2017 AMC 10B Problems/Problem 11
At Typico High School, of the students like dancing, and the rest dislike it. Of those who like dancing, say that they like it, and the rest say that they dislike it. Of those who dislike dancing, say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
of the people that claim that they like dancing actually dislike it, and of the people that claim that they dislike dancing actually dislike it. Therefore, the answer is .
Assume WLOG that there are 100 people. Then 60 of them like dancing, 40 dislike dancing. Of the ones that like dancing, 48 say they like dancing and 12 say they dislike it. Of the ones who dislike dancing, 36 say they dislike dancing and 4 say they like it. We want the ratio of students like it but say they dislike it to the total amount of students that say they dislike it. This is . We choose
Solution 3 (Solution 2 but organized w table)
WLOG, assume that there are a total of students at Typico High School. We make a chart:
We know that of the students like dancing (since of is ), so we fill that in:
of those kids say that they like dancing, so that's kids who like dancing and say that they like dancing. The other kids like dancing and say that they do not.
students do not like dancing. of those students say that they do not like it, which is of them.
At this point, one can see that there are total students who say that they do not like dancing. of those actually like it, so that is
For the sake of completeness, let us fill out the rest of the table.
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