Pipes A and B can fill a tank in 6 hours and 8 hours respectively and pipe C can empty the full tank in 12 hours. All three pipes are opened together, A is closed after 3 hours. In how many hours will the remaining part of the tank be filled?

Options

A. 9
B. 12
C. 11
D. 10

Answer

To solve the problem, we need to determine how much of the tank is filled after all three pipes are opened together for a certain time and then how long it will take to fill the remaining part of the tank after one of the pipes is closed.

### Step-by-Step Solution:

1. **Determine the capacity of the tank**:
- Let’s assume the capacity of the tank is 24 liters (this is a common multiple of the times given for the pipes).
- Pipe A fills the tank in 6 hours, so its rate is \( \frac{24 \text{ liters}}{6 \text{ hours}} = 4 \text{ liters/hour} \).
- Pipe B fills the tank in 8 hours, so its rate is \( \frac{24 \text{ liters}}{8 \text{ hours}} = 3 \text{ liters/hour} \).
- Pipe C empties the tank in 12 hours, so its rate is \( \frac{24 \text{ liters}}{12 \text{ hours}} = 2 \text{ liters/hour} \).

2. **Calculate the combined rate of all three pipes**:
- When all three pipes are opened together, the effective filling rate is:
\[ \text{Rate of A} + \text{Rate of B} - \text{Rate of C} = 4 + 3 - 2 = 5 \text{ liters/hour} \]

3. **Calculate the amount filled in the first 3 hours**:
- In 3 hours, the amount of water filled by all three pipes is:
\[ 5 \text{ liters/hour} \times 3 \text{ hours} = 15 \text{ liters} \]

4. **Determine the remaining capacity of the tank**:
- The total capacity of the tank is 24 liters, and after 3 hours, 15 liters have been filled. Therefore, the remaining capacity is:
\[ 24 - 15 = 9 \text{ liters} \]

5. **Calculate the rate of filling after pipe A is closed**:
- After 3 hours, pipe A is closed, so only pipe B and pipe C are left. The effective rate now is:
\[ \text{Rate of B} - \text{Rate of C} = 3 - 2 = 1 \text{ liter/hour} \]

6. **Calculate the time required to fill the remaining part of the tank**:
- To fill the remaining 9 liters at the rate of 1 liter/hour, the time required is:
\[ \frac{9 \text{ liters}}{1 \text{ liter/hour}} = 9 \text{ hours} \]

### Final Answer:
The remaining part of the tank will be filled in **9 hours**.