In a firm, the average salary of male employees is Rs 5200 and the average salary of female employees is Rs 4200. The overall average salary of all employees together is Rs 5000. What percentage of the employees are female?

Introduction / Context:
This question involves weighted averages and percentage composition of a group. The firm has male and female employees with different average salaries and a combined overall average. From these averages, we must determine what fraction or percentage of the employees are female. Such problems are very important in data interpretation, mixture, and statistics type questions where groups are combined with different averages or concentrations.

Given Data / Assumptions:

• Average salary of male employees = Rs 5200.
• Average salary of female employees = Rs 4200.
• Overall average salary of all employees = Rs 5000.
• Let the number of male employees be M and the number of female employees be F.
• We need to find the percentage of employees who are female, that is (F / (M + F)) * 100.

Concept / Approach:
The total salary paid to males is 5200 * M and to females is 4200 * F. The overall average of 5000 means that the combined total salary is 5000 * (M + F). So we can write an equation: 5200M + 4200F = 5000(M + F). Solving this equation for the ratio F:M allows us to compute the fraction of female employees. The approach is directly based on the definition of weighted average.

Step-by-Step Solution:
Step 1: Let number of males be M and number of females be F. Step 2: Total salary of males = 5200M. Step 3: Total salary of females = 4200F. Step 4: Total number of employees = M + F and overall average salary is Rs 5000. Step 5: So total combined salary = 5000(M + F). Step 6: Form equation: 5200M + 4200F = 5000(M + F). Step 7: Expand right side: 5200M + 4200F = 5000M + 5000F. Step 8: Rearrange terms: 5200M − 5000M = 5000F − 4200F. Step 9: This gives 200M = 800F, so dividing both sides by 200 gives M = 4F. Step 10: Therefore, the ratio M:F = 4:1 and F:M = 1:4. Step 11: Fraction of employees who are female = F / (M + F) = 1 / (4 + 1) = 1/5. Step 12: Percentage of female employees = (1/5) * 100% = 20%.

Verification / Alternative check:
Assume there are 4 males and 1 female as per the ratio. Total salary of males = 4 * 5200 = 20800. Total salary of females = 1 * 4200 = 4200. Combined salary = 20800 + 4200 = 25000. Average for all 5 employees = 25000 / 5 = 5000, which matches the given overall average. So the assumed numbers follow the condition perfectly and confirm that 20% of the employees are female.

Why Other Options Are Wrong:

• 80% female would imply far more females than males, which would pull the overall average salary much closer to Rs 4200, not Rs 5000.
• 40% or 30% are intermediate incorrect guesses that do not satisfy the weighted average equation and would give a combined average different from Rs 5000.

Common Pitfalls:
Many students attempt to handle this type of problem intuitively and misjudge the required female percentage. Another error is miswriting the weighted average equation, for instance using 5000M + 5000F on the wrong side or forgetting that the average must be multiplied by the total number of employees. Some aspirants also confuse the ratio M:F with the percentage directly, without converting properly. Writing the equation carefully and solving step by step avoids such mistakes.

Final Answer:
The percentage of female employees in the firm is 20%.