2025 AMC 12A Problem 3

Problem 3 of 25EasierAlgebra

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 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,

a = 2(7)+14 = 28.

Thus, the correct answer is A.