Arun and Satyam can complete a work individually in ,12 working days and 15 working days respectively with their full efficiencies. Arun does work only on Monday, Wednesday and Friday while Satyam does the work on Tuesday, Thursday and Saturday. Sunday is always off. But Arun and Satyam both works with half of their efficiencies on Friday and Saturday respectively. If Arun started the work on 1st January which falls on Monday followed by Satyam on the next day and so on (i.e., they work collectively in alternate days), then on which day work will be completed?

Options

A. Tuesday
B. Wednesday
C. Thursday
D. Friday

Answer

To solve the problem, we need to determine how much work Arun and Satyam complete in a week and then find out when the total work will be completed. Let's break it down step by step.

### Step 1: Calculate the daily work efficiency of Arun and Satyam.

- Arun can complete the work in 12 days. Therefore, his work efficiency is:
\[ \text{Arun's efficiency} = \frac{1}{12} \text{ of the work per day} \]

- Satyam can complete the work in 15 days. Therefore, his work efficiency is:
\[ \text{Satyam's efficiency} = \frac{1}{15} \text{ of the work per day} \]

### Step 2: Determine their effective work on specific days.

- Arun works on Monday, Wednesday, and Friday:
- On Monday and Wednesday, he works at full efficiency:
\[ \text{Work on Monday} = \frac{1}{12}, \quad \text{Work on Wednesday} = \frac{1}{12} \]
- On Friday, he works at half efficiency:
\[ \text{Work on Friday} = \frac{1}{2} \times \frac{1}{12} = \frac{1}{24} \]

- Satyam works on Tuesday, Thursday, and Saturday:
- On Tuesday and Thursday, he works at full efficiency:
\[ \text{Work on Tuesday} = \frac{1}{15}, \quad \text{Work on Thursday} = \frac{1}{15} \]
- On Saturday, he works at half efficiency:
\[ \text{Work on Saturday} = \frac{1}{2} \times \frac{1}{15} = \frac{1}{30} \]

### Step 3: Calculate the total work done in one week.

Now, let's sum up the work done in one complete week (Monday to Sunday):

- Total work in one week:
\[ \text{Total work} = \left(\frac{1}{12} + \frac{1}{15} + \frac{1}{12} + \frac{1}{15} + \frac{1}{24} + \frac{1}{30}\right) \]

Calculating each term:
- Convert to a common denominator (the least common multiple of 12, 15, 24, and 30 is 120):
\[ \frac{1}{12} = \frac{10}{120}, \quad \frac{1}{15} = \frac{8}{120}, \quad \frac{1}{24} = \frac{5}{120}, \quad \frac{1}{30} = \frac{4}{120} \]

Now substituting these values:
\[ \text{Total work} = \left(\frac{10}{120} + \frac{8}{120} + \frac{10}{120} + \frac{8}{120} + \frac{5}{120} + \frac{4}{120}\right) = \frac{45}{120} = \frac{3}{8} \]

### Step 4: Determine how many weeks are needed to complete the work.

Since the total work is 1 (whole work), we need to find out how many weeks it takes to complete the work:
\[ \text{Weeks required} = \frac{1}{\frac{3}{8}} = \frac{8}{3} \text{ weeks} \approx 2.67 \text{ weeks} \]

### Step 5: Calculate the total days to complete the work.

Since 2.67 weeks equals 2 weeks and 4 days (0.67 weeks is approximately 4 days), we can calculate the completion date:
- 2 weeks from January 1st (Monday) is January 15th (Monday).
- Adding 4 more days:
- January 16th (Tuesday)
- January 17th (Wednesday)
- January 18th (Thursday)
- January 19th (Friday)

### Conclusion

The work will be completed on **Friday, January 19th**.