Problem 33 - A Complete Resource Book in Mathematics for JEE Main

Question

The average weight of students in a class of 35 students is \(40 \mathrm{~kg}\). If the weight of the teacher be included, the average rises by \(\frac{1}{2} \mathrm{~kg}\); the weight of the teacher is (A) \(40.5 \mathrm{~kg}\) (B) \(50 \mathrm{~kg}\) (C) \(41 \mathrm{~kg}\) (D) \(58 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The weight of the teacher is 58 kg.

Step 1: Identify the Given Information

The average weight of 35 students is \(40\,\mathrm{kg}\). When the teacher's weight is included, the average weight increases by \(\frac{1}{2}\,\mathrm{kg}\). The total number of people is then 36.

Step 2: Calculate the Total Weight of 35 Students

Since the average weight of the students is \(40\,\mathrm{kg}\), the total weight for the 35 students is \(35 \times 40 = 1400\,\mathrm{kg}\).

Step 3: Calculate New Average Weight with Teacher Included

The new average weight with the teacher included is \(40 + \frac{1}{2} = 40.5\,\mathrm{kg}\).

Step 4: Determine the Total Weight Including the Teacher

Multiply the new average weight by the total number of people (students plus teacher): \(36 \times 40.5 = 1458\,\mathrm{kg}\).

Step 5: Find the Weight of the Teacher

Subtract the total weight of the students from the total weight including the teacher: \(1458 - 1400 = 58\,\mathrm{kg}\).

Key Concepts

Arithmetic Mean

To understand how the arithmetic mean works, think of it as a way to find the 'central' value of a set of numbers, often called an average. Calculating this involves a straightforward process:

• Add up all the numbers in the set.
• Count how many numbers are in the set.
• Divide the total sum by the quantity of numbers.

For example, if you have 35 students with an average weight of 40 kg, you find the total weight by multiplying: \(35 \times 40 = 1400\,\text{kg}\). This total is then divided by the number of students to verify the average: \(\frac{1400}{35} = 40\,\text{kg}\). It’s important to see how this arithmetic tool can simplify understanding a bigger picture of data in both small and large sets.

Classroom Problem Solving

In any classroom problem-solving scenario, it's essential to break the problem down into manageable parts. By identifying and understanding what you are given, what you need to find out, and how the data is interconnected, solving the problem becomes much less daunting.

• Start with what you know. Here, it was the average weight of 35 students (40 kg).
• Account for changes made to the scenario, such as including the teacher, which altered the average weight.
• Use these calculations to progress step by step to the unknown, namely the teacher's weight.

Problem-solving in the classroom involves using known data to find the unknown, breaking down the steps to make sense of unfamiliar or complex information. Over time, this method can enable students to apply similar strategies to a variety of mathematical challenges.

Mathematical Reasoning

Mathematical reasoning is about making logical deductions from given information to reach a conclusion. It plays a critical role in solving problems like this weight calculation example. This calculation demonstrates mathematical reasoning through steps:
First, the initial condition (average weight without the teacher) needs to be understood clearly. Next, compute how this average changes with the addition of the teacher.

• Calculate the new average of 40.5 kg when the teacher is included.
• Determine the new total weight (1458 kg) for both students and the teacher.
• Subtract the known total weight of the students (1400 kg) from this to reveal the teacher's weight (58 kg).

Mathematics often involves working backwards from the result we want to find, using reasoning to verify each step until the conclusion is reached accurately.