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$\text{AMC 12A 2017 Question 3}$

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 of these statements necessarily follows logically?

$(A)$ If Lewis did not receive an $A$, then he got all of the multiple choice wrong.

$(B)$ If Lewis did not receive an $A$, then he got at least one of the multiple choice questions wrong.

$(C)$ If Lewis got at least one of the multiple choice questions wrong, then he did not receive an $A$.

$(D)$ If Lewis received an $A$, then he got all of the multiple choice questions right.

$(E)$ If Lewis received an $A$, then he got at least one of the multiple choice questions right.

According to the answer key the answer is $(B)$ since the statement is: $(p)$ If you get all questions correct $\implies (q)$ You get an $A$ and $(B)$ is the contrapositive of $p \implies q$

While solving this question I considered whether it is $p \implies q$ or $q \implies p$ since solving all questions correctly is the necessary condition for getting an $A$. So my answer was $(D)$. Can someone show me why my logic is incorrect. Thank you in advance

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Solving all the MCQ questions correctly is a sufficient condition for getting an $A$, but not a necessary one.

This can happen in tests with more than one component. The MCQ component carries enough weight such that if you score full marks in that part, you get an $A$ regardless of how you do in the rest of the test, so getting a full score in the MCQ component is sufficient for an $A$. But it may not be necessary because you might still be able to do well enough in the remainder of the paper that you could offset one or more incorrect MCQ responses.

You know $(B)$ is the correct option because it is given that the student did not receive an $A$ grade. He could not have got this with a full MCQ score (because that, by itself, guarantees an $A$ grade), so it is necessary that he got at least one MCQ response wrong.

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  • $\begingroup$ The second paragraph just cleared all my doubts. Thanks!!! $\endgroup$
    – HDatta
    Sep 30 '17 at 9:49
  • $\begingroup$ Most welcome. :) $\endgroup$
    – Deepak
    Sep 30 '17 at 9:50

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