In my office the average age of all the female employees is 21 years and that of male employees is 32 years, where the average age of all the (male and female) employees is 28 years. The total no. of employees in my office could be :

Options

A. 35
B. 78
C. 231
D. 90

Answer

To solve the problem step by step, we will use the concept of alligation to find the total number of employees in the office based on the average ages provided.

### Step-by-Step Solution:

1. **Identify the Average Ages**:
- Average age of female employees (A_f) = 21 years
- Average age of male employees (A_m) = 32 years
- Overall average age of all employees (A) = 28 years

2. **Set Up the Alligation Formula**:
- We will use the alligation method to find the ratio of female to male employees.
- The formula for alligation is:
\[ \text{Ratio of females to males} = \frac{A_m - A}{A - A_f} \]

3. **Calculate the Differences**:
- Difference between the average age of males and the overall average:
\[ A_m - A = 32 - 28 = 4 \]
- Difference between the overall average and the average age of females:
\[ A - A_f = 28 - 21 = 7 \]

4. **Form the Ratio**:
- Now, we can form the ratio of female to male employees:
\[ \text{Ratio of females to males} = \frac{4}{7} \]
- This means for every 4 female employees, there are 7 male employees.

5. **Express the Number of Employees**:
- Let the number of female employees be \(4k\) and the number of male employees be \(7k\).
- Therefore, the total number of employees (T) is:
\[ T = 4k + 7k = 11k \]

6. **Determine Possible Values for k**:
- The total number of employees must be a multiple of 11 (since \(T = 11k\)).
- We need to check the options given in the question to see which ones are multiples of 11.

7. **Check Each Option**:
- Option 1: 35 (not a multiple of 11)
- Option 2: 78 (not a multiple of 11)
- Option 3: 231 (is a multiple of 11, since \(231 \div 11 = 21\))
- Option 4: (not provided, but assumed not a multiple of 11)

8. **Conclusion**:
- The only option that is a multiple of 11 is 231.
- Therefore, the total number of employees in the office could be **231**.