Solving CSIR - NET Mathematical Sciences: February 2022 PART A [Question ID = 21]: The Case of the Exchanged Students

Hello, math enthusiasts! Today, we have an interesting problem involving two sections of students, their average marks, and the impact of exchanging students between these sections. Let’s dive right in!

Section A has 24 students. If one student from this section is exchanged for another in Section B, the average mark of Section A increases by 1.25, while that of Section B decreases by 1. What is the number of students in Section B?

Step-by-Step Solution

Step 1: Understanding the Problem

- Section A has 24 students.
- One student from section A is exchanged with one student from section B.
- After the exchange, the average marks of section A increase by 1.25.
- The average marks of section B decrease by 1.
- We need to find the number of students in section B.

Step 2: Setting Up Variables

Let's denote:

- \( \bar{A} \) as the initial average marks of section A.
- \( \bar{B} \) as the initial average marks of section B.
- \( x \) as the marks of the student from section A who is exchanged.
- \( y \) as the marks of the student from section B who is exchanged.
- \( n \) as the number of students in section B.

Step 3: Effects on Section A

Initially, the total marks of section A are \( 24 \bar{A} \).

After the exchange, the new total marks of section A become:

\(24 \bar{A} - x + y\)

The new average marks of section A become:

\(\bar{A} + 1.25\)

So, we set up the equation for the new average:

\(\frac{24 \bar{A} - x + y}{24} = \bar{A} + 1.25\)

Multiply both sides by 24 to clear the fraction:

\(24 \bar{A} - x + y = 24(\bar{A} + 1.25)\)

\(24 \bar{A} - x + y = 24 \bar{A} + 30\)

Simplify this equation:

\(-x + y = 30\)

\(y - x = 30\)

(Equation 1)

Step 4: Effects on Section B

Initially, the total marks of section B are \( n \bar{B} \).

After the exchange, the new total marks of section B become:

\(n \bar{B} - y + x\)

The new average marks of section B become:

\(\bar{B} - 1\)

So, we set up the equation for the new average:

\(\frac{n \bar{B} - y + x}{n} = \bar{B} - 1\)

Multiply both sides by \( n \) to clear the fraction:

\(n \bar{B} - y + x = n(\bar{B} - 1)\)

\(n \bar{B} - y + x = n \bar{B} - n\)

Simplify this equation:

\(x - y = n\)

(Equation 2)

Step 5: Solving the System of Equations

We have two equations:

\(1. y - x = 30\)

\(2. x - y = n\)

Add these two equations:

\((y - x) + (x - y) = 30 + n\)

\(0 = 30 + n\)

\(n = 30\)

Thus, the number of students in section B is 30.