Two pipes can fill a tank in 6 hours and 8 hours respectively. A third pipe can empty the same tank in 12 hours. If all the pipes start working together, how long it will take to fill the tank?

Options

A. 5.2 hours
B. 4.8 hours
C. 4.5 hours
D. 4 hours

Answer

To solve the problem, we need to determine how long it will take for the three pipes (two filling and one emptying) to fill the tank when they are working together.

### Step-by-Step Solution:

1. **Determine the filling rate of each pipe:**
- Pipe A can fill the tank in 6 hours. Therefore, in 1 hour, it fills:
\[ \text{Rate of A} = \frac{1}{6} \text{ of the tank} \]
- Pipe B can fill the tank in 8 hours. Therefore, in 1 hour, it fills:
\[ \text{Rate of B} = \frac{1}{8} \text{ of the tank} \]
- Pipe C can empty the tank in 12 hours. Therefore, in 1 hour, it empties:
\[ \text{Rate of C} = \frac{1}{12} \text{ of the tank} \]

2. **Combine the rates of the pipes:**
- When all pipes are working together, the net rate of filling the tank is:
\[ \text{Net Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
- Substituting the rates we found:
\[ \text{Net Rate} = \frac{1}{6} + \frac{1}{8} - \frac{1}{12} \]

3. **Find a common denominator:**
- The least common multiple (LCM) of 6, 8, and 12 is 24. We convert each fraction:
\[ \frac{1}{6} = \frac{4}{24}, \quad \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24} \]
- Now substituting these values into the net rate:
\[ \text{Net Rate} = \frac{4}{24} + \frac{3}{24} - \frac{2}{24} = \frac{5}{24} \]

4. **Calculate the time to fill the tank:**
- If the net rate of filling the tank is \(\frac{5}{24}\) of the tank per hour, then the time \(T\) to fill the entire tank is given by:
\[ T = \frac{1 \text{ tank}}{\text{Net Rate}} = \frac{1}{\frac{5}{24}} = \frac{24}{5} \text{ hours} \]
- Converting \(\frac{24}{5}\) hours into hours and minutes:
\[ \frac{24}{5} = 4.8 \text{ hours} = 4 \text{ hours and } 0.8 \times 60 \text{ minutes} = 4 \text{ hours and } 48 \text{ minutes} \]

### Final Answer:
The time taken to fill the tank when all three pipes are working together is **4 hours and 48 minutes**.