Calculate Initial Total Marks
The initial group has 16 students with an average score of 20. The total marks for these 16 students can be calculated as:
Total Marks (Initial) = Number of students $\times$ Average marks
Total Marks (Initial) = $16 \times 20 = 320$
Calculate Marks of the Student Who Left
When one student left, the group size became 15, and the average marks increased to 21. The total marks for the remaining 15 students are:
Total Marks (15 students) = $15 \times 21 = 315$
The marks obtained by the student who left is the difference between the initial total marks and the total marks after they left:
Marks (Student Left) = Total Marks (Initial) - Total Marks (15 students)
Marks (Student Left) = $320 - 315 = 5$
Calculate Marks of the Student Who Joined
After the student left, another student joined the group, bringing the total number of students back to 16. The average marks of this group of 16 students became 20.5. The new total marks are:
Total Marks (New 16 students) = $16 \times 20.5 = 328$
The marks obtained by the student who joined is the difference between the new total marks (with 16 students) and the total marks of the 15 students (before the new one joined):
Marks (Student Joined) = Total Marks (New 16 students) - Total Marks (15 students)
Marks (Student Joined) = $328 - 315 = 13$
Calculate the Average Marks of Both Students
The question asks for the average marks obtained by the student who left and the one who joined. We have the marks for both:
- Marks of the student who left = 5
- Marks of the student who joined = 13
The average is calculated as:
Average Marks = $\frac{\text{Marks (Student Left)} + \text{Marks (Student Joined)}}{2}$
Average Marks = $\frac{5 + 13}{2} = \frac{18}{2} = 9$
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