2017 AMC 10 A
Complete problem set with solutions and individual problem pages
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an on the exam. Which of these statements necessarily follows logically? (2017 AMC 10A Problem, Question#6)
- A.
If Lewis did not receive an , then he got all of the multiple choice questions wrong.
- B.
If Lewis did not receive an , 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 .
- D.
If Lewis received an , then he got all of the multiple choice questions right.
- E.
If Lewis received an , then he got at least one of the multiple choice questions right.
Rewriting the given statement: "if someone got all the multiple choice questions right on the upcoming exam then he or she would receive an on the exam." If that someone is Lewis the statement becomes: "if Lewis got all the multiple choice questions right, then he got an on the exam." The contrapositive: "If Lewis did not receive an , then he got at least one of the multiple choice questions wrong (did not get all of them right)" must also be true leaving as the correct answer. is also equivalent to the contrapositive of the original statement, which implies that it must be true, so the answer is
| If Lewis did not receive an , then he got at least one of the multiple choice questions wrong. |
Note that Answer choice is the contrapositive of the given statement. Weknow that
contrapositive is always true if the given statement is true.
