Section A has 24 students. If one student of this section is exchanged for another in section B, then the average mark of A is increased by 1.25, while that of B reduced by 1. The number of students in section B is
30
Students in Section B: Finding the Number of Students
Let's break down this problem involving the exchange of students between two sections, A and B, and the resulting changes in their average marks. We are given information about Section A and the changes that occur when one student is swapped between the sections. We need to find the initial number of students in Section B.
Let:
- \(n_A\) be the number of students in Section A. Initially, \(n_A = 24\).
- \(n_B\) be the number of students in Section B. This is what we need to find.
- \(S_A\) be the initial total marks of students in Section A.
- \(S_B\) be the initial total marks of students in Section B.
- \(Avg_A = \frac{S_A}{n_A}\) be the initial average mark of Section A.
- \(Avg_B = \frac{S_B}{n_B}\) be the initial average mark of Section B.
Now, consider the exchange:
- Let \(m_A\) be the mark of the student moved from Section A to Section B.
- Let \(m_B\) be the mark of the student moved from Section B to Section A.
After the exchange, Section A still has \(n_A = 24\) students, and Section B still has \(n_B\) students.
The new total marks for Section A will be the initial total minus the mark of the student who left, plus the mark of the student who joined: \(S_A - m_A + m_B\).
The new total marks for Section B will be the initial total minus the mark of the student who left, plus the mark of the student who joined: \(S_B - m_B + m_A\).
Average Mark Change in Section A
We are told the average mark of Section A increased by 1.25.
New Average of A = Initial Average of A + 1.25
$$ \frac{S_A - m_A + m_B}{n_A} = \frac{S_A}{n_A} + 1.25 $$Substitute \(n_A = 24\):
$$ \frac{S_A - m_A + m_B}{24} = \frac{S_A}{24} + 1.25 $$Multiply both sides by 24:
$$ S_A - m_A + m_B = S_A + 24 \times 1.25 $$ $$ S_A - m_A + m_B = S_A + 30 $$Subtract \(S_A\) from both sides:
$$ -m_A + m_B = 30 $$ $$ m_B - m_A = 30 \quad (\text{Equation } 1) $$This equation tells us the difference in marks between the student coming into A and the student leaving A is 30.
Average Mark Change in Section B
We are told the average mark of Section B reduced by 1.
New Average of B = Initial Average of B - 1
$$ \frac{S_B - m_B + m_A}{n_B} = \frac{S_B}{n_B} - 1 $$Multiply both sides by \(n_B\):
$$ S_B - m_B + m_A = S_B - n_B $$Subtract \(S_B\) from both sides:
$$ -m_B + m_A = -n_B $$Multiply both sides by -1:
$$ m_B - m_A = n_B \quad (\text{Equation } 2) $$This equation tells us the difference in marks between the student leaving B and the student coming into B is equal to the number of students in Section B.
Finding the Number of Students in Section B
Now we have two equations for the difference \(m_B - m_A\):
$$ m_B - m_A = 30 \quad (\text{Equation } 1) $$ $$ m_B - m_A = n_B \quad (\text{Equation } 2) $$Since both expressions are equal to \(m_B - m_A\), they must be equal to each other:
$$ n_B = 30 $$Therefore, the number of students in Section B is 30.
Summary of Calculation
| Section | Initial Students | Mark Change (\(m_B - m_A\)) | Average Change | Equation |
|---|---|---|---|---|
| A | 24 | \(m_B - m_A\) | +1.25 | \(\frac{m_B - m_A}{24} = 1.25 \implies m_B - m_A = 30\) |
| B | \(n_B\) | \(m_A - m_B\) | -1 | \(\frac{m_A - m_B}{n_B} = -1 \implies m_A - m_B = -n_B \implies m_B - m_A = n_B\) |
Equating the expressions for \(m_B - m_A\):
$$ 30 = n_B $$So, the number of students in Section B is 30.
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