To solve the problem, we need to determine how much work A, B, and C can complete together, given their individual work rates, and how the work progresses over the days according to the conditions provided.
### Step 1: Determine the work rates of A, B, and C.
- A can complete the work in 10 days, so A's work rate is \( \frac{1}{10} \) of the work per day.
- B can complete the work in 15 days, so B's work rate is \( \frac{1}{15} \) of the work per day.
- C can complete the work in 30 days, so C's work rate is \( \frac{1}{30} \) of the work per day.
### Step 2: Calculate the combined work rate of A, B, and C.
On the days when B and C assist A (every third day), we need to calculate their combined work rate:
- Combined work rate of A, B, and C on the third day:
\[
\text{Work rate} = \frac{1}{10} + \frac{1}{15} + \frac{1}{30}
\]
To add these fractions, we need a common denominator. The least common multiple of 10, 15, and 30 is 30.
\[
\frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}, \quad \frac{1}{30} = \frac{1}{30}
\]
So,
\[
\text{Combined work rate} = \frac{3}{30} + \frac{2}{30} + \frac{1}{30} = \frac{6}{30} = \frac{1}{5}
\]
### Step 3: Calculate the work done in a 3-day cycle.
- On the first two days, only A works:
\[
\text{Work done by A in 2 days} = 2 \times \frac{1}{10} = \frac{2}{10} = \frac{1}{5}
\]
- On the third day, A, B, and C work together:
\[
\text{Work done on the third day} = \frac{1}{5}
\]
- Total work done in 3 days:
\[
\text{Total work in 3 days} = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}
\]
### Step 4: Determine how many cycles are needed to complete the work.
To complete the entire work (1 unit of work), we need to find out how many 3-day cycles are required:
- In each cycle of 3 days, \( \frac{2}{5} \) of the work is completed.
- Let \( n \) be the number of cycles needed:
\[
n \times \frac{2}{5} = 1 \implies n = \frac{5}{2} = 2.5
\]
This means we need 2 complete cycles (6 days) and a part of another cycle.
### Step 5: Calculate the remaining work after 2 cycles.
- Work done in 2 cycles (6 days):
\[
2 \times \frac{2}{5} = \frac{4}{5}
\]
- Remaining work:
\[
1 - \frac{4}{5} = \frac{1}{5}
\]
### Step 6: Calculate how much time is needed to complete the remaining work.
- On the 7th day, A works alone:
\[
\text{Work done by A on 7th day} = \frac{1}{10}
\]
- On the 8th day, A works alone again:
\[
\text{Work done by A on 8th day} = \frac{1}{10}
\]
- On the 9th day, A, B, and C work together:
\[
\text{Work done on 9th day} = \frac{1}{5}
\]
After the 8th day, A has done \( \frac{1}{5} \) of the work, and on the 9th day, they complete the remaining \( \frac{1}{5} \) of the work.
### Total Days Taken
- Total days = 6 days (2 cycles) + 2 days (7th and 8th) + 1 day (9th) = 9 days.
Thus, the total time taken to complete the work is **9 days**.
