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?

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

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 $\boxed{\textbf{(C)}\ 29}$.

Solution 2

We can plug in each answer choice and find our what is the greatest one that works. We start of with Answer Choice $E$ since it is the largest one.

E is right, Jesse got $33$ questions right.

If Jesse got $33$ questions right, then he gains $132$ points, he then needs to get another $33$ wrong to achieve a score of $99$. However, this is impossible as the test only contains $50$ questions, and he needs $66$ questions in order to achieve this.

D is right, Jesse got $31$ questions right.

If Jesse got $31$ questions right, then he gains $124$ points, he then needs to get another $25$ wrong in order to achieve a score of $99$. However, this is impossible as the test only contains $50$ questions, and he needs $56$ questions in order to achieve this.

C is right, Jesse got $29$ questions right.

If Jesse got $29$ questions right, then he gains $116$ points, he then needs to get another $17$ wrong in order to achieve a score of $99$. This is possible since this only requires $46$ questions. The other $4$ questions remain blank and earn him $0$ points.

- SAMANTAP

Video Solution

https://youtu.be/vYXz4wStBUU?t=549

~IceMatrix

See Also

2010 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 10 Problems and Solutions

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