2010 AMC 10B Problems/Problem 15

Problem

On a $50$ -question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$ . What is the maximum number of questions that Jesse could have answered correctly?

$\mathrm{(A)}\ 25 \qquad \mathrm{(B)}\ 27 \qquad \mathrm{(C)}\ 29 \qquad \mathrm{(D)}\ 31 \qquad \mathrm{(E)}\ 33$

Solution

Let $a$ be the amount of questions Jesse answered correctly, $b$ be the amount of questions Jesse left blank, and $c$ be the amount of questions Jesse answered incorrectly. Since there were $50$ questions on the contest, $a+b+c=50$ . Since his total score was $99$ , $4a-c=99$ . Also, $a+c\leq50 \Rightarrow c\leq50-a$ . We can substitute this inequality into the previous equation to obtain another inequality: $4a-(50-a)\leq99 \Rightarrow 5a\leq149 \Rightarrow a\leq \frac{149}5=29.8$ . Since $a$ is an integer, the maximum value for $a$ is $\boxed{\mathrm {(C)}\ 29}$ .

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2010 AMC 10B Problems/Problem 15

Problem

Solution