The average salary of male employees in a firm was Rs. 5200 and that of female was Rs. 4200 . The mean salary of all the employees was Rs. 5000 . What is the percentage of female employees ?

Options

A. `80%`
B. `20%`
C. `40%`
D. `30%`

Answer

To solve the problem, we need to find the percentage of female employees in the firm based on the given average salaries. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- \( M \) = number of male employees
- \( F \) = number of female employees

### Step 2: Calculate Total Salaries
From the information provided:
- Average salary of male employees = Rs. 5200
- Average salary of female employees = Rs. 4200
- Mean salary of all employees = Rs. 5000

The total salary of male employees can be expressed as:
\[ \text{Total salary of males} = 5200 \times M \]

The total salary of female employees can be expressed as:
\[ \text{Total salary of females} = 4200 \times F \]

### Step 3: Calculate Overall Total Salary
The total salary of all employees can be expressed as:
\[ \text{Total salary of all employees} = \text{Total salary of males} + \text{Total salary of females} \]
This can be written as:
\[ 5200M + 4200F \]

### Step 4: Set Up the Equation
Since the mean salary of all employees is given as Rs. 5000, we can express this as:
\[ \frac{5200M + 4200F}{M + F} = 5000 \]

### Step 5: Cross Multiply to Eliminate the Fraction
Cross multiplying gives us:
\[ 5200M + 4200F = 5000(M + F) \]

### Step 6: Expand the Right Side
Expanding the right side, we get:
\[ 5200M + 4200F = 5000M + 5000F \]

### Step 7: Rearrange the Equation
Rearranging the equation to group like terms:
\[ 5200M - 5000M = 5000F - 4200F \]
This simplifies to:
\[ 200M = 800F \]

### Step 8: Simplify the Ratio
Dividing both sides by 200 gives us:
\[ M = 4F \]

### Step 9: Express Total Employees
Now we can express the total number of employees:
\[ M + F = 4F + F = 5F \]

### Step 10: Calculate the Percentage of Female Employees
The percentage of female employees can be calculated as:
\[ \text{Percentage of female employees} = \left( \frac{F}{M + F} \right) \times 100 = \left( \frac{F}{5F} \right) \times 100 = \frac{1}{5} \times 100 = 20\% \]

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

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