The following is the age distribution of 1,000 persons working in a large industrial 
house:
Age Group No. of Persons
20-25 30
25-30 160
30-35 210
35-40 180
40-45 145
45-50 105
50-55 70
55-60 60
60-65 40
Due to continuous losses, it is desired to bring down the strength to 30% of the present 
number according to the following scheme:
(i) To retrench the first 15% from the lower group. (ii) To absorb the next 45% in other 
branches. (iii) To make 10% from the highest age group retire permanently, if necessary.
Calculate the age limits of the persons retained
and those to be transferred to other departments. Also find the average age of those 
retained.

Let's analyze the problem step by step:

### Step 1: Calculate the total number of employees currently working.
The total number of employees is 1,000.

### Step 2: Determine the target number of employees to retain.
The company wants to bring down the strength to 30% of the current number.
\[
\text{Target number of employees} = 30\% \text{ of } 1000 = 0.3 \times 1000 = 300 \text{ employees}
\]

### Step 3: Determine how many employees need to be retrenched.
The first 15% are to be retrenched from the lower age group.
\[
\text{Number of employees to retrench} = 15\% \text{ of } 1000 = 0.15 \times 1000 = 150 \text{ employees}
\]

### Step 4: Determine how many employees will be transferred to other departments.
The next 45% will be absorbed into other branches.
\[
\text{Number of employees to transfer} = 45\% \text{ of } 1000 = 0.45 \times 1000 = 450 \text{ employees}
\]

### Step 5: Determine how many employees will retire.
The remaining employees in the highest age group may be asked to retire, but this will only be done if necessary.

### Step 6: Identify the age limits of the employees to be retrenched, transferred, and retained.

**Retrenchment (First 15%):**

- The first 150 employees to be retrenched should come from the lowest age groups:
  - 20-25: 30 employees
  - 25-30: 160 employees (add 120 more to reach 150)
  
The employees retrenched will be from the age group **20-30**.

**Transfer to Other Departments (Next 45%):**

- After retrenching, 850 employees remain. From these, the next 450 should be transferred.
  - From the remaining employees:
    - 25-30: 40 employees (remainder after retrenchment)
    - 30-35: 210 employees
    - 35-40: 180 employees
    - 40-45: 145 employees (remaining = 450)

So, employees to be transferred to other departments are from the age group **25-45**.

**Retention (Remaining 30%):**

- The remaining 300 employees will be retained. These employees are from the age groups:
  - 45-50: 105 employees
  - 50-55: 70 employees
  - 55-60: 60 employees
  - 60-65: 40 employees
  - Total = 275 (Shortfall of 25, so we need to retain 25 more from the previous group).
  
Thus, employees from the age group **40-65** will be retained.

### Step 7: Calculate the average age of the retained employees.

Let's calculate the average age using a weighted average method:

- Age Group 45-50: 105 employees, midpoint = 47.5
- Age Group 50-55: 70 employees, midpoint = 52.5
- Age Group 55-60: 60 employees, midpoint = 57.5
- Age Group 60-65: 40 employees, midpoint = 62.5
- Age Group 40-45: 25 employees, midpoint = 42.5 (to fill the gap).

\[
\text{Average Age} = \frac{(105 \times 47.5) + (70 \times 52.5) + (60 \times 57.5) + (40 \times 62.5) + (25 \times 42.5)}{300}
\]

\[
\text{Average Age} = \frac{4987.5 + 3675 + 3450 + 2500 + 1062.5}{300} = \frac{15675}{300} = 52.25 \text{ years}
\]

### Summary of the Results:
1. **Retained Employees Age Group:** 40-65
2. **Transferred Employees Age Group:** 25-45
3. **Retrenched Employees Age Group:** 20-30
4. **Average Age of Retained Employees:** 52.25 years

---

Do you want more details or have any questions? Here are some related questions you might find interesting:

1. How would the situation change if the percentage of employees to be retained was 40% instead of 30%?
2. What would be the impact on the age distribution if the lowest 20% of employees were retrenched?
3. How would the average age change if only the oldest employees were retrenched instead?
4. Can we determine the median age of the retained employees?
5. What would the distribution look like if the company retrenched by age groups proportionally?

**Tip:** Understanding the concept of weighted averages can help in making more accurate predictions when dealing with diverse data sets.

Explore how to optimize a workforce based on age distribution using statistical analysis. Calculate retained age limits, transferred employees, and average age of retained employees. Learn about weighted averages and percentage calculations. Suitable for college-level analysis and workforce management.

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