Question

At the beginning of the school year, $50\%$ of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ 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 $x?$

$0$

$20$

$40$

$60$

$80$

Answer: D

Concepts:percentage extremal argument

Difficulty rating: 1410

Solution:

Assume $100$ students. The number of "Yes" answers rises from $50$ to $70,$ so at least $70-50=20$ students switched from "No" to "Yes"; thus $x\ge20.$

Since only $30$ students answer "No" at the end, at least $50-30=20$ of the original "Yes" students still answer "Yes," so at most $80$ students switched; thus $x\le80.$

Both extremes are achievable, so the difference is $80-20=60.$

Thus, the correct answer is D.

2010 AMC 12B Problem 6

Answer: D

Difficulty rating: 1410

Solution:

Problem 6 in Other Years