AMC 8 2019 Problem 11
AMC8 2019 年第 11 题
基本计数原理★★★☆☆
The eighth grade class at Lincoln Middle School has 93 students. Each student takes a math class or a foreign language class or both. There are 70 eighth graders taking a math class, and there are 54 eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?
林肯中学的八年级总共有 93 个学生。每个学生都会上一门数学课或者一门外语课或者两者都上。已知有 70 个八年级学生上一门数学课，54 个八年级学生上一门外语课。问有多少个八年级学生只上数学课而不上外语课？
- A.16
- B.23
- C.31
- D.39
- E.70
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let be the number of students taking both a math and a foreign language class. By P-I-E, we get 70 + 54 - x = 93. Solving gives us x = 31. But we want the number of students taking only a math class, which is 70 - 31 = 39. 39 ):):):):):)
We have 70 + 54 = 124 people taking classes. However, we over-counted the number of people who take both classes. If we subtract the original amount of people who take classes we get that 31 people took the two classes. To find the amount of people who took only math class, we subtract the people who didn't take only one math class, so we get 70 - 31 = 39. -fath2012
We are looking for students in math only, which is the complement (exactly the rest of the students) compared to those taking a language class. Since 54 students take a language (with or without math), we subtract that from the total number of students. Since 93-54= 39, our answer is 39. (It's not necessary to know that 70 students take math.)
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