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 make the tank full?

Options

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

Answer

To solve the problem, we need to calculate the rate at which each pipe fills or empties the tank, and then combine these rates to find the total time taken to fill the tank when all pipes are opened together.

### Step-by-Step Solution:

1. **Calculate the filling rate of Pipe A:**
- Pipe A can fill the tank in 30 minutes.
- Therefore, in 1 minute, Pipe A fills \( \frac{1}{30} \) of the tank.

2. **Calculate the filling rate of Pipe B:**
- Pipe B can fill the tank in 10 minutes.
- Therefore, in 1 minute, Pipe B fills \( \frac{1}{10} \) of the tank.

3. **Calculate the emptying rate of Pipe C:**
- Pipe C can empty the tank in 40 minutes.
- Therefore, in 1 minute, Pipe C empties \( \frac{1}{40} \) of the tank.

4. **Combine the rates of all pipes:**
- When all pipes are opened together, the effective filling rate is:
\[ \text{Effective Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
- Substituting the rates we calculated:
\[ \text{Effective Rate} = \frac{1}{30} + \frac{1}{10} - \frac{1}{40} \]

5. **Find a common denominator:**
- The least common multiple of 30, 10, and 40 is 120.
- Convert each rate to have a denominator of 120:
\[ \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{10} = \frac{12}{120}, \quad \frac{1}{40} = \frac{3}{120} \]

6. **Calculate the effective rate:**
\[ \text{Effective Rate} = \frac{4}{120} + \frac{12}{120} - \frac{3}{120} = \frac{4 + 12 - 3}{120} = \frac{13}{120} \]

7. **Calculate the time to fill the tank:**
- If the effective rate is \( \frac{13}{120} \) of the tank per minute, then the time \( T \) to fill the tank is the reciprocal of the effective rate:
\[ T = \frac{1}{\frac{13}{120}} = \frac{120}{13} \text{ minutes} \]

8. **Convert the time to a mixed fraction:**
- \( \frac{120}{13} \) can be calculated as:
\[ 120 \div 13 = 9 \quad \text{(remainder 3)} \]
- Thus, \( \frac{120}{13} = 9 \frac{3}{13} \) minutes.

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