AMC10 2010 B

AMC10 2010 B · Q12

AMC10 2010 B · Q12. It mainly tests Averages (mean), Probability (basic).

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 $x$?

在学年初，Wells 先生数学班上 50% 的学生回答“爱数学”，50% 回答“不爱”。学年末，70% 回答“是”，30% 回答“否”。总共有 $x$% 的学生前后答案不同。$x$ 的最大和最小可能值的差是多少？

(A) 0 0

(B) 20 20

(D) 60 60

(E) 80 80

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Assume there are 100 students in Mr. Wells’ class. Then at least $70-50=20$ students answered “No” at the beginning of the school year and “Yes” at the end, so $x\ge 20$. Because only 30 students answered “No” at the end of the school year, at least $50-30=20$ students who answered “Yes” at the beginning of the year gave the same answer at the end, so $x\le 80$. The difference between the maximum and minimum possible values of $x$ is $80-20=60$. The minimum $x=20$ is achieved if exactly 20 students answered “No” at the beginning and “Yes” at the end of the school year. The maximum $x=80$ is achieved if exactly 20 students answered “Yes” at the beginning and the end.

答案（D）：假设威尔斯先生的班上有 100 名学生。那么至少有 $70-50=20$ 名学生在学年开始时回答“否”，学年结束时回答“是”，因此 $x\ge 20$。由于学年结束时只有 30 名学生回答“否”，那么在学年开始时回答“是”的学生中，至少有 $50-30=20$ 人在学年结束时也给出了相同的答案，因此 $x\le 80$。$x$ 的最大可能值与最小可能值之差为 $80-20=60$。当恰好有 20 名学生学年开始回答“否”、学年结束回答“是”时，最小值 $x=20$ 取得。当恰好有 20 名学生在学年开始和结束都回答“是”时，最大值 $x=80$ 取得。

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