Question

一队学生将和一队老师进行知识竞赛。学生和老师的总人数为 $15$ 。Ash 是其中一名学生的表亲，他想加入比赛。如果 Ash 加入学生队，该队的平均年龄会从 $12$ 岁增加到 $14$ 岁。如果 Ash 加入老师队，该队的平均年龄会从 $55$ 岁降低到 $52$ 岁。Ash 多少岁？

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?

$28$

$29$

$30$

$32$

$33$

答案：A

知识点：平均数方程组年龄问题

难度评级：1160

解答：

设学生人数为 $s$ 。如果 Ash 加入学生队，他的年龄为 $14(s+1) - 12s = 2s + 14$ 。如果他加入老师队，老师有 $15 - s$ 人，他的年龄为 $52(16 - s)$ $- 55(15 - s) = 3s + 7$ 。两式描述的是同一个 Ash，所以 $2s + 14 = 3s + 7$ 。解得 $s = 7$ ，Ash 的年龄为 $2 \cdot 7 + 14 = 28$ 。因此正确答案是 A。

Let $s$ be the number of students. If Ash joins them, his age is $14(s+1) - 12s = 2s + 14.$ If he joins the teachers instead (there are $15 - s$ of them), his age is $52(16 - s)$ $- 55(15 - s) = 3s + 7.$ Both describe the same Ash, so $2s + 14 = 3s + 7.$ That gives $s = 7,$ and Ash is $2 \cdot 7 + 14 = 28.$ Therefore, the answer is A.

2025 AMC 10A 第 4 题

答案：A

难度评级：1160

解答：

其他年份的第 4 题