The ratio of the average age of the class without the teacher to the

The ratio of the average age of the class without the teacher to the average age including the teacher is 8 : 9. If the ratio of the teacher’s age and the average age of the students is 3 : 1, find the number of students.

Let number of students in the class = x
Let the average age of the students = y
Let the age of the teacher = z

Given,
1) ratio of the average age of the class without the teacher to the average age including the teacher = 8 : 9
Average age of students and teacher = \((No.Of Students*Avg Age + Age of teacher)/(No.of students + 1)\)
= \((x*y + z)/(x+1)\)
Given this ratio = 8:9
\(y/((x*y + z)/(x+1)) = 8/9\)
\(y(x+1)/(x*y + z) = 8/9\)
\(xy + 9y = 8z\) -----------------------equation 1

2) Ratio of the teacher’s age and the average age of the students is 3 : 1
\(z/y = 3/1\)
\(z = 3y\) --------------------------------- equation 2
equation 2 in equation 1 implies,
\(xy + 9y = 8(3z)\)
\(xy = 15y\)
\(y(x-15) =0\)
Since y avg age can't be zero
\(x = 15\)
Number of students = 15

Option E is correct