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.
Important Questions from Average
- Find the mean marks of the following distribution.
Marks obtained No. of students 25 5 30 6 35 3 40 1 - The average of first 169 even numbers is
- The average weight (in kg) of a family of five members whose weights are 40 kg, 49 kg, 56 kg, 74 kg and 35 kg is:
- The average of first 152 odd natural numbers, is:
The average runs scored by a batsman in 23 matches is 40. In the next 10 matches, the batsman scored an average of 16 runs. Find his average runs scored (rounded off to two decimals) in all the 33 matches.
