A and B can complete the work individually in 24 days and 30 days respectively, working 10 hours a day. Work is to be done in two shift. Morning shift lasts for 6 hours and evening shift lasts for 4 hours. On the first day A works in the morning shift while B works in the evening shift. Next day A works in the evening shift while B works in the morning shift and so on. It means they work alternatively with respect to their shifts. Thus they work on this pattern till the work is completed. On which day the work got completed?

Options

A. 26th day
B. 27th day
C. 28th day
D. 30th day

Answer

To solve the problem, we need to determine how much work A and B can complete in a day and then find out how many days it will take to complete the entire work.

### Step-by-Step Solution:

1. **Calculate the Work Done by A and B in One Day:**
- A can complete the work in 24 days. Therefore, A's work rate is:
\[ \text{Work Rate of A} = \frac{1}{24} \text{ of the work per day} \]
- B can complete the work in 30 days. Therefore, B's work rate is:
\[ \text{Work Rate of B} = \frac{1}{30} \text{ of the work per day} \]

2. **Convert Work Rates to Work Done in Hours:**
- Since they work 10 hours a day, we need to find out how much work they can do in 1 hour:
- A's work in 1 hour:
\[ \text{A's work in 1 hour} = \frac{1}{24 \times 10} = \frac{1}{240} \text{ of the work} \]
- B's work in 1 hour:
\[ \text{B's work in 1 hour} = \frac{1}{30 \times 10} = \frac{1}{300} \text{ of the work} \]

3. **Calculate Work Done in Each Shift:**
- **Morning Shift (6 hours):**
- A works in the morning shift for 6 hours:
\[ \text{Work done by A in morning} = 6 \times \frac{1}{240} = \frac{1}{40} \text{ of the work} \]
- **Evening Shift (4 hours):**
- B works in the evening shift for 4 hours:
\[ \text{Work done by B in evening} = 4 \times \frac{1}{300} = \frac{4}{300} = \frac{1}{75} \text{ of the work} \]

4. **Total Work Done in One Day:**
- On the first day, A works in the morning and B works in the evening:
\[ \text{Total work on Day 1} = \frac{1}{40} + \frac{1}{75} \]
- To add these fractions, find a common denominator (which is 600):
\[ \frac{1}{40} = \frac{15}{600}, \quad \frac{1}{75} = \frac{8}{600} \]
\[ \text{Total work on Day 1} = \frac{15}{600} + \frac{8}{600} = \frac{23}{600} \]

5. **Work Done on the Second Day:**
- On the second day, B works in the morning and A works in the evening:
\[ \text{Total work on Day 2} = \frac{1}{75} + \frac{1}{40} = \frac{8}{600} + \frac{15}{600} = \frac{23}{600} \]

6. **Total Work Done in Two Days:**
- In two days, the total work done is:
\[ \text{Total work in 2 days} = \frac{23}{600} + \frac{23}{600} = \frac{46}{600} = \frac{23}{300} \]

7. **Calculate Total Days to Complete the Work:**
- The total work is 1 (whole work). To find out how many such 2-day cycles are needed to complete the work:
\[ \text{Total cycles} = \frac{1}{\frac{23}{300}} = \frac{300}{23} \approx 13.04 \text{ cycles} \]
- This means it will take 13 full cycles (26 days) and a little more work on the next day.

8. **Calculate Remaining Work After 26 Days:**
- Work done in 26 days:
\[ \text{Total work done in 26 days} = 13 \times \frac{23}{300} = \frac{299}{300} \]
- Remaining work:
\[ \text{Remaining work} = 1 - \frac{299}{300} = \frac{1}{300} \]

9. **Determine Work Done on Day 27:**
- On Day 27, A works in the morning (6 hours):
\[ \text{Work done by A on Day 27} = \frac{6}{240} = \frac{1}{40} \]
- Since \(\frac{1}{40} > \frac{1}{300}\), A will complete the remaining work on Day 27.

### Conclusion:
The work will be completed on **Day 27**.