \frac{1}{17/60} = \frac{60}{17}$ hours