A tank is filled by a pipe A in 32 minutes and pipe B in 36 minutes,When full ,it can be emptied by a pipe C in 20 minutes.If all the three pipes are opned simultaneously,half of the tank will be filled in -----

Options

A. 16 minutes
B. 24 minutes
C. 48 minutes
D. None of these

Answer

To solve the problem, we need to determine how long it will take to fill half of the tank when all three pipes (A, B, and C) are opened simultaneously.

### Step-by-Step Solution:

1. **Determine the filling and emptying rates of the pipes:**
- Pipe A fills the tank in 32 minutes. Therefore, the rate of Pipe A is:
\[ \text{Rate of A} = \frac{1}{32} \text{ tank/minute} \]
- Pipe B fills the tank in 36 minutes. Therefore, the rate of Pipe B is:
\[ \text{Rate of B} = \frac{1}{36} \text{ tank/minute} \]
- Pipe C empties the tank in 20 minutes. Therefore, the rate of Pipe C is:
\[ \text{Rate of C} = -\frac{1}{20} \text{ tank/minute} \]

2. **Find a common time frame for calculations:**
- To simplify calculations, we can find the least common multiple (LCM) of the times taken by the pipes. The LCM of 32, 36, and 20 is 720 minutes.

3. **Calculate the total work done by each pipe in 720 minutes:**
- Amount filled by Pipe A in 720 minutes:
\[ \text{Amount filled by A} = 720 \times \frac{1}{32} = 22.5 \text{ tanks} \]
- Amount filled by Pipe B in 720 minutes:
\[ \text{Amount filled by B} = 720 \times \frac{1}{36} = 20 \text{ tanks} \]
- Amount emptied by Pipe C in 720 minutes:
\[ \text{Amount emptied by C} = 720 \times \left(-\frac{1}{20}\right) = -36 \text{ tanks} \]

4. **Calculate the net effect of all three pipes:**
- Total amount filled when all pipes are opened:
\[ \text{Total filled} = 22.5 + 20 - 36 = 6.5 \text{ tanks} \]

5. **Determine the time taken to fill half the tank:**
- Since we need to fill half of the tank (0.5 tanks), we can set up the equation:
\[ \text{Time} = \frac{\text{Amount to fill}}{\text{Net rate}} = \frac{0.5 \text{ tanks}}{6.5 \text{ tanks in 720 minutes}} \times 720 \text{ minutes} \]
- Calculating this gives:
\[ \text{Time} = \frac{0.5}{6.5} \times 720 \approx 55.38 \text{ minutes} \]

### Final Answer:
Half of the tank will be filled in approximately **55 minutes**.