Step-by-step explanation:
Let \(A\) represent the average age of the students.
The total age of the 15 students is \(15 \times 12 = 180\) (as the average age of the students is 12).
The total age of the entire group, considering 25 students (15 + 10), is \(25 \times 14 = 350\) (as the average age of the entire group is 14).
Now, let's find the total age of the 10 students. The total age of the entire group minus the total age of the 15 students equals the total age of the 10 students.
\(350 - 180 = 170\)
Therefore, the total age of the 10 students is 170.
The average age of the 10 students can be found by dividing their total age by the number of students:
\(B = \frac{{170}}{{10}} = 17\)
So, the average age of the 10 students is 17.
Now, let's find the average age of all 25 students (15 students + 10 students):
\(C = \frac{{350}}{{25}} = 14\)
The average age of all 25 students is 14.
Now, to find the average age of just the 15 students, we can use the formula:
Total age of 15 students = Total age of all 25 students - Total age of 10 students
\(180 = 350 - 170\)
\(180 = 180\)
This checks out, which means the total age of the 15 students is correct. Therefore, the average age of the 15 students is:
\(A = \frac{{180}}{{15}} = 12\)
So, the average age of the students is 12 years old.
