Question

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 $12$ to $14.$ If Ash plays with the teachers, the average age on that team will decrease from $55$ to $52.$ How old is Ash?

$28$

$29$

$30$

$32$

$33$

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.

2025 AMC 10A Problem 4

Solution:

Problem 4 in Other Years