Question

一队学生将和一队老师进行知识竞赛。学生和老师总共有 $15$ 人。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：方程组平均数年龄问题

难度评级：1130

解答：

设学生人数为 $s$ ，Ash 的年龄为 $a$ 。学生年龄总和为 $12s$ ，加入 Ash 后得到 $\begin{aligned} 12s + a &= 14(s+1) \\ &\implies a = 2s + 14. \end{aligned}$

老师有 $15 - s$ 人，年龄总和为 $55(15-s)$ ，加入 Ash 后得到 $\begin{aligned} 55(15-s) + a &= 52(16-s) \\ &\implies a = 3s + 7. \end{aligned}$

令 $2s+14 = 3s+7$ ，得 $s = 7$ ，所以 $a = 2(7)+14 = 28$ 。

因此，正确答案是 A。

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 $s = 7,$ so $a = 2(7)+14 = 28.$

Thus, the correct answer is A.

2025 AMC 12A 第 3 题

答案：A

难度评级：1130

解答：

其他年份的第 3 题