2017 AMC 10A Problem 6

P1-10~1 minPrint

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

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?

Show solution without answering

Key insight

All right means an A. So no A means not all right: at least one answer was wrong. That flipped-and-reversed statement is the only one that must be true.

Solution

The promise says one thing: if you get every multiple choice question right, then you get an A. Test each choice against a simple question: could it be false while Ms. Carroll still keeps her promise?

Think about a student with no A. Could that student have gotten every question right? No. If he had, the promise would have given him an A. So a student without an A must have missed at least one question. That is exactly choice (B), so (B) must be true.
Choice (A) says a student with no A got every question wrong. Too strong. Missing just one question is enough to explain the missing A, so (A) does not have to be true.
Choice (C) says one wrong answer means no A. The promise never says that. Ms. Carroll might still give an A to a student who missed one question and did great on the rest of the exam.
Choice (D) says an A means every answer was right. Again the promise does not say this. It gives one way to earn an A, not the only way.
Choice (E) says an A means at least one answer was right. The promise says nothing about students who get an A in some other way, so this is not forced either.

Only (B) is guaranteed.

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

A statement "if this, then that" also guarantees "if not that, then not this." That flipped-and-reversed form (called the contrapositive) is always true whenever the original is. Two traps sit next to it: just reversing the statement, or just flipping both parts, does not have to be true. Also be careful with the opposite of "all right": it is "at least one wrong," not "all wrong."

Alternative approach

Imagine a teacher who keeps the promise but is also generous: she gives an A to everyone, whatever their score. This teacher keeps her promise, yet a student who got every question wrong still gets an A. That one example breaks (C), (D) and (E). (A) is broken by a strict teacher who gives an A only for a perfect paper: a student who misses one question gets no A, but did not get every question wrong. Only (B) survives.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)