AMC12 2025 A

AMC12 2025 A · Q3

AMC12 2025 A · Q3. 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知识竞赛。学生和教师总数为15人。阿什是其中一名学生的表亲，想加入竞赛。如果阿什加入学生队，该队的平均年龄将从12岁增加到14岁。如果阿什加入教师队，该队的平均年龄将从55岁减少到52岁。阿什多大年龄？

(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 $A= \boxed{28}.$

当阿什加入一支队伍时，该队的平均年龄会向他的年龄拉动。设 $A$ 为阿什的年龄，$N$ 为学生队人数。这意味着教师队有 $15-N$ 人。我们为每支队伍的平均年龄变化写出表达式。学生队原平均年龄为12岁，阿什加入后变为14岁，增加了2。项 $A-12$ 表示阿什比学生平均年龄大多少。将此除以 $N+1$（阿什加入后的学生队人数），得到每人平均变化。因此， \[\frac{A-12}{N+1} = 2.\] 类似地，教师队原平均年龄为55岁，阿什加入后减少3变为52岁。直观上，$55-A$ 表示阿什比教师平均年龄小多少。将此除以 $(15-N)+1$（教师队新总人数），表示每人平均变化。我们得到方程 \[\frac{55-A}{16-N} = 3.\] 解方程组，将方程(1)乘以 $N+1$，方程(2)乘以 $16-N$，然后相加，$A$ 抵消，得到 $43=50-N$。由此 $N=7$，$A= \boxed{28}$。

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