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The average marks of some students in a class is 75. When five students are included, the average becomes 78 and the marks of those five students are 80, 85, 89, 81 and 82. Find the total number of students.
6
9
8
10
9
Use the deviation method: Total deviation caused by 5 new students = (78-75) * InitialCount + 5 * (78 - New_Avg_of_5). Here, (78-75) * N = Sum(NewMarks) - 5 * 78.
Average of some students = 75. Marks of 5 new students = 80, 85, 89, 81, 82. New average after inclusion = 78.
Average=Number of termsSum of terms
Use the deviation method: Total deviation caused by 5 new students = (78-75) * InitialCount + 5 * (78 - New_Avg_of_5). Here, (78-75) * N = Sum(NewMarks) - 5 * 78.
Students often forget that the new average of 78 applies to the total number of students (N + 5), not just the new ones.
Define variables
Let the initial number of students be N. The sum of their marks is 75N.
Suminitial=75N
Calculate sum of new students
The sum of the marks of the five new students is 80+85+89+81+82=417.
Sumnew=417
Set up the average equation
The new total number of students is N+5. The new average is given as 78, so the new sum is 78(N+5).
75N+417=78(N+5)
Solve for N
Expanding the equation: 75N+417=78N+390. Rearranging gives 3N=417−390=27. Thus, N=9.
N=9
B is correct because the calculated value of N is 9.
This concept of 'change in average due to inclusion' is frequently tested in Data Interpretation and Probability sets involving population means.
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