¶ 2019 AMC8 Problem 11
Problem:
The eighth grade class at Lincoln Middle School has students. Each student takes a math class or a foreign language class or both. There are eighth graders taking a math class, and there are eight graders taking a foreign language class. How many eight graders take only a math class and not a foreign language class?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The total enrollment in math classes and foreign language classes is students. There are students, so there must be students who are counted twice because they are taking both a math class and a foreign language class. There are therefore students taking only a math class and not a foreign language class.
In the Venn diagram below, the circle on the left represents the students taking a math class and the circle on the right represents the students taking a foreign language class. Let represent the number of students taking both a math class and a foreign language class. Then students take only a math class and students take only a foreign language class. Because , it follows that . There are therefore students who take only a math class and not a foreign language class.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions