Pipe A can fill 50% of the tank in 6 hours and Pipe B can fill 100% capacity of tank in 18 hours. If Pipe C can empty the full tank in 9 hours, then find the time taken by the all the pipes to fill the tank together (in hours).

The correct option is 36.

Let's solve the problem step-by-step by determining the individual rates of each pipe and then calculating their combined effect.

Step 1: Calculate the time taken by each pipe to fill or empty the tank completely.
- Pipe A: It can fill 50% of the tank in 6 hours. Therefore, the time taken by Pipe A to fill 100% of the tank is:
$6\times 2=12\text{hours}$
- Pipe B: It can fill 100% of the tank in 18 hours.
- Pipe C: It can empty the full tank in 9 hours.

Step 2: Determine the work rate of each pipe per hour.
- Hourly rate of Pipe A (filling) = $\frac{1}{12}$
- Hourly rate of Pipe B (filling) = $\frac{1}{18}$
- Hourly rate of Pipe C (emptying) = $-\frac{1}{9}$ (we represent emptying with a negative rate)

Step 3: Calculate the combined rate when all three pipes are opened together.
The combined hourly rate is:
$\text{Combined Rate}=\frac{1}{12}+\frac{1}{18}-\frac{1}{9}$
Find the Least Common Multiple (LCM) of 12, 18, and 9, which is 36:
$\text{Combined Rate}=\frac{3+2-4}{36}=\frac{1}{36}$

Step 4: Find the total time taken to fill the tank.
The total time taken is the reciprocal of the combined rate:
$\text{Total Time}=36\text{hours}$