2025 AMC 10A Problem 4

A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is

15.

Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from

12

14.

If Ash plays with the teachers, the average age on that team will decrease from

55

52.

How old is Ash?

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Solution

Let

s

be the number of students. If Ash joins them, his age is

14(s+1) - 12s = 2s + 14.

If he joins the teachers instead (there are

15 - s

of them), his age is

52(16 - s)

- 55(15 - s) = 3s + 7.

Both describe the same Ash, so

2s + 14 = 3s + 7.

That gives

s = 7,

and Ash is

2 \cdot 7 + 14 = 28.

Therefore, the answer is A.

Tagged: mean · system of equations · ages