A, B and C can do a piece of work in 30, 20 and 10 days respectively. A is assisted by B on one day and by C on the next day, alternately. How long would the work take to finish ?

Options

A. `9(3)/(8)` days
B. `4(8)/(8)` days
C. `8(4)/(13)` days
D. `3(9)/(13)` days

Answer

To solve the problem, we need to determine how long it will take for A, B, and C to complete the work when A is assisted by B on one day and by C on the next day, alternately.

### Step 1: Calculate the work done by A, B, and C in one day.
- A can complete the work in 30 days, so in one day, A does:
\[ \text{Work done by A in one day} = \frac{1}{30} \]

- B can complete the work in 20 days, so in one day, B does:
\[ \text{Work done by B in one day} = \frac{1}{20} \]

- C can complete the work in 10 days, so in one day, C does:
\[ \text{Work done by C in one day} = \frac{1}{10} \]

### Step 2: Calculate the combined work done by A and B on the first day.
On the first day, A is assisted by B:
\[ \text{Work done by A and B in one day} = \frac{1}{30} + \frac{1}{20} \]
To add these fractions, we find a common denominator, which is 60:
\[ \frac{1}{30} = \frac{2}{60} \quad \text{and} \quad \frac{1}{20} = \frac{3}{60} \]
So,
\[ \text{Work done by A and B} = \frac{2}{60} + \frac{3}{60} = \frac{5}{60} = \frac{1}{12} \]

### Step 3: Calculate the combined work done by A and C on the second day.
On the second day, A is assisted by C:
\[ \text{Work done by A and C in one day} = \frac{1}{30} + \frac{1}{10} \]
Again, using a common denominator of 30:
\[ \frac{1}{10} = \frac{3}{30} \]
So,
\[ \text{Work done by A and C} = \frac{1}{30} + \frac{3}{30} = \frac{4}{30} = \frac{2}{15} \]

### Step 4: Calculate the total work done in two days.
Now, we can find the total work done in two days:
\[ \text{Total work in 2 days} = \frac{1}{12} + \frac{2}{15} \]
Finding a common denominator for 12 and 15, which is 60:
\[ \frac{1}{12} = \frac{5}{60} \quad \text{and} \quad \frac{2}{15} = \frac{8}{60} \]
So,
\[ \text{Total work in 2 days} = \frac{5}{60} + \frac{8}{60} = \frac{13}{60} \]

### Step 5: Calculate how many such cycles are needed to complete the work.
Let’s find out how many cycles of 2 days are needed to complete the total work of 1 (or 60 units):
\[ \text{Number of cycles} = \frac{1}{\frac{13}{60}} = \frac{60}{13} \approx 4.615 \]
This means that 4 complete cycles (8 days) will be done, and a fraction of the work will remain.

### Step 6: Calculate the work done in 8 days.
In 8 days (4 cycles), the work done is:
\[ \text{Work done in 8 days} = 4 \times \frac{13}{60} = \frac{52}{60} = \frac{26}{30} = \frac{13}{15} \]

### Step 7: Calculate the remaining work.
The remaining work after 8 days is:
\[ \text{Remaining work} = 1 - \frac{13}{15} = \frac{2}{15} \]

### Step 8: Determine who works on the 9th day.
On the 9th day, A works with B:
\[ \text{Work done on 9th day} = \frac{1}{12} \]

### Step 9: Calculate how much of the remaining work can be done on the 9th day.
We need to find out how much of the remaining work can be completed:
- On the 9th day, A and B together can do \(\frac{1}{12}\) of the work.

### Step 10: Calculate the total work left after the 9th day.
After the 9th day:
- Work done on 9th day = \(\frac{1}{12}\)
- Remaining work after 9th day:
\[ \text{Remaining work} = \frac{2}{15} - \frac{1}{12} \]
Finding a common denominator (60):
\[ \frac{2}{15} = \frac{8}{60} \quad \text{and} \quad \frac{1}{12} = \frac{5}{60} \]
So,
\[ \text{Remaining work} = \frac{8}{60} - \frac{5}{60} = \frac{3}{60} = \frac{1}{20} \]

### Step 11: Determine who works on the 10th day.
On the 10th day, A works with C:
\[ \text{Work done on 10th day} = \frac{1}{30} + \frac{1}{10} = \frac{1}{30} + \frac{3}{30} = \frac{4}{30} = \frac{2}{15} \]

### Step 12: Calculate how much time is needed to finish the remaining work.
To finish the remaining \(\frac{1}{20}\) of work on the 10th day, we need to find out how much of the day is required:
- Work done on the 10th day is \(\frac{2}{15}\), which is more than \(\frac{1}{20}\).

To find the exact time taken to complete \(\frac{1}{20}\):
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Work done on 10th day}} = \frac{\frac{1}{20}}{\frac{2}{15}} = \frac{1}{20} \times \frac{15}{2} = \frac{15}{40} = \frac{3}{8} \text{ days} \]

### Final Calculation:
Total time taken to complete the work:
\[ \text{Total time} = 8 \text{ days} + 1 \text{ day} + \frac{3}{8} \text{ day} = 9 + \frac{3}{8} = 9 \frac{3}{8} \text{ days} \]

Thus, the total time taken to finish the work is:
\[ \boxed{9 \frac{3}{8} \text{ days}} \]