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2017 AMC 10A Problem 6

2017 AMC 10A Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Logic Puzzles · difficulty 1 of 5 · about 1 min · position P1-10

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

Answer

Key insight

The promise is 'all right implies an A'; the only statement forced by it is the contrapositive: no A implies at least one question wrong.

Solution

Write $P$ for "Lewis got every multiple choice question right" and $Q$ for "Lewis received an A." The promise is exactly $P \Rightarrow Q$ .

An implication is logically equivalent to its contrapositive, $\text{not } Q \Rightarrow \text{not } P$ . Here "not $P$ " means "at least one multiple choice question was wrong," so the contrapositive reads: if Lewis did not receive an A, then he missed at least one question. That is statement (B).

Check the others. (A) strengthens "not all right" to "all wrong," which is not justified. (C) is the inverse ( $\text{not } P \Rightarrow \text{not } Q$ ) and (D) is the converse ( $Q \Rightarrow P$ ); neither follows, since the promise says nothing about how else an A might be earned. (E) is also a claim about what happens given an A, which the promise does not address.

The answer is $\boxed{\textbf{(B)}\ \text{If Lewis did not receive an A, then he got at least one question wrong.}}$

Why this works

From $P \Rightarrow Q$ you may conclude only $P \Rightarrow Q$ itself and its contrapositive. Translating the negation correctly matters: the opposite of "all right" is "at least one wrong," never "all wrong." When a logic problem offers converse, inverse and contrapositive as choices, the contrapositive is the one that must be true.

The trap

Choosing the converse (D): receiving an A does not force all answers right, because the teacher never said an A can be earned only that way.

Common mistakes

Choosing the converse (D): receiving an A does not force all answers right, because the teacher never said an A can be earned only that way.
Negating "all right" as "all wrong" and picking (A).

Techniques

Set up the equation/formula and compute; no special trick needed