Question

The average age of boys in class is \(16.66\), while the average age of girls is \(18.75 .\) Thus the average age of all the 40 students of the class is \(17.5\). If the difference between the no. of boys and girls is 8 , then the no. of girls in the class is : (a) 12 (b) 16 (c) 18 (d) data insufficient

Short answer

Based on the given information, we found out that there are 16 girls in the class. Therefore, the correct answer is (b) 16.

Step-by-step solution

Write down the given information

We have the following information: - Average age of boys: \(16.66\) - Average age of girls: \(18.75\) - Average age of all students: \(17.5\) - Total number of students: \(40\) - Difference between number of boys and girls: \(8\)

Define the variables

Let's define the variables: - \(B\): Number of boys in the class - \(G\): Number of girls in the class We are given that \(B - G = 8\) and \(B + G = 40\).

Solve the system of equations

We need to solve the following system of equations: 1. \(B - G = 8\) 2. \(B + G = 40\) Adding equation (1) and equation (2), we get: \(2B = 48\), which means that \(B = 24\). Now, substituting the value of \(B\) in equation (2), we have: \(24 + G = 40\), which means that \(G = 16\).

Check the answer with the given options

Since the number of girls in the class is \(16\), let's check this value with the given options: (a) 12 (b) 16 (c) 18 (d) data insufficient The correct option is (b) 16, as it matches our calculated value.

Key concepts

System of Equations

The system of equations is a set of two or more equations that have common variables. Solving a system means finding the values for the variables that make all the equations true simultaneously. In our average age problem, we use a system of linear equations to determine the number of boys (B) and the number of girls (G) in a class.

Linear equations are the simplest type of system where each equation can be represented as a line on a graph. The solution of a system of linear equations is the point where these lines intersect. For the given problem, we start with equations that represent the difference and the total of boys and girls:
1. The difference in the number of boys and girls is 8: \(B - G = 8\).2. The sum of boys and girls equals the total number of students, which is 40: \(B + G = 40\).
By using algebraic methods such as substitution or elimination (in this case, addition), the values of B and G can be calculated. These techniques enable students to arrive at the precise number of boys and girls in the class, illustrating the practical application of a system of equations.

Arithmetic Mean

The arithmetic mean is one of the most fundamental concepts of statistics, often referred to as the average. It is computed by adding all the numbers in a set and then dividing by the count of those numbers. In our average age problem, separate arithmetic means are given for boys, girls, and the entire class. These means are essential to creating a system of equations to solve for the number of boys and girls.

The formula for arithmetic mean is: \[\text{Arithmetic Mean} = \frac{\text{Sum of all terms}}{\text{Number of terms}}\].
In a classroom setting, if we have a certain number of boys with an average age of \(16.66\) years and a certain number of girls with an average age of \(18.75\) years, we can calculate the total ages for boys and girls when multiplied by their respective counts. Combined with the total average age for the entire class, this data sets up a situation where we can apply the mean to deduce the number of students split between genders.

Quantitative Aptitude

Quantitative aptitude refers to the ability to handle numerical and logical problem solving. This aptitude is crucial for performance in competitive exams, academics, and various professional fields. In the context of the average age problem, quantitative aptitude encompasses understanding and applying mathematical concepts to derive a solution.

To excel in problems dealing with quantitative aptitude, a grasp of the following is essential:

• Basic arithmetic operations and properties
• Algebraic equation solving, like the system of equations used in the problem
• Understanding of averages and the concept of the arithmetic mean
• Analytical skills to translate word problems into mathematical expressions
• Logical reasoning to determine which equations or formulas apply

With a strong foundation in these areas, students can adeptly navigate through the complexities of quantitative questions, such as determining the number of girls in a class based on average ages and the total student count. Such problems not only test the conceptual understanding but also the application of those concepts in real-world scenarios.