A can do a work in 6 days, B can do it in 8 days and C can complete it in 12 days. If A starts the work and they work in alternatively, then in how many days, the work will be completed?

Options

A. `6(3)/(5)`
B. `7(2)/(3)`
C. `5(2)/(5)`
D. `6(2)/(3)`

Answer

To solve the problem, we need to determine how many days A, B, and C will take to complete the work when they work alternately. Here's the step-by-step solution:

### Step 1: Determine the work done by A, B, and C in one day.
- A can complete the work in 6 days, so A's work rate is \( \frac{1}{6} \) of the work per day.
- B can complete the work in 8 days, so B's work rate is \( \frac{1}{8} \) of the work per day.
- C can complete the work in 12 days, so C's work rate is \( \frac{1}{12} \) of the work per day.

### Step 2: Calculate the total work done in a 3-day cycle.
In a 3-day cycle:
- Day 1: A works and completes \( \frac{1}{6} \) of the work.
- Day 2: B works and completes \( \frac{1}{8} \) of the work.
- Day 3: C works and completes \( \frac{1}{12} \) of the work.

Now, we find the total work done in these 3 days:
\[ \text{Total work in 3 days} = \frac{1}{6} + \frac{1}{8} + \frac{1}{12} \]

### Step 3: Find a common denominator and sum the fractions.
The least common multiple (LCM) of 6, 8, and 12 is 24. We convert each fraction:
- \( \frac{1}{6} = \frac{4}{24} \)
- \( \frac{1}{8} = \frac{3}{24} \)
- \( \frac{1}{12} = \frac{2}{24} \)

Now, adding these fractions:
\[ \text{Total work in 3 days} = \frac{4}{24} + \frac{3}{24} + \frac{2}{24} = \frac{9}{24} = \frac{3}{8} \]

### Step 4: Calculate how many such cycles are needed to complete the work.
To complete the entire work (1 unit of work), we need to find how many cycles of 3 days are required:
\[ \text{Number of cycles} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \]
This means they will complete the work in \( 2 \) full cycles (which is \( 6 \) days) and will have some work remaining.

### Step 5: Calculate the remaining work after 2 cycles.
In 2 cycles (6 days), the work done is:
\[ \text{Work done in 6 days} = 2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4} \]
The remaining work is:
\[ \text{Remaining work} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 6: Determine who will work next and how long it will take to finish the remaining work.
On the 7th day, A will work again. A can complete \( \frac{1}{6} \) of the work in one day. To find out how much time A will take to complete the remaining \( \frac{1}{4} \) of the work:
\[ \text{Time taken by A} = \frac{\text{Remaining work}}{\text{A's work rate}} = \frac{\frac{1}{4}}{\frac{1}{6}} = \frac{1}{4} \times 6 = \frac{6}{4} = 1.5 \text{ days} \]

### Step 7: Calculate the total time taken.
Total time taken = Time for 2 cycles (6 days) + Time taken by A (1.5 days):
\[ \text{Total time} = 6 + 1.5 = 7.5 \text{ days} \]

### Final Answer:
The work will be completed in **7.5 days**.