2015 AMC 8 Problems/Problem 15
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?
We can see that this is a Venn Diagram Problem.
First, we analyze the information given. There are students. Let's use A as the first issue and B as the second issue.
students were for A, and students were for B. There were also students against both A and B.
Solving this without a Venn Diagram, we subtract away from the total, . Out of the remaining , we have people for A and
people for B. We add this up to get . Since that is more than what we need, we subtract from to get
There are people. We know that people voted against both the first issue and the second issue. That leaves us with people who voted for at least one of them. If people voted for both of them, then that would leave people out of the vote, because is less than people. is , so to make it even; we have to take away from the people, which leaves us with .
Divide the students into four categories:
- A. Students who voted in favor of both issues.
- B. Students who voted against both issues.
- C. Students who voted in favor of the first issue and against the second issue.
- D. Students who voted in favor of the second issue and against the first issue.
We are given that:
- students voted in favor of the first issue.
- students voted in favor of the second issue.
We can quickly find that:
- students voted against the second issue.
- students voted against the first issue.
The answer is .
Solution 4 (PIE)
Using PIE (Principle of Inclusion-Exclusion), we find that the students who voted in favor of both issues are .
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