The Relation Between Average Weight and Individual Weight Changes in a Class

Understanding the relationship between the average weight of a class and the individual weight changes can be crucial in various real-world applications. This article aims to explore the concept through a specific problem involving a class of students and demonstrate how to calculate the weight of a new student after an existing one is replaced. We will use mathematical reasoning to solve the problem and provide a step-by-step guide for similar scenarios.

Introduction

The average weight of a group of objects (or individuals) can change when a specific object (or individual) is added or removed. This change can be quantified and computed using basic algebra. In this case, the problem involves a class of students, where the average weight of the class is impacted by the replacement of one student.

The Problem

A class of 20 students has an average weight. When a student weighing 40 kg is replaced by a new student, the class average weight decreases by 500 grams (0.5 kg).

Solution

Let's denote the original average weight of the 20 students as A (in kg).

Step 1: Calculate the total weight of the original 20 students

The total weight of the 20 students is given by:

Total weight 20A

Step 2: Calculate the new average weight

After the replacement, the average weight of the new group of students is:

New average weight A - 0.5 kg

The total weight of the new group of students is then:

New total weight 20A - 0.5 * 20 20A - 10 kg

Step 3: Relate the total weights

When the student weighing 40 kg is removed and the new student is added, the total weight changes according to:

New total weight 20A - 40 weight of new boy

Equating the two expressions for the new total weight:

20A - 10 20A - 40 weight of new boy

Simplifying the equation:

30 weight of new boy

Hence, the weight of the new boy is 30 kg.

Alternative Solutions and Real-Life Applications

Similar problems can arise in various fields such as statistics, economics, and sports. Understanding these concepts can help in making more informed decisions based on the average metrics of a group.

The problem can also be modified to involve different numbers of students and different weight changes. For example:

When a student weighing 25 kg leaves a class of 30 students, and the average weight decreases by 200 grams (0.2 kg) when a new student joins, what is the weight of the new student?

By applying the same method:

Total weight of 30 students initially 30 * 25 kg 750 kg

Weight of 29 students after one student leaves 750 kg - 25 kg 725 kg

Total weight after the new student joins 725 kg weight of new student - 0.2 * 30 725 kg weight of new student - 6 kg

New total weight 750 kg - 0.2 * 30 750 kg - 6 kg 744 kg

Equating and solving:

725 kg weight of new student - 6 kg 744 kg

Weight of new student - 6 kg 19 kg

Weight of new student 25 kg - 6 kg 19 kg

Therefore, the weight of the new student in this case is 19 kg.

Conclusion

Understanding the relationship between the average weight and individual weight changes in a class can help in solving various real-world problems. The steps involved in the solution include calculating the total weight, relating the total weights, and solving the equations to find the unknown weight. This approach is not only useful for academic purposes but also in fields such as sports and health science.