2010 AMC 10B Problems/Problem 12

Problem

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 $30\%$ answerws "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 $x$ ?

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80$

Solution

The minimum possible value occurs when $20\%$ of the students who originally answered "No." answer "Yes." In this case, $x=20$

The maximum possible value occurs when $30\%$ of the students who originally answered "Yes." answer "No." and the $50\%$ of the students who originally answered "No." answer "Yes." In this case, $x=50+30=80$

Subtract $80-20$ to obtain an answer of $\boxed{\mathrm{(D)} 60}$

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2010 AMC 10B Problems/Problem 12

Problem

Solution