2010 AMC 12B 第 6 题

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：

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