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Difference between revisions of "2010 AMC 10B Problems/Problem 12"
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== Problem ==
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At the beginning of the school year, <math>50\%</math> of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and <math>50\%</math> answered "No." At the end of the school year, <math>70\%</math> answered "Yes" and <math>30\%</math> answerws "No." Altogether, <math>x\%</math> 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 <math>x</math>?
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<math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 80</math>
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== Solution ==
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The minimum possible value occurs when <math>20\%</math> of the students who originally answered "No." answer "Yes." In this case, <math>x=20</math>
 
The minimum possible value occurs when <math>20\%</math> of the students who originally answered "No." answer "Yes." In this case, <math>x=20</math>
  

Revision as of 14:07, 24 January 2011

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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