2010 AMC 12B Problem 6
Attempt Problem 6 of the 2010 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12B solutions, or check the answer key.
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6.
At the beginning of the school year, of all students in Mr. Wells’ math class answered “Yes” to the question “Do you love math”, and answered “No.” At the end of the school year, answered “Yes” and answered “No.” Altogether, 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
Answer: D
Small Hint:
Assume students; the “Yes” count rises from to
Big Hint:
At least must switch, and only end with “No,” bounding how many can switch
Solution:
Assume students. The number of “Yes” answers rises from to so at least students switched from “No” to “Yes”; thus
Since only students answer “No” at the end, at least of the original “Yes” students still answer “Yes,” so at most students switched; thus
Both extremes are achievable, so the difference is
Thus, the correct answer is D.
Problem 6 in Other Years
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