A and B can complete a piece of work in 4 days and 8 days, respctively. They work on alternate days and A starts the work. In how work in may days will the work be completed

Options

A. 3
B. 4
C. 5
D. 6

Explanation

To solve the problem step by step, we will determine how much work A and B can do in one day, and then calculate how many days it will take for them to complete the work when they work alternately.

### Step-by-Step Solution:

1. **Determine the work done by A in one day:**
- A can complete the work in 4 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{4} \]

2. **Determine the work done by B in one day:**
- B can complete the work in 8 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{8} \]

3. **Calculate the total work done in 2 days (one cycle of A and B working):**
- On the first day, A works and completes \(\frac{1}{4}\) of the work.
- On the second day, B works and completes \(\frac{1}{8}\) of the work.
- Therefore, the total work done in 2 days is:
\[ \text{Total work in 2 days} = \frac{1}{4} + \frac{1}{8} \]
- To add these fractions, we need a common denominator:
\[ \frac{1}{4} = \frac{2}{8} \quad \text{(converting to a common denominator)} \]
- Now, add the fractions:
\[ \text{Total work in 2 days} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8} \]

4. **Determine how many cycles are needed to complete the work:**
- Since \(\frac{3}{8}\) of the work is done in 2 days, we need to find out how many such cycles are needed to complete the entire work (1 unit of work).
- Let \(n\) be the number of cycles (2 days each) needed:
\[ n \cdot \frac{3}{8} \geq 1 \]
- Solving for \(n\):
\[ n \geq \frac{8}{3} \approx 2.67 \]
- This means we need 3 cycles (since we can't have a fraction of a cycle).

5. **Calculate the work done after 3 cycles (6 days):**
- In 6 days (3 cycles), the work done will be:
\[ 3 \cdot \frac{3}{8} = \frac{9}{8} \]
- This means the work is complete after 6 days, and we have \(\frac{1}{8}\) of the work remaining.

6. **Determine how much more time is needed to complete the remaining work:**
- On the 7th day, A works again and completes \(\frac{1}{4}\) of the work. Since only \(\frac{1}{8}\) of the work is left, A will complete the remaining work on the 7th day.
- Thus, the total time taken to complete the work is:
\[ 6 \text{ days} + 1 \text{ day} = 7 \text{ days} \]

### Conclusion:
The total time taken to complete the work is **7 days**.