AMC8 2015

AMC8 2015 · Q15

AMC8 2015 · Q15. It mainly tests Inclusion–exclusion (basic), Probability (basic).

At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?

在欧拉中学，有198名学生在学校公投中对两个议题进行了投票，结果如下：149人赞成第一个议题，119人赞成第二个议题。如果恰好有29名学生反对两个议题，那么有多少名学生赞成两个议题？

(A) 49 49

(B) 70 70

(D) 99 99

(E) 149 149

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The sum 149 + 119 is related to the Venn diagram shown. The top number in the left set is the number who voted for the first issue but not the second; and the number at the bottom in the left set is the number who voted for both issues. Similarly, the top number in the right set is the number who voted for the second issue but not the first; and 29 students voted against both issues. The sum of the numbers in the diagram must be 198, so \[ (149 - x) + x + x + (119 - x) + 29 = 198, \] \[ 297 - 2x = 198, \] \[ x = 99. \]

答案（D）：149 与 119 的和与下图所示的维恩图有关。左侧集合上方的数字表示只赞成第一项而不赞成第二项的人数；左侧集合下方的数字表示两项都赞成的人数。类似地，右侧集合上方的数字表示只赞成第二项而不赞成第一项的人数；有 29 名学生两项都反对。图中各部分数字之和应为 198，因此 \[ (149 - x) + x + x + (119 - x) + 29 = 198, \] \[ 297 - 2x = 198, \] \[ x = 99. \]

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