To solve the problem step by step, we need to determine how long A and C will take to complete the work together based on the information given about A, B, and C.
### Step 1: Understand the given information
- A and B together can complete the work in 8 days.
- B and C together can complete the work in 12 days.
- A, B, and C together can complete the work in 6 days.
### Step 2: Calculate the total work in terms of units
To find a common measure for the work, we can use the least common multiple (LCM) of the days taken by the groups.
- The LCM of 8, 12, and 6 is 24.
- We can consider the total work to be 24 units.
### Step 3: Calculate the efficiencies of each group
- **Efficiency of A and B**: Since A and B complete the work in 8 days, their combined efficiency is:
\[
\text{Efficiency of A + B} = \frac{24 \text{ units}}{8 \text{ days}} = 3 \text{ units/day}
\]
- **Efficiency of B and C**: Since B and C complete the work in 12 days, their combined efficiency is:
\[
\text{Efficiency of B + C} = \frac{24 \text{ units}}{12 \text{ days}} = 2 \text{ units/day}
\]
- **Efficiency of A, B, and C**: Since A, B, and C complete the work in 6 days, their combined efficiency is:
\[
\text{Efficiency of A + B + C} = \frac{24 \text{ units}}{6 \text{ days}} = 4 \text{ units/day}
\]
### Step 4: Calculate the efficiency of A and C
Now, we can find the individual efficiencies of A and C.
- **Efficiency of A**: We can find A's efficiency by subtracting the efficiency of B and C from the efficiency of A, B, and C:
\[
\text{Efficiency of A} = \text{Efficiency of A + B + C} - \text{Efficiency of B + C} = 4 - 2 = 2 \text{ units/day}
\]
- **Efficiency of C**: Similarly, we can find C's efficiency by subtracting the efficiency of A and B from the efficiency of A, B, and C:
\[
\text{Efficiency of C} = \text{Efficiency of A + B + C} - \text{Efficiency of A + B} = 4 - 3 = 1 \text{ unit/day}
\]
### Step 5: Calculate the efficiency of A and C together
Now we can find the combined efficiency of A and C:
\[
\text{Efficiency of A + C} = \text{Efficiency of A} + \text{Efficiency of C} = 2 + 1 = 3 \text{ units/day}
\]
### Step 6: Calculate the time taken by A and C to complete the work
Finally, we can calculate the time taken by A and C to complete the total work of 24 units:
\[
\text{Time taken by A and C} = \frac{\text{Total Work}}{\text{Efficiency of A + C}} = \frac{24 \text{ units}}{3 \text{ units/day}} = 8 \text{ days}
\]
### Final Answer
A and C will take **8 days** to complete the work together.
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