Pipes A and B can fill a tank in 5 and 6 hours, respectively.
Pipe C can empty it in 12 hours. The tank is half full. All the
three pipes are in operation simultaneously. After how much
time, the tank will be full?

Options

A. `3 9/17 h`
B. 11 h
C. `2 8/11 h`
D. `1 13/17 h`

Explanation

To solve the problem, we need to determine how long it will take for pipes A, B, and C to fill the tank when they are all working together, given that the tank is initially half full.

### Step-by-Step Solution:

1. **Determine the filling and emptying rates of the pipes:**
- Pipe A can fill the tank in 5 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{5} \text{ tank/hour} \]
- Pipe B can fill the tank in 6 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{6} \text{ tank/hour} \]
- Pipe C can empty the tank in 12 hours. Therefore, its rate is:
\[ \text{Rate of C} = -\frac{1}{12} \text{ tank/hour} \quad (\text{negative because it empties}) \]

2. **Calculate the combined rate of all pipes:**
- The combined rate when all three pipes are working together is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
- Substituting the rates:
\[ \text{Combined Rate} = \frac{1}{5} + \frac{1}{6} - \frac{1}{12} \]

3. **Finding a common denominator:**
- The least common multiple of 5, 6, and 12 is 60. We convert each rate:
\[ \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
- Now, substituting these values:
\[ \text{Combined Rate} = \frac{12}{60} + \frac{10}{60} - \frac{5}{60} = \frac{17}{60} \text{ tank/hour} \]

4. **Determine the amount of work needed to fill the tank:**
- Since the tank is half full, we need to fill the remaining half:
\[ \text{Work needed} = \frac{1}{2} \text{ tank} \]

5. **Calculate the time required to fill the remaining half of the tank:**
- Using the formula:
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} = \frac{\frac{1}{2}}{\frac{17}{60}} = \frac{1}{2} \times \frac{60}{17} = \frac{30}{17} \text{ hours} \]

6. **Convert the time into a more understandable format:**
- \( \frac{30}{17} \) hours is approximately 1.76 hours, which can be converted into minutes:
\[ 1.76 \text{ hours} \approx 1 \text{ hour and } 45 \text{ minutes} \]

### Final Answer:
The tank will be full after approximately **1 hour and 45 minutes**.

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