Problem 32 - A Complete Resource Book in Mathematics for JEE Main

Question

The average salary of male employees in a firm was Rs. 520 and that of females was Rs. 420 . The mean salary of all the employees was Rs. 500 . The percentage of male employees is (A) 80 (B) 60 (B) 40 (D) 20

Step-by-Step Solution

Answer

The percentage of male employees is 80%.

Step 1: Define Variables

Let's denote the number of male employees as \( m \) and the number of female employees as \( f \). We know the average salary for all males is Rs. 520 and for all females, it's Rs. 420.

Step 2: Establish the Mean Salary Equation

We know that the total salary for all employees is the sum of the salaries of all male and female employees. The mean salary for all employees is given as Rs. 500. Therefore, we can write the equation for the mean salary:\[\frac{520m + 420f}{m + f} = 500\]

Step 3: Simplify the Mean Salary Equation

To eliminate the denominator, multiply the whole equation by \( m + f \):\[ 520m + 420f = 500(m + f) \] Expand the right side of the equation:\[ 520m + 420f = 500m + 500f \]

Step 4: Rearrange and Solve for Ratios

Rearrange the terms to group variables together:\[ 520m - 500m = 500f - 420f \] This simplifies to:\[ 20m = 80f \] Divide both sides by 20:\[ m = 4f \]This implies the ratio of males to females is 4:1.

Step 5: Calculate Percentage of Male Employees

Since the ratio \( m : f = 4 : 1 \), it means for every 5 employees (4 males + 1 female), 4 are male. Therefore, the percentage of male employees is:\[ \frac{4}{4+1} \times 100 = 80\% \]

Key Concepts

Ratio and Proportion

Sometimes when dealing with different groups, like male and female employees in a company, we use ratios to express the relationship between their numbers. A ratio indicates how many times one number contains another. In this exercise, the number of male employees is 4 times the number of female employees. We express this as a ratio of 4:1.

Ratio and proportion are powerful tools for comparing quantities that are interrelated. To find a ratio, break down the relationship into simpler parts. If you can express quantities as multiples of each other, you've established a ratio.

To solve practical problems, employ the following steps:

• Identify the quantities to compare.
• Determine the relationship between them.
• Express the relationship as a ratio.

In our exercise, since for every 4 males, there is 1 female, the male to female ratio is 4:1.

Percentage Calculation

Percentages are helpful for understanding parts of a whole in percentage terms. A percentage represents a number out of 100. This exercise seeks to find what percentage of the employees are male.

Once you have a ratio, converting it into a percentage is straightforward. Take the part you are interested in (in this case, male employees) and divide by the total number of parts, then multiply by 100.

In this case, the male employees take up 4 parts for every total of 5 parts when combining males and females. Therefore, the percentage calculation for male employees is:

• Calculate the proportion: \( \frac{4}{4+1} \)
• Convert to percentage: \( \frac{4}{5} \times 100 = 80\% \)

Working through these steps will help clarify how many employees out of every 100 are male.

Mean Salary Equation

The mean salary equation is a way to determine the average salary given by taking the total of all salaries and dividing by the number of employees. With this exercise, we want to balance the total salary contributions of both male and female employees to reach the overall average.

Here's how the mean salary equation works:

• Establish the salaries: Males earn Rs. 520 and females Rs. 420 per head.
• Create the equation for total salaries: \( 520m + 420f \).
• Set up the equation for the mean salary by dividing total salaries by total employees: \[ \frac{520m + 420f}{m + f} = 500 \]

This equation helps determine how the salaries contribute to the overall average.

To simplify, multiply through by the total number of employees, rearrange to group like terms, and solve for the unknowns. By achieving a simplified relationship between m and f, such as \( m = 4f \), you can easily deduce the ratios and subsequently the percentage of employees for each category.