Problem:
At Euler Middle School, students voted on two issues in a school referendum with the following results: voted in favor of the first issue and voted in favor of the second issue. If there were exactly students who voted against both issues, how many students voted in favor of both issues?
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Solution:
The sum counts the number of students who voted for both issues twice. So the number who voted in favor of both issues is or .
In the diagram below, the left circle represents the students who voted for the first issue, and the right circle represents the students who voted for the second issue. Let be the number of students who voted for both issues. Then students voted for the first issue but not the second, students voted for the second issue but not the first and students voted against both issues. The sum of the numbers in the diagram must be , so
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions