2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 4 Easy

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?

  • A.

    28

  • B.

    29

  • C.

    30

  • D.

    32

  • E.

    33

Answer:A

Let n denote the number of students, making (15-n) the number of teachers. Let a represent Ash's age.

From the students' average age: 12n + a = (n+1) \times 14

From the teachers' average age: 55 \cdot (15-n) + a = 52 \cdot (15-n+1)

Solving this system: - From the first equation: a = 2n + 14 - From the second equation: a = 3n + 7

Therefore 2n + 14 = 3n + 7, which gives n = 7 and a = 28.

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