To solve the problem, we need to determine how many days A can complete the work if he is assisted by B and C on every third day. Let's break it down step by step.
### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 20 days, so his work rate is:
\[
\text{Work rate of A} = \frac{1}{20} \text{ (work per day)}
\]
- B can complete the work in 40 days, so his work rate is:
\[
\text{Work rate of B} = \frac{1}{40} \text{ (work per day)}
\]
- C can complete the work in 80 days, so his work rate is:
\[
\text{Work rate of C} = \frac{1}{80} \text{ (work per day)}
\]
### Step 2: Calculate the combined work rate of A, B, and C
On the third day, A is assisted by B and C. Therefore, the combined work rate of A, B, and C is:
\[
\text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} + \text{Work rate of C}
\]
Calculating this:
\[
\text{Combined work rate} = \frac{1}{20} + \frac{1}{40} + \frac{1}{80}
\]
To add these fractions, we need a common denominator. The least common multiple of 20, 40, and 80 is 80:
\[
\text{Combined work rate} = \frac{4}{80} + \frac{2}{80} + \frac{1}{80} = \frac{7}{80}
\]
### Step 3: Calculate the work done in a cycle of 3 days
In the first two days, only A works:
\[
\text{Work done by A in 2 days} = 2 \times \frac{1}{20} = \frac{2}{20} = \frac{1}{10}
\]
On the third day, A, B, and C work together:
\[
\text{Work done by A, B, and C on the 3rd day} = \frac{7}{80}
\]
Thus, the total work done in 3 days is:
\[
\text{Total work in 3 days} = \frac{1}{10} + \frac{7}{80}
\]
Converting \(\frac{1}{10}\) to a fraction with a denominator of 80:
\[
\frac{1}{10} = \frac{8}{80}
\]
So,
\[
\text{Total work in 3 days} = \frac{8}{80} + \frac{7}{80} = \frac{15}{80} = \frac{3}{16}
\]
### Step 4: Determine how many cycles are needed to complete the work
To complete 1 unit of work, we can find out how many cycles of 3 days are needed:
\[
\text{Number of cycles} = \frac{1}{\frac{3}{16}} = \frac{16}{3} \approx 5.33 \text{ cycles}
\]
This means it takes 5 complete cycles (15 days) and a portion of the next cycle.
### Step 5: Calculate the work done in 15 days
In 15 days, the total work done is:
\[
\text{Total work done in 15 days} = 5 \times \frac{3}{16} = \frac{15}{16}
\]
### Step 6: Determine the remaining work
The remaining work after 15 days is:
\[
\text{Remaining work} = 1 - \frac{15}{16} = \frac{1}{16}
\]
### Step 7: Calculate the work done on the 16th and 17th days
On the 16th and 17th days, A works alone:
- Work done by A in 2 days:
\[
\text{Work done by A in 2 days} = 2 \times \frac{1}{20} = \frac{2}{20} = \frac{1}{10}
\]
Now, we need to check if the remaining work can be completed in these 2 days:
- Convert \(\frac{1}{10}\) to a fraction with a denominator of 16:
\[
\frac{1}{10} = \frac{1.6}{16}
\]
Since \(\frac{1.6}{16} > \frac{1}{16}\), A can complete the remaining work on the 17th day.
### Conclusion
Thus, the total time taken to complete the work is:
\[
15 \text{ days} + 1 \text{ day} = 16 \text{ days}
\]
### Final Answer
**A can complete the work in 16 days.**
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