PT141.S2.Q21 > Explanation

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Most of the students who took Spanish 101 at the university last semester attended every class session. However, each student who received a grade lower than B minus missed at least one class session.

Most + All

The stimulus gives us two facts about Spanish 101 students last semester.

Fact 1: Most of the students attended every class session.

Fact 2: Each student who received a grade lower than B minus missed at least one class session.

As with many Must Be True questions, the key move is pushing these facts together to see whether we can make an inference from the combination of them. The connection point is attending every class session, which appears in both facts. But to link them, we need the contrapositive of Fact 2:

Contrapositive of Fact 2: If a student attended every class session, then that student did not receive a grade lower than B minus. In other words, they got a B minus or higher.

Now chain the facts together:

most students	→	attended every session		→	grade ≥ B−
Fact 1			Contrapositive of Fact 2

Most students attended every session (Fact 1), and everyone who attended every session got a B minus or higher (contrapositive of Fact 2). So most students got a B minus or higher. We should look for that inferences in the answers.

Traps to Watch For

One thing to watch out for: the chain tells us only that these students got a grade of B minus or higher. We can't pin down exactly where in that range they landed. Maybe they all got B minuses. Maybe they all got A's. The stimulus doesn't say.

Also watch for answers that try to get us to confuse sufficient and necessary conditions. Fact 2 tells us that everyone below B minus missed a session. That does not mean everyone who missed a session got below B minus. "All X are Y" doesn't imply "all Y are X" or even "most Y are X."

21.

Which one of the following statements about the students who took Spanish 101 at the university last semester can be properly inferred from the information above?

At least some of the students who received a grade of A minus or higher attended every class session.

We have no information about students who received A minus or higher specifically. The contrapositive of Fact 2 tells us that students who attended every session got B minus or higher, but "B minus or higher" is a wide bucket. We can't break it down further. Some of those students might have gotten a B minus, some a B, some an A. We just don't know, which means we can't guarantee that any particular sub-group within "B minus or higher" attended every session.

13%

Most, if not all, of the students who missed at least one class session received a grade lower than B minus.

This reverses the second statement. We know everyone who got a grade lower than B minus missed at least one class session. This doesn’t imply that most people who missed at least one class session got a grade lower than B minus. “All X are Y” does not imply “Most Y are X.”

11%

Most of the students received a grade higher than B minus.

We can prove that most students got a grade of B minus or higher, but (C) says higher than B minus. That rules out B minus itself. What if every student who attended every session got exactly a B minus? If (C) had said "B minus or higher" instead of "higher than B minus," it would be correct.

31%

At least one student who received a grade of B minus or higher missed one or more class sessions.

We know that all students who got a grade lower than a B minus missed at least one class session. This doesn’t imply that some students who got a B minus or higher missed a class session. Maybe the ones who got a B minus or higher attended every class.

More than half of the students received a grade of B minus or higher.

Must be true based on the first statement and the contrapositive of the second statement. Over half of the students attended every class session. Everyone who attended every class session did NOT get a grade lower than a B minus (in other words, B minus or higher).

37%

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