Pipe A can fill a tank in 6 hrs while pipe B alone can fill it in 5 hrs and pipe C can empty the full tank in 8 hrs. If all the pipes are opened together, how much time will be needed to completely fill the tank?

Answer

To solve the problem, we need to determine the rate at which each pipe fills or empties the tank and then combine these rates to find out how long it will take to fill the tank when all pipes are opened together.

### Step-by-Step Solution:

1. **Determine the rates of each pipe:**
- Pipe A can fill the tank in 6 hours. Therefore, the rate of Pipe A is:
\[ \text{Rate of A} = \frac{1}{6} \text{ tank per hour} \]
- Pipe B can fill the tank in 5 hours. Therefore, the rate of Pipe B is:
\[ \text{Rate of B} = \frac{1}{5} \text{ tank per hour} \]
- Pipe C can empty the tank in 8 hours. Therefore, the rate of Pipe C (which is negative since it empties) is:
\[ \text{Rate of C} = -\frac{1}{8} \text{ tank per hour} \]

2. **Combine the rates:**
To find the combined rate when all pipes are opened together, we add the rates of A and B and subtract the rate of C:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
\[ \text{Combined Rate} = \frac{1}{6} + \frac{1}{5} - \frac{1}{8} \]

3. **Find a common denominator:**
The least common multiple (LCM) of 6, 5, and 8 is 120. We convert each fraction:
\[ \frac{1}{6} = \frac{20}{120}, \quad \frac{1}{5} = \frac{24}{120}, \quad \frac{1}{8} = \frac{15}{120} \]

4. **Combine the fractions:**
Now we can combine the fractions:
\[ \text{Combined Rate} = \frac{20}{120} + \frac{24}{120} - \frac{15}{120} = \frac{20 + 24 - 15}{120} = \frac{29}{120} \]

5. **Calculate the time to fill the tank:**
The combined rate of \(\frac{29}{120}\) means that together, the pipes fill \(\frac{29}{120}\) of the tank in one hour. To find the time \(t\) taken to fill the entire tank (1 tank), we set up the equation:
\[ t = \frac{1 \text{ tank}}{\text{Combined Rate}} = \frac{1}{\frac{29}{120}} = \frac{120}{29} \text{ hours} \]

6. **Convert to mixed fraction:**
To convert \(\frac{120}{29}\) into a mixed fraction:
- Divide 120 by 29, which gives 4 with a remainder of 4.
- Thus, \(\frac{120}{29} = 4 \frac{4}{29}\) hours.

### Final Answer:
The time needed to completely fill the tank when all pipes are opened together is:
\[ 4 \frac{4}{29} \text{ hours} \]