2017 AMC 10A Problem 6

Question

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?

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

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

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

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

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

Answer: B

Concepts:logical deduction

Difficulty rating: 900

Small hint:

Translate the promise as an implication

Big hint:

Use the contrapositive of the statement

Solution:

The promise says that getting every multiple choice question right guarantees an A. Its contrapositive is that anyone who did not receive an A must not have gotten every multiple choice question right. In other words, that person got at least one multiple choice question wrong.

Thus, B is the correct answer.