Pipe A can fill a tank in 30 min, while pipe B can fill the same tank in 10 min and pipe C can empty the full tank in 40 min. If all the pipes are opened together, how much time will be needed to fill the tank full?

Options

A. `9(3)/(13)h`
B. `9(4)/(13)h`
C. `9(7)/(13)h`
D. `9(9)/(13)h`

Answer

To solve the problem step by step, we will calculate the time taken to fill the tank when all three pipes are opened together.

### Step-by-Step Solution:

1. **Identify the Filling and Emptying Rates:**
- Pipe A fills the tank in 30 minutes.
- Pipe B fills the tank in 10 minutes.
- Pipe C empties the tank in 40 minutes.

2. **Calculate the Work Done by Each Pipe:**
- Work done by Pipe A in 1 minute = \( \frac{1}{30} \) of the tank.
- Work done by Pipe B in 1 minute = \( \frac{1}{10} \) of the tank.
- Work done by Pipe C in 1 minute = \( -\frac{1}{40} \) of the tank (negative because it empties the tank).

3. **Find the Least Common Multiple (LCM):**
- The LCM of 30, 10, and 40 is 120 minutes. This will help us to calculate the total work done in a common time frame.

4. **Calculate the Total Work Done by Each Pipe in 120 Minutes:**
- Work done by Pipe A in 120 minutes = \( \frac{120}{30} = 4 \) tanks.
- Work done by Pipe B in 120 minutes = \( \frac{120}{10} = 12 \) tanks.
- Work done by Pipe C in 120 minutes = \( \frac{120}{40} = 3 \) tanks (emptying).

5. **Calculate the Combined Work Done:**
- Total work done by A and B (filling) = \( 4 + 12 = 16 \) tanks.
- Total work done by C (emptying) = \( 3 \) tanks.
- Net work done = \( 16 - 3 = 13 \) tanks.

6. **Calculate the Time to Fill One Tank:**
- Since the net work done is 13 tanks in 120 minutes, the time taken to fill 1 tank is:
\[ \text{Time} = \frac{\text{Total Time}}{\text{Net Work}} = \frac{120 \text{ minutes}}{13} = 9 \text{ minutes} \text{ and } \frac{3}{13} \text{ minutes}. \]

### Final Answer:
The time needed to fill the tank full when all pipes are opened together is \( 9 \frac{3}{13} \) minutes.