To solve the problem, we will first determine the efficiencies of A, B, and C based on the information given.
1. **Determine the total work done**:
We can assume the total work is represented by the least common multiple (LCM) of the days taken by A and B, B and C, and A, B, and C.
- A and B can complete the work in 8 days, so their combined efficiency is \( \frac{1}{8} \) of the work per day.
- B and C can complete the work in 12 days, so their combined efficiency is \( \frac{1}{12} \) of the work per day.
- A, B, and C together can complete the work in 6 days, so their combined efficiency is \( \frac{1}{6} \) of the work per day.
The LCM of 8, 12, and 6 is 24. Therefore, we can assume the total work is 24 units.
2. **Calculate the efficiencies**:
- Let the efficiency of A be \( a \), B be \( b \), and C be \( c \).
- From A and B:
\[ a + b = \frac{24}{8} = 3 \quad \text{(Equation 1)} \]
- From B and C:
\[ b + c = \frac{24}{12} = 2 \quad \text{(Equation 2)} \]
- From A, B, and C:
\[ a + b + c = \frac{24}{6} = 4 \quad \text{(Equation 3)} \]
3. **Solve the equations**:
- From Equation 1: \( a + b = 3 \)
- From Equation 2: \( b + c = 2 \)
- From Equation 3: \( a + b + c = 4 \)
We can add Equations 1 and 2:
\[ (a + b) + (b + c) = 3 + 2 \]
\[ a + 2b + c = 5 \quad \text{(Equation 4)} \]
Now we can subtract Equation 3 from Equation 4:
\[ (a + 2b + c) - (a + b + c) = 5 - 4 \]
\[ 2b - b = 1 \]
\[ b = 1 \]
4. **Substitute back to find A and C**:
- Substitute \( b = 1 \) into Equation 1:
\[ a + 1 = 3 \]
\[ a = 2 \]
- Substitute \( b = 1 \) into Equation 2:
\[ 1 + c = 2 \]
\[ c = 1 \]
5. **Find the efficiency of A and C together**:
- The efficiency of A and C together:
\[ a + c = 2 + 1 = 3 \]
6. **Calculate the time taken by A and C to finish the work**:
- Time taken = Total Work / Efficiency
\[ \text{Time} = \frac{24}{3} = 8 \text{ days} \]
Thus, A and C together will complete the work in **8 days**.
