AMC10 2025 A

AMC10 2025 A · Q4

AMC10 2025 A · Q4. It mainly tests Systems of equations, Averages (mean).

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?

一支学生队将与一支教师队进行琐碎知识竞赛。学生和教师总数为 15 人。Ash 是其中一名学生的表亲，想加入竞赛。如果 Ash 加入学生队，该队的平均年龄将从 12 岁增加到 14 岁。如果 Ash 加入教师队，该队的平均年龄将从 55 岁下降到 52 岁。Ash 多大年龄？

(A) 28 28

(B) 29 29

(D) 32 32

(E) 33 33

Answer

Correct choice: (A)

正确答案：（A）

Solution

When Ash joins a team, the team's average is pulled towards his age. Let $A$ be Ash's age and $N$ be the number of people on the student team. This means that there are $15-N$ people in the teacher team. Let us write an expression for the change in the average for each team. The students originally had an average of $12$, which became $14$ when Ash joined, so there was an increase of $2$. The term $A-12$ represents how much older Ash is compared to the average of the students'. If we divide this by $N+1$, which is the number of people on the student team when Ash joins, we get the average change per team member once Ash is added. Therefore, \[\frac{A-12}{N+1} = 2.\] Similarly, for teachers, the average was originally $55$, which decreased by $3$ to become $52$ when Ash joined. Intuitively, $55-A$ represents how much younger Ash is than the average age of the teachers. Dividing this by the expression $(15-N)+1$, which is the new total number of people on the teacher team, represents the average change per team member once Ash joins. We can write the equation \[\frac{55-A}{16-N} = 3.\] To solve the system, multiply equation (1) by $N+1$, and similarly multiply equation (2) by $16-N$. Then add the equations together, canceling $A$, leaving equation $43=50-N$. From this we get $N=7$ and $A= \boxed{28}.$

当 Ash 加入一支队伍时，该队的平均年龄会向他的年龄拉近。设 $A$ 为 Ash 的年龄，$N$ 为学生队人数，则教师队有 $15-N$ 人。我们为每支队伍的平均变化写表达式。学生队原平均年龄 12 岁，Ash 加入后变为 14 岁，增加 2 岁。$A-12$ 表示 Ash 比学生平均年龄大多少。除以 $N+1$（Ash 加入后的学生队人数），得到每个队员的平均变化。因此， \[\frac{A-12}{N+1} = 2.\] 类似地，教师队原平均年龄 55 岁，Ash 加入后下降 3 岁至 52 岁。$55-A$ 表示 Ash 比教师平均年龄小多少。除以 $(15-N)+1 = 16-N$（Ash 加入后的教师队人数），得到平均变化。我们得到方程 \[\frac{55-A}{16-N} = 3.\] 解系统：将方程 (1) 乘以 $N+1$，方程 (2) 乘以 $16-N$，然后相加，消去 $A$，得到 $43=50-N$。从而 $N=7$，$A= \boxed{28}$。

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