Understanding the Student Average Marks Problem

This problem involves calculating the total marks of a group of students at different stages as students leave and join. We start with an initial group, then one student leaves, changing the average, and finally, another student joins, changing the average again. The question asks for the average marks of the specific student who left and the student who joined.

Step-by-Step Calculation of Marks

Let's break down the problem into stages and calculate the total marks at each point.

Stage 1: Initial Group

• Number of students = 10
• Average marks = 20

The total marks for the initial group of 10 students can be calculated using the formula:

$$ \text{Total Marks} = \text{Number of Students} \times \text{Average Marks} $$

So, the total marks initially were:

$$ \text{Total Marks}_{\text{Initial}} = 10 \times 20 = 200 $$

Stage 2: One Student Leaves

One student leaves the group. The details for the remaining group are:

• Number of remaining students = 10 - 1 = 9
• Average marks of remaining students = 21

The total marks for these 9 remaining students are:

$$ \text{Total Marks}_{\text{Remaining 9}} = 9 \times 21 = 189 $$

The marks of the student who left can be found by subtracting the total marks of the remaining students from the initial total marks:

$$ \text{Marks of Student who Left} = \text{Total Marks}_{\text{Initial}} - \text{Total Marks}_{\text{Remaining 9}} $$

$$ \text{Marks of Student who Left} = 200 - 189 = 11 $$

Stage 3: Another Student Joins

A new student joins the group. The details for the group now are:

• Number of students = 9 + 1 = 10 (the group is back to 10 students)
• Average marks of the new group = 20.6

The total marks for this new group of 10 students are:

$$ \text{Total Marks}_{\text{Final 10}} = 10 \times 20.6 = 206 $$

This new total includes the total marks of the 9 students who remained (189) plus the marks of the student who joined. So, the marks of the student who joined can be found by subtracting the total marks of the 9 remaining students from the total marks of the final 10 students:

$$ \text{Marks of Student who Joined} = \text{Total Marks}_{\text{Final 10}} - \text{Total Marks}_{\text{Remaining 9}} $$

$$ \text{Marks of Student who Joined} = 206 - 189 = 17 $$

Calculating the Average of the Two Students' Marks

The question asks for the average marks obtained by the student who left and the one who joined. We have found their individual marks:

• Marks of Student who Left = 11
• Marks of Student who Joined = 17

To find the average of these two marks, we add them together and divide by 2:

$$ \text{Average Marks of the Two Students} = \frac{\text{Marks of Student who Left} + \text{Marks of Student who Joined}}{2} $$

$$ \text{Average Marks of the Two Students} = \frac{11 + 17}{2} = \frac{28}{2} = 14 $$

Therefore, the average marks obtained by the student who left and the one who joined is 14.

From the table: Marks of Student who Left = Initial Total - Total of Remaining 9 = $200 - 189 = 11$.

Marks of Student who Joined = Total of Final 10 - Total of Remaining 9 = $206 - 189 = 17$.

Average of the two students' marks = $(11 + 17) / 2 = 28 / 2 = 14$.

Revision Table: Key Concepts in Average Problems

Additional Information on Average Calculations

Understanding how the average changes when an element is added or removed is crucial for solving these types of problems. When a value is added to a group, the new total sum is the old total sum plus the value of the added item. When a value is removed, the new total sum is the old total sum minus the value of the removed item. The new average is then calculated by dividing the new total sum by the new number of items in the group.

In this problem, we used these principles twice:

• To find the marks of the student who left: Total sum decreased, and the number of students decreased. The amount of decrease in the total sum is the marks of the student who left.
• To find the marks of the student who joined: Total sum increased, and the number of students increased back to the original number. The amount of increase in the total sum (relative to the 9 students) is the marks of the student who joined.

Finally, we calculated the average of these two individual marks, which is a standard average calculation for two numbers.