The number of days required by A, B and C to work individually is 6, 12 and 8 respectively. They started a work doing it alternatively. If A has started then followed by B and so on, how many days are needed to complete the whole work?

Options

A. 8
B. 7.5
C. 8.5
D. `9(1)/(2)`

Answer

To solve the problem, we need to find out how many days A, B, and C will take to complete the work when they work alternatively. Let's break it down step by step.

### Step 1: Calculate the work efficiency of A, B, and C
- A can complete the work in 6 days, so A's work efficiency is:
\[ \text{Efficiency of A} = \frac{1}{6} \text{ (work per day)} \]
- B can complete the work in 12 days, so B's work efficiency is:
\[ \text{Efficiency of B} = \frac{1}{12} \text{ (work per day)} \]
- C can complete the work in 8 days, so C's work efficiency is:
\[ \text{Efficiency of C} = \frac{1}{8} \text{ (work per day)} \]

### Step 2: Calculate the total work done in 3 days
Since they work alternatively, the work done in 3 days (A works on Day 1, B on Day 2, and C on Day 3) is:
\[ \text{Total work in 3 days} = \text{Efficiency of A} + \text{Efficiency of B} + \text{Efficiency of C} \]
Substituting the efficiencies:
\[ = \frac{1}{6} + \frac{1}{12} + \frac{1}{8} \]

### Step 3: Find the least common multiple (LCM) to add the fractions
The LCM of 6, 12, and 8 is 24. We convert each fraction:
- For A:
\[ \frac{1}{6} = \frac{4}{24} \]
- For B:
\[ \frac{1}{12} = \frac{2}{24} \]
- For C:
\[ \frac{1}{8} = \frac{3}{24} \]

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

### Step 4: Calculate how many sets of 3 days are needed to complete the work
Since \( \frac{3}{8} \) of the work is done in 3 days, we can find out how many such sets are needed to complete 1 whole work:
\[ \text{Number of sets required} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \]

### Step 5: Calculate the total number of days
Each set takes 3 days, so the total number of days required to complete the work is:
\[ \text{Total days} = 3 \times \frac{8}{3} = 8 \text{ days} \]

### Final Answer
Thus, the total number of days needed to complete the whole work is **8 days**.

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