AMC8 2019

AMC8 2019 · Q11

AMC8 2019 · Q11. It mainly tests Arithmetic misc, Inclusion–exclusion (basic).

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?

林肯中学八年级有$93$名学生。每名学生都上数学课或外语课或两者兼上。有$70$名八年级学生上数学课，有$54$名八年级学生上外语课。有多少名八年级学生只上数学课而不上外语课？

(A) 16 16

(B) 53 53

(D) 39 39

(E) 70 70

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let $x$ 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 $70 - 31 = 39$. $\boxed{\textbf{(D)}\ 39}$

设$x$为同时上数学和外语课的学生数。根据容斥原理，$70 + 54 - x = 93$。解得$x = 31$。只上数学课的学生数为$70 - 31 = 39$。 $\boxed{\textbf{(D)}\ 39}$

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