To solve the problem, we need to determine how many days A, B, and C will take to complete the work when A works alone on the first day, is assisted by B on the second day, and is assisted by C on the third day. This cycle continues until the work is completed.
### Step-by-Step Solution:
1. **Determine the Work Rate of Each Person:**
- A can complete the work in 400 days, so A's work rate is:
\[
\text{Work rate of A} = \frac{1}{400} \text{ work/day}
\]
- B can complete the work in 600 days, so B's work rate is:
\[
\text{Work rate of B} = \frac{1}{600} \text{ work/day}
\]
- C can complete the work in 900 days, so C's work rate is:
\[
\text{Work rate of C} = \frac{1}{900} \text{ work/day}
\]
2. **Calculate the Total Work Done in One Cycle (3 Days):**
- On Day 1, A works alone:
\[
\text{Work done on Day 1} = \frac{1}{400}
\]
- On Day 2, A and B work together:
\[
\text{Work done on Day 2} = \frac{1}{400} + \frac{1}{600}
\]
To add these fractions, find a common denominator (1200):
\[
\frac{1}{400} = \frac{3}{1200}, \quad \frac{1}{600} = \frac{2}{1200} \quad \Rightarrow \quad \text{Total for Day 2} = \frac{3}{1200} + \frac{2}{1200} = \frac{5}{1200}
\]
- On Day 3, A and C work together:
\[
\text{Work done on Day 3} = \frac{1}{400} + \frac{1}{900}
\]
Again, find a common denominator (3600):
\[
\frac{1}{400} = \frac{9}{3600}, \quad \frac{1}{900} = \frac{4}{3600} \quad \Rightarrow \quad \text{Total for Day 3} = \frac{9}{3600} + \frac{4}{3600} = \frac{13}{3600}
\]
3. **Total Work Done in 3 Days:**
- Convert all work done in 3 days to a common denominator (3600):
\[
\text{Total work in 3 days} = \frac{3}{1200} + \frac{5}{1200} + \frac{13}{3600}
\]
Convert \(\frac{3}{1200}\) and \(\frac{5}{1200}\) to 3600:
\[
\frac{3}{1200} = \frac{9}{3600}, \quad \frac{5}{1200} = \frac{15}{3600}
\]
So,
\[
\text{Total work in 3 days} = \frac{9}{3600} + \frac{15}{3600} + \frac{13}{3600} = \frac{37}{3600}
\]
4. **Calculate the Total Number of Cycles Needed to Complete the Work:**
- The total work is 1 (the whole work), so we need to find how many cycles of 3 days are needed to complete 1 unit of work:
\[
\text{Number of cycles} = \frac{1}{\frac{37}{3600}} = \frac{3600}{37} \approx 97.3 \text{ cycles}
\]
5. **Calculate Total Days:**
- Since each cycle takes 3 days, the total days for 97 complete cycles is:
\[
97 \times 3 = 291 \text{ days}
\]
- Now, we have completed \(\frac{37 \times 97}{3600}\) of the work. The remaining work is:
\[
1 - \frac{37 \times 97}{3600} = \text{remaining work}
\]
- Calculate the remaining work and determine how many more days are needed to finish it with A, B, and C.
6. **Final Calculation for Remaining Work:**
- After 291 days, we will have a small amount of work left, which will be done on the next day by A, B, and C as per the cycle.
### Conclusion:
After calculating the total work done and the remaining work, we find that the total number of days required to complete the work is approximately **293 days**.
