Problem 19 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

In a class of 100 students there are 70 boys whose average marks in a subject are 75 . If the average marks of the complete class is 72 , then what is the average of the girls? [2002] (a) 73 (b) 65 (c) 68 (d) 74

Step-by-Step Solution

Answer

The average marks of the girls is 65.

Step 1: Find Total Marks of Boys

The average marks of 70 boys is 75, so the total marks of boys can be calculated by multiplying the number of boys with their average marks: \[ \text{Total Marks of Boys} = 70 \times 75 = 5250 \]

Step 2: Calculate Total Marks for the Whole Class

The total marks of the entire class is determined by multiplying the number of students by the average marks of the class:\[ \text{Total Marks of Class} = 100 \times 72 = 7200 \]

Step 3: Determine Total Marks of Girls

Subtract the total marks of boys from the total marks of the class to find the total marks obtained by the girls: \[ \text{Total Marks of Girls} = 7200 - 5250 = 1950 \]

Step 4: Find Number of Girls in the Class

Since there are a total of 100 students and 70 are boys, the number of girls can be calculated as follows:\[ \text{Number of Girls} = 100 - 70 = 30 \]

Step 5: Calculate Average Marks of Girls

The average marks of the girls can be found by dividing the total marks obtained by the girls by the number of girls:\[ \text{Average Marks of Girls} = \frac{1950}{30} = 65 \]

Key Concepts

Arithmetic Mean

The arithmetic mean, often referred to as the average, is a fundamental concept in statistics and mathematics. It is used to find a central value of a set of numbers. In this problem, we observed the average marks of boys, the whole class, and subsequently calculated the average for girls. Let’s understand how the arithmetic mean works:

- It's calculated by adding all the numbers in a data set and dividing by the number of data points.- Formula: \( \text{Arithmetic Mean} = \frac{\text{Sum of all values}}{\text{Total number of values}} \)- For instance, if the average marks of boys is 75, and there are 70 boys, the total marks of boys would be: \( 70 \times 75 = 5250 \). The arithmetic mean for girls was found using similar logic.

Using arithmetic mean helps in analyzing the performance across different subsets such as defining average performance levels for different groups in a class.

Class Performance

Class performance is an insightful way to evaluate the learning and development of students in an educational setting. It provides a snapshot of how students are doing by averaging their marks. In our problem, the class of 100 students had an overall average score of 72. This average gives teachers and students an idea of where they stand collectively.

- The class performance can guide educators to identify areas needing improvement and adapt teaching methods accordingly. - Comparing the average scores of boys and girls helps highlight performance trends. - A higher or lower average for sub-groups can lead to targeted interventions and resource allocation.

Overall, class performance evaluation through averages is vital for educational advancements, aiding teachers in promoting better learning outcomes.

Mathematical Problem Solving

Mathematical problem solving involves using mathematical techniques and strategies to unravel solutions systematically. This problem is a great example to illustrate how steps and logical reasoning are applied in math.

- Establishing the known quantities: We knew the average marks of boys and the total class. - Determining unknowns by breaking the problem into smaller steps: Total marks of girls were found by subtracting the boys' total from the class total. - Using arithmetic operations to enact calculations: Multiplication was used to find total marks, and division to find averages.
Effective problem solvers follow structured approaches, using mathematical tools or logic sequences tailored to specific needs. Practice makes proficiency in translating real-world problems into mathematical terms and successfully solving them.