Question

学年开始时，Wells 老师数学班中 $50\%$ 的学生对问题“你喜欢数学吗？”回答“喜欢”， $50\%$ 回答“不喜欢”。学年结束时， $70\%$ 回答“喜欢”， $30\%$ 回答“不喜欢”。共有 $x\%$ 的学生在学年开始和结束时给出了不同回答。 $x$ 的最大可能值与最小可能值相差多少？

At the beginning of the school year, $50\%$ of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and $50\%$ answered “No.” At the end of the school year, $70\%$ answered “Yes” and $30\%$ answered “No.” Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x?$

$0$

$20$

$40$

$60$

$80$

答案：D

知识点：百分数极端原理

难度评级：1410

小提示：

假设有 $100$ 名学生；回答“喜欢”的人数从 $50$ 增加到 $70$

Assume $100$ students; the “Yes” count rises from $50$ to $70$

大提示：

至少有 $20$ 人必须改变回答，而最终只有 $30$ 人回答“不喜欢”，这会限制最多有多少人改变回答

At least $20$ must switch, and only $30$ end with “No,” bounding how many can switch

解答：

假设有 $100$ 名学生。回答“喜欢”的人数从 $50$ 增加到 $70$ ，所以至少有 $70-50=20$ 名学生从“不喜欢”改为“喜欢”；因此 $x\ge20$ 。

因为结束时只有 $30$ 名学生回答“不喜欢”，原来回答“喜欢”的五十人中至少有 $50-30=20$ 人仍回答“喜欢”，所以最多有 $80$ 名学生改变回答；因此 $x\le80$ 。

两个极端都可以实现，所以差为 $80-20=60$ 。

因此，正确答案是 D。

Assume $100$ students. The number of “Yes” answers rises from $50$ to $70,$ so at least $70-50=20$ students switched from “No” to “Yes”; thus $x\ge20.$

Since only $30$ students answer “No” at the end, at least $50-30=20$ of the original “Yes” students still answer “Yes,” so at most $80$ students switched; thus $x\le80.$

Both extremes are achievable, so the difference is $80-20=60.$

Thus, the correct answer is D.

2010 AMC 12B 第 6 题

答案：D

难度评级：1410

小提示：

大提示：

解答：

其他年份的第 6 题