2015 AIME II Problems/Problem 2
In a new school percent of the students are freshmen, percent are sophomores, percent are juniors, and percent are seniors. All freshmen are required to take Latin, and percent of the sophomores, percent of the juniors, and percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is , where and are relatively prime positive integers. Find .
We see that of students are learning Latin. In addition, of students are sophomores learning Latin. Thus, our desired probability is and our answer is .
Assume that there are 100 students in the school. There are 40 freshman taking Latin, 24 sophomores taking Latin, 10 juniors taking Latin, and 2 seniors taking Latin. We get the probability to be the number of sophomores taking Latin over the total number of students taking Latin, or . Simplifying, we get . Adding, we get .
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