Difference between revisions of "2017 AMC 10A Problems/Problem 6"
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<math>\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.}</math> | <math>\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.}</math> | ||
| − | ==Solution== | + | ==Solution 1 (Using the Contrapositive)== |
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." | 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 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 would receive an A on the exam." | ||
Revision as of 12:56, 8 June 2021
Contents
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?
Solution 1 (Using the Contrapositive)
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 would receive an A on the exam."
The contrapositive: "If Lewis not receive an A, then he did not get all of them right"
The contrapositive (in other words): "If Lewis not get an A, then he got at least one of the multiple choice questions wrong".
B is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is 
.
- 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.
~minor edits by BakedPotato66
Solution 2 (Elimination of Answer Choices)
False. If Lewis did not receive an A, that does not mean he got all the multiple-choice section wrong. For example, he might have only gotten 
multiple-choice question wrong.
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 all the multiple-choice section right.
False. There might be other sections as well. For example, his teacher can dictate that any student who answers all the Writing problems incorrectly will get an A. Or perhaps, a student who answers at least 5/8 of all questions correctly will receive an A. Or perhaps a student who draws a smiley face will receive an A. There may be multiple other ways to receive an A.
False. He might've messed up on the multiple-choice section, but still get an A using the example methods above.
False. Lewis could have gotten 0 multiple-choice correct but still receive an A using the example methods above.
Therefore, our answer is 
~minor edits by Terribleteeth
~edits by BakedPotato66
Video Solution
~savannahsolver
See Also
| 2017 AMC 10A (Problems • Answer Key • Resources) | ||
| 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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