2017 AMC 10A Problems/Problem 6

Problem

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?

$\textbf{(A)}\ \text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}\\\textbf{(B)}\ \text{If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}\\\textbf{(C)}\ \text{If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. }\\\textbf{(D)}\ \text{If Lewis received an A, then he got all of the multiple choice questions right.}\\\textbf{(E)}\ \text{If Lewis received an A, then he got at least one of the multiple choice questions right.}$

Solution

Rewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an A on the exam." If that someone is Lewis the statement becomes: "if Lewis got all the multiple choice questions right, then he got an A on the exam." The contrapositive: "If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong (did not get all of them right)" must also be true leaving B as the correct answer. B is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is $\boxed{\textbf{(B)}\text{ If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}}$ .

Note that answer choice (B) is the contrapositive of the given statement. (That is, it has been negated as well as reversed.) We know that the contrapositive is always true if the given statement is true.

Solution 2 (Logic it out!)

$(A)$ False. This can easily be identified as wrong, as Lewis might've only gotten $1$ multiple-choice questions wrong, but still not received an A

$(B)$ This is true. If Lewis got all the multiple choice questions right, he would've gotten an A. If he didn't get an A, then he didn't get all of them right, since Lewis would have gotten an A if he got the multiple-choice section all right.

$(C)$ False. There might be another section, call it the writing section. Then Lewis still might get an A if the writing section can carry his grade.

$(D)$ False. He might've messed up on the multiple-choice section, but got extra credit to still level his grade up to an A.

$(E)$ False. Lewis could've gotten 0 multi-choice correct but still receive an A.

Therefore, our answer is $(B)$

~minor edits Terribleteeth

Video Solution

https://youtu.be/pxg7CroAt20

https://youtu.be/7j5JigR0pbs

~savannahsolver

2017 AMC 10A (Problems • Answer Key • Resources)
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2017 AMC 10A Problems/Problem 6

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Problem

Solution

Solution 2 (Logic it out!)

Video Solution

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