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2010 AMC 12B Problem 6

Problem 6 of 25EasierArithmeticProblem-Solving Techniques

At the beginning of the school year, 50%50\% of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and 50%50\% answered “No.” At the end of the school year, 70%70\% answered “Yes” and 30%30\% answered “No.” Altogether, x%x\% of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of x?x?

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Solution

Assume 100100 students. The number of “Yes” answers rises from 5050 to 70,70, so at least 70−50=2070-50=20 students switched from “No” to “Yes”; thus x≥20.x\ge20. Since only 3030 students answer “No” at the end, at least 50−30=2050-30=20 of the original “Yes” students still answer “Yes,” so at most 8080 students switched; thus x≤80.x\le80. Both extremes are achievable, so the difference is 80−20=60.80-20=60. Thus, the correct answer is D.
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Tagged: percentage · extremal argument

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