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?

$28$

$29$

$30$

$32$

$33$

Answer: A

Concepts:system of equations mean ages

Difficulty rating: 1130

Solution:

Let $s$ be the number of students and $a$ be Ash's age. The students' ages total $12s,$ and adding Ash gives $\begin{aligned} 12s + a &= 14(s+1) \\ &\implies a = 2s + 14. \end{aligned}$

There are $15 - s$ teachers with ages totaling $55(15-s),$ and adding Ash gives $\begin{aligned} 55(15-s) + a &= 52(16-s) \\ &\implies a = 3s + 7. \end{aligned}$

Setting $2s+14 = 3s+7$ gives $s = 7,$ so $a = 2(7)+14 = 28.$

Thus, the correct answer is A.

2025 AMC 12A Problem 3

Answer: A

Difficulty rating: 1130

Solution:

Problem 3 in Other Years