Let's break down the problem systematically.

Step 1: Define the rates at which each person works.

• Let $A$'s rate of work be $a$ (portion of the task completed per day).
• Let $B$'s rate of work be $b$.
• Let $C$'s rate of work be $c$.

We are given:

1. $A$ and $B$ together can complete the task in 12 days, so their combined rate is: $a + b = \frac{1}{12}$
2. $B$ and $C$ together can complete the task in 16 days, so their combined rate is: $b + c = \frac{1}{16}$
3. $A$, $B$, and $C$ together can complete the task in 8 days, so their combined rate is: $a + b + c = \frac{1}{8}$

Step 2: Solve for the individual rates.

We have the following system of equations:

1. $a + b = \frac{1}{12}$
2. $b + c = \frac{1}{16}$
3. $a + b + c = \frac{1}{8}$

From equation 3, we can substitute $a + b$ from equation 1: $a + b + c = \frac{1}{8} \implies \frac{1}{12} + c = \frac{1}{8}$ Solving for $c$: $c = \frac{1}{8} - \frac{1}{12} = \frac{3 - 2}{24} = \frac{1}{24}$

Now that we know $c = \frac{1}{24}$, substitute this into equation 2: $b + c = \frac{1}{16} \implies b + \frac{1}{24} = \frac{1}{16}$ Solving for $b$: $b = \frac{1}{16} - \frac{1}{24} = \frac{3 - 2}{48} = \frac{1}{48}$

Finally, substitute $b = \frac{1}{48}$ into equation 1: $a + b = \frac{1}{12} \implies a + \frac{1}{48} = \frac{1}{12}$ Solving for $a$: $a = \frac{1}{12} - \frac{1}{48} = \frac{4 - 1}{48} = \frac{3}{48} = \frac{1}{16}$

Step 3: Find the combined rate of $A$ and $C$.

Now that we know $a = \frac{1}{16}$ and $c = \frac{1}{24}$, the combined rate of $A$ and $C$ is: $a + c = \frac{1}{16} + \frac{1}{24} = \frac{3 + 2}{48} = \frac{5}{48}$

Step 4: Calculate the time for $A$ and $C$ to complete the task.

The time taken for $A$ and $C$ to complete the task is the reciprocal of their combined rate: $\text{Time} = \frac{1}{a + c} = \frac{1}{\frac{5}{48}} = \frac{48}{5} = 9.6 \, \text{days}$

Thus, $A$ and $C$ together will take 9.6 days to complete the task.

Would you like more details on any step?

Related Questions:

1. How can we generalize this method to more complex work problems?
2. What happens if A and C together had a different working efficiency?
3. How does the concept of rates apply to other areas like speed and distance?
4. What if all workers take breaks, how do we factor that in?
5. How do we solve a work problem when the rates are in different units?

Tip: Always break down work problems by focusing on individual rates and using the total time as the sum of rates.