Question

学年初，Wells 老师数学课上 $50\%$ 的学生对“你热爱数学吗？”回答“是”， $50\%$ 回答“否”。学年末， $70\%$ 回答“是”， $30\%$ 回答“否”。总共有 $x\%$ 的学生在学年初和学年末给出了不同回答。 $x$ 的最大可能值与最小可能值之差是多少？

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：百分数极限情形界定

难度评级：1420

解答：

为了最小化 $x$ ，尽量让学生保持原答案。

由于回答“是”的比例从百分之五十增到百分之七十，至少有 $70 - 50 = 20$ 个百分点所对应的学生必须改变答案。

为了最大化 $x$ ，可以让所有原来回答“否”的学生都改成“是”，但最终仍需要百分之七十回答“是”。

这要求原来回答“是”的学生中有 $70 - 50 = 20$ 个百分点仍回答“是”，所以原来回答“是”的百分之五十中，最多有 $50 - 20 = 30$ 个百分点可以改变。

因此最多有 $50 + 30 = 80$ 个百分点的学生改变答案。最大值与最小值之差为 $80 - 20 = 60$ 。

所以正确答案是 D。

To minimize $x,$ we want to have as many kids as possible maintain their answer.

We then need at least $70 - 50 = 20$ percent of the students to change their answer.

To maximize $x,$ we can have everybody that answered no change their answer, but some who answered yes must stay yes.

We need $70 - 50 = 20$ percent of the people who said yes to stay yes, which means only $50 - 20 = 30$ percent can switch.

This makes the maximum percent of students that can switch $50 + 30 = 80$ percent. The difference is then $80 - 20 = 60.$

Thus, D is the correct answer.

2010 AMC 10B 第 12 题

答案：D

难度评级：1420

解答：

其他年份的第 12 题