Difference between revisions of "2017 AMC 10A Problems/Problem 6"

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==Problem==
 
==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?
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Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an <math>A</math> on the exam. Which one of these statements necessarily follows logically?
  
 
<math>\textbf{(A)}\ \text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}\\\qquad\textbf{(B)}\ \text{If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}\\\qquad\textbf{(C)}\ \text{If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.}\\\qquad\textbf{(D)}\ \text{If Lewis received an A, then he got all of the multiple choice questions right.}\\\qquad\textbf{(E)}\ \text{If Lewis received an A, then he got at least one of the multiple choice questions right.}</math>
 
<math>\textbf{(A)}\ \text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}\\\qquad\textbf{(B)}\ \text{If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}\\\qquad\textbf{(C)}\ \text{If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.}\\\qquad\textbf{(D)}\ \text{If Lewis received an A, then he got all of the multiple choice questions right.}\\\qquad\textbf{(E)}\ \text{If Lewis received an A, then he got at least one of the multiple choice questions right.}</math>

Revision as of 19:16, 23 December 2019

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?

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

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. We know that contrapositive is always true if the given statement is true.

See Also

2017 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions

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