Mr. Ankit Bansal operates a roofing company in Jaipur and has been asked by the local government authority of Jaipur to repair the roofs of several of their properties damaged in a recent storm. The job must be completed during the next four weeks (twenty working days), and government authority of Jaipur has offered Rs.1,65,000 for the job. Mr. Bansal has estimated that the job requires seventy-five work days, but he can only use his regular three workers for the job because it is a very busy period for the industry as a whole. Fortunately, Mr. Bansal's son, Vinay. can take time off from his regular job (paying Rs. 1100 per day) to help complete the work. Mr. Bansal has estimated that, for his regular employees, the cost per work day is Rs.1500. This consists of a wage of Rs.1000 (which is only paid if the employee is working) and Rs. 500 in contributions to the government (which are paid annually regardless of how many days the employees work). How mnay workers should join and what is the opportunity cost 

To determine how many workers should join and calculate the opportunity cost, we can break down the problem step by step.

### 1. **Work Requirement and Workers Available:**
   - The job requires **75 workdays**.
   - Mr. Bansal has **3 regular workers** available for **20 days**.

### 2. **Work Done by Regular Workers:**
   - Each worker can work 20 days.
   - Total workdays completed by the 3 regular workers: 
     \[
     \text{Total Workdays} = 3 \times 20 = 60 \text{ workdays}
     \]
   - Remaining workdays needed: 
     \[
     \text{Remaining Workdays} = 75 - 60 = 15 \text{ workdays}
     \]

### 3. **Need for Additional Worker (Vinay):**
   - Since there are 15 workdays left and Mr. Bansal’s regular workers have no more time available, Vinay, Mr. Bansal's son, should join to complete the remaining work.

### 4. **Opportunity Cost Calculation:**
   - Vinay’s regular job pays him Rs. 1100 per day.
   - By taking time off to help with the roofing job, he forgoes this income.
   - The opportunity cost of Vinay helping with the roofing job for 15 days is:
     \[
     \text{Opportunity Cost} = 15 \times 1100 = Rs. 16,500
     \]

### **Conclusion:**
- **Vinay should join to complete the remaining 15 workdays.**
- The **opportunity cost** of Vinay joining the roofing work is **Rs. 16,500**.

Would you like any additional details or have any questions?

Here are 5 related questions you might consider:
1. What would be the total cost incurred if Vinay doesn’t join and extra workers are hired instead?
2. How does the cost of hiring an external worker compare to the opportunity cost of Vinay?
3. What is the total expected profit Mr. Bansal would earn after paying for all costs including Vinay’s opportunity cost?
4. If Vinay works for fewer days, how does it affect the project completion and overall costs?
5. What happens if the government authority of Jaipur decides to extend the deadline?

**Tip:** Always consider the opportunity cost when deciding to allocate resources, as it represents the potential benefits lost when choosing one alternative over another.

Explore how to optimize a roofing job using linear programming and calculate opportunity costs. This problem involves determining the optimal number of workers and assessing the cost implications of different workforce scenarios. Suitable for higher education and professionals.

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