To solve the problem step by step, let's denote the work done by A, B, and C in one day as \( A \), \( B \), and \( C \) respectively.
### Step 1: Determine the work done by A and B together
Given that A and B can complete the work in 16 days, the amount of work they do in one day is:
\[
A + B = \frac{1}{16} \text{ (work per day)}
\]
### Step 2: Determine the work done by B and C together
Given that B and C can complete the work in 24 days, the amount of work they do in one day is:
\[
B + C = \frac{1}{24} \text{ (work per day)}
\]
### Step 3: Determine the work done by A, B, and C together
Given that A, B, and C can complete the work in 12 days, the amount of work they do in one day is:
\[
A + B + C = \frac{1}{12} \text{ (work per day)}
\]
### Step 4: Set up the equations
From the above information, we have the following three equations:
1. \( A + B = \frac{1}{16} \)
2. \( B + C = \frac{1}{24} \)
3. \( A + B + C = \frac{1}{12} \)
### Step 5: Solve for A, B, and C
We can express \( C \) in terms of \( A \) and \( B \) using the third equation:
\[
C = (A + B + C) - (A + B) = \frac{1}{12} - \frac{1}{16}
\]
To subtract these fractions, find a common denominator (which is 48):
\[
\frac{1}{12} = \frac{4}{48}, \quad \frac{1}{16} = \frac{3}{48}
\]
Thus,
\[
C = \frac{4}{48} - \frac{3}{48} = \frac{1}{48}
\]
### Step 6: Substitute C back to find A and B
Now substitute \( C \) back into the second equation:
\[
B + \frac{1}{48} = \frac{1}{24}
\]
This gives:
\[
B = \frac{1}{24} - \frac{1}{48}
\]
Finding a common denominator (which is 48):
\[
\frac{1}{24} = \frac{2}{48}
\]
Thus,
\[
B = \frac{2}{48} - \frac{1}{48} = \frac{1}{48}
\]
Now substitute \( B \) back into the first equation:
\[
A + \frac{1}{48} = \frac{1}{16}
\]
This gives:
\[
A = \frac{1}{16} - \frac{1}{48}
\]
Finding a common denominator (which is 48):
\[
\frac{1}{16} = \frac{3}{48}
\]
Thus,
\[
A = \frac{3}{48} - \frac{1}{48} = \frac{2}{48} = \frac{1}{24}
\]
### Step 7: Calculate the work done by A and C together
Now we have:
- \( A = \frac{1}{24} \)
- \( B = \frac{1}{48} \)
- \( C = \frac{1}{48} \)
Now, we calculate the work done by A and C together:
\[
A + C = \frac{1}{24} + \frac{1}{48}
\]
Finding a common denominator (which is 48):
\[
\frac{1}{24} = \frac{2}{48}
\]
Thus,
\[
A + C = \frac{2}{48} + \frac{1}{48} = \frac{3}{48} = \frac{1}{16}
\]
### Step 8: Find the time taken by A and C to complete the work together
If A and C together do \(\frac{1}{16}\) of the work in one day, then the time taken to complete the entire work is:
\[
\text{Time} = \frac{1}{\frac{1}{16}} = 16 \text{ days}
\]
### Final Answer
The time taken by A and C to complete the work together is **16 days**.
