To solve the problem, we will first determine the work rates of A, B, and C, and then calculate how much work is done before C leaves and how much is done after that.
### Step 1: Determine the work rates of A, B, and C.
- A can complete the work in 16 days, so A's work rate is \( \frac{1}{16} \) of the work per day.
- B can complete the work in 24 days, so B's work rate is \( \frac{1}{24} \) of the work per day.
- C can complete the work in 32 days, so C's work rate is \( \frac{1}{32} \) of the work per day.
### Step 2: Calculate the combined work rate of A, B, and C.
To find the combined work rate, we add their individual work rates:
\[
\text{Combined work rate} = \frac{1}{16} + \frac{1}{24} + \frac{1}{32}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 16, 24, and 32 is 96.
Now, we convert each fraction:
- \( \frac{1}{16} = \frac{6}{96} \)
- \( \frac{1}{24} = \frac{4}{96} \)
- \( \frac{1}{32} = \frac{3}{96} \)
Adding these together:
\[
\text{Combined work rate} = \frac{6}{96} + \frac{4}{96} + \frac{3}{96} = \frac{13}{96}
\]
### Step 3: Calculate the work done in the first 4 days.
In the first 4 days, A, B, and C work together:
\[
\text{Work done in 4 days} = 4 \times \frac{13}{96} = \frac{52}{96} = \frac{13}{24}
\]
### Step 4: Determine the remaining work.
The total work is 1 (whole work), so the remaining work after 4 days is:
\[
\text{Remaining work} = 1 - \frac{13}{24} = \frac{24 - 13}{24} = \frac{11}{24}
\]
### Step 5: Calculate the work rate after C leaves.
After 4 days, C leaves, so only A and B continue to work. Their combined work rate is:
\[
\text{A's work rate} + \text{B's work rate} = \frac{1}{16} + \frac{1}{24}
\]
Using the common denominator of 48:
- \( \frac{1}{16} = \frac{3}{48} \)
- \( \frac{1}{24} = \frac{2}{48} \)
Thus, the combined work rate of A and B is:
\[
\frac{3}{48} + \frac{2}{48} = \frac{5}{48}
\]
### Step 6: Calculate the time taken to finish the remaining work.
Let \( t \) be the time taken by A and B to complete the remaining work \( \frac{11}{24} \):
\[
\frac{5}{48} \times t = \frac{11}{24}
\]
To solve for \( t \):
\[
t = \frac{11}{24} \div \frac{5}{48} = \frac{11}{24} \times \frac{48}{5} = \frac{11 \times 2}{5} = \frac{22}{5} = 4.4 \text{ days}
\]
### Step 7: Calculate the total time taken to complete the work.
The total time taken is the initial 4 days plus the time taken by A and B to finish the remaining work:
\[
\text{Total time} = 4 + 4.4 = 8.4 \text{ days}
\]
### Final Answer:
The work will be completed in **8.4 days**.
