Logic Questions on the SAT Math Section

Logic Questions on the SAT Math Section
Photograph by Sigrid Olsson/Corbis

This tip on improving your SAT score was provided by Veritas Prep.

The SAT at its core is a test of reasoning, not a test of knowledge. Reasoning is technique that people use to create conclusions from premises or evidence or determine whether a conclusion is valid or invalid, based on premises. The most fundamental building blocks of reasoning come from logic, the formal study of valid reasoning.

While the SAT won’t call you to expound on the teachings of Aristotle, you will see instances 0f math questions on the SAT that purely test your logical reasoning ability. In order to tackle these, it’s essential to know a few fundamental terms from formal logic. First, one of the fundamental building blocks of logic is the syllogism: a logical argument in which a conclusion is derived from two or more premises. A classic example of a syllogism is:

All men are mortal.
Aristotle is a man.
Therefore, Aristotle is mortal.

An additional set of concepts that is useful for SAT logic questions include contrapositive, converse, and inverse statements. Here’s a quick definition of these terms:

Statement—If A, then B
Contrapositive—If not B, then not A
Converse—If B, then A
Inverse—If not A, then not B

You should also remember the following keys that relate to the validity of each of the statements above:

If the statement is true, then the contrapositive must also be true—and vice-versa.
If the converse is true, then the inverse must also be true—and vice-versa.

Let’s look at an example in practice:

Statement—“If I studied for the SAT, I will get a high score.”
Contrapositive—“If I did not get a high score, I did not study for the SAT.”
Converse— “If I get a high score, I studied for the SAT.”
Inverse—“If I did not study for the SAT, I will not get a high score.”

As you can see, if the statement is true, statement, the contrapositive must be true, and if the converse is true, then the inverse must be true. However, just because the statement is true, it does not mean that that converse and inverse are true. In this case, it’s possible that you could get a high score without studying for the SAT; maybe you were already very good at SAT-type questions.

Now let’s see how we can apply these concepts to an SAT problem. The following problem appeared on the official 2010-2011 SAT pretest:

All of Kay’s brothers can swim.

If the statement above is true, which of the following must also be true?

(A) If Dave can swim, then he is not Kay’s brother.
(B) If Walt can swim, then he is Kay’s brother.
(C) If Fred cannot swim, then he is not Kay’s brother.
(D) If Pete is Kay’s brother, then he cannot swim.
(E) If Mark is not Kay’s brother, then he cannot swim.

Since we are looking for a statement that is also true, given the statement, we are looking for the contrapositive statement. It may also be useful to restate the original statement in the form of:

If you are Kay’s brother, you can swim. Or simply, “If brother, can swim” or just “If B, then S.”

Now let’s look at the form of the answer choices and determine which one is the contrapositive and therefore the true statement:

A) If S, then not B.—this is nonsensical since it isn’t the inverse, converse, or the contrapositive.
B) If S, then B—this is the converse, which we know is not necessarily true, given the statement.
C) If not S, then not B—this is the contrapositive and must be true. (But let’s check the other answer choices, too.)
D) If B, then not S—this is nonsensical as well because it does not match inverse, converse, or contrapositive.
E) If not B, then not S—This is the inverse, which is not necessarily true, given the statement.

As you can see, by applying the concepts of logic and knowing the relationship between the statement, contrapositive, converse, and inverse, we can systematically identify which answer choice must be true, given the statement. This was an easier example and you may have been able to solve it just by reasoning it out on the fly. If you encounter a more difficult logic question on the SAT, you may need to fall back on the definitions we have covered here.

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