Two pipes A and B can fill a tank in 24 and 32 min, respectively. If both the pipes are opened together, after how much time pipeB should be closed so that the tank is full in 9 min?

Options

A. 40 min
B. 30 min
C. 10 min
D. 20 min

Answer

To solve the problem, we need to determine how long pipe B should remain open while both pipes A and B are filling the tank together, so that the tank is completely filled in 9 minutes.

### Step-by-Step Solution:

1. **Determine the filling rates of pipes A and B**:
- Pipe A can fill the tank in 24 minutes.
- Pipe B can fill the tank in 32 minutes.
- Let's assume the capacity of the tank is 96 units (this is a common multiple of 24 and 32 for easier calculations).

- The rate of pipe A filling the tank:
\[ \text{Rate of A} = \frac{96 \text{ units}}{24 \text{ minutes}} = 4 \text{ units/minute} \]

- The rate of pipe B filling the tank:
\[ \text{Rate of B} = \frac{96 \text{ units}}{32 \text{ minutes}} = 3 \text{ units/minute} \]

2. **Set up the equation for the total filling**:
- Let pipe B be closed after \( x \) minutes. This means:
- Pipe A will be open for 9 minutes.
- Pipe B will be open for \( x \) minutes.

- The total amount filled by both pipes in 9 minutes can be expressed as:
\[ \text{Total filled} = (\text{Amount filled by A}) + (\text{Amount filled by B}) \]
\[ \text{Total filled} = (4 \text{ units/minute} \times 9 \text{ minutes}) + (3 \text{ units/minute} \times x \text{ minutes}) \]
\[ \text{Total filled} = 36 + 3x \]

3. **Set the equation equal to the tank's capacity**:
- Since the tank's total capacity is 96 units, we can set up the equation:
\[ 36 + 3x = 96 \]

4. **Solve for \( x \)**:
- Rearranging the equation gives:
\[ 3x = 96 - 36 \]
\[ 3x = 60 \]
\[ x = \frac{60}{3} = 20 \]

5. **Conclusion**:
- Pipe B should be closed after 20 minutes to ensure that the tank is full in 9 minutes.

### Final Answer:
Pipe B should be closed after **20 minutes**.

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