SAT Test Prep

CHAPTER 6
WHAT THE SAT MATH IS REALLY TESTING

Lesson 7: Thinking Logically

Numerical and Algebraic Proof

Logical proofs aren”t just for geometry class. They apply to arithmetic and algebra, too. In arithmetic, you often need to apply the laws of arithmetic (such as odd × even = even, negative÷positive = negative—see Chapter 9, Lesson 3) to prove what you”re looking for. When you solve an algebraic equation, you use logical laws of equality (such as the addition law of equality) to prove the equation you want.

“Must Be True” Questions

Logic is especially useful in solving SAT “must be true” questions. You know them and hate them—they usually have those roman numerals I, II, and III. To prove that a statement “must be true,” apply the laws of equality or the laws of arithmetic. To prove that a statement doesn”t have to be true, just find one counterexample, a valid example for which the statement is false.

If a and b are positive integers such that and , which of the following must be true?

I. is an integer.

II. b is an even number.

III. ab is 6 greater than a.

(A) I only

(B) I and II only

(C) I and III only

(D) I and III only

(E) I, II, and III

This requires both numerical and algebraic logic. First, let”s see how far we can get trying to solve the equation for a and b.

Okay, we”ve got a problem. We have two unknowns but only one equation, which means we can”t solve it uniquely. Fortunately, we know that a and b must be positive integers, so the equation basically says that the product of two positive integers, a and , is 6. The only positive integer pairs with a product of 6 are and , so one possibility is that and . This gives , and it satisfies the condition that . Now check the statements. Statement I is true here because , which is an integer. Statement II is also true here because 4 is an even number. Statement III is also true because , which is 6 greater than 2. So the answer is (E) I, II and III, right?

Wrong. Remember that the question asks what must be true, not just what can be true. We”ve only shown that the statements can be true. We can prove that statement I must be true by testing all the possible cases. Since there is only one other possible solution that satisfies the conditions: and , and since is an integer, we can say with confidence that statement I must be true. But statement II doesn”t have to be true because b can equal 7, which is not even. We have found a counterexample. Next, we can prove that statement III must be true by checking both cases: is 6 greater than 2, and is 6 greater than 1. (We can prove it algebraically too! If we add a to both sides of the original equation, we get , which proves that ab is 6 greater than a.)

Process of Elimination (POE)

On multiple-choice questions (and especially “must be true” questions), it helps to cross off wrong answers right away. Sometimes POE simplifies the problem dramatically.

What if, in the preceding question, the first solution we found was and . For this solution, statements I and III are true, but statement II is not. Therefore, we could eliminate those choices containing II—(B), (D), and (E). Since the two remaining choices contain statement I, it must be true—we don”t even need to prove it!

Concept Review 7: Thinking Logically

1. What is a proof, and why is understanding proofs helpful on the SAT?

2. How can POE help on the SAT?

3. What is the difference between geometric, algebraic, and numerical proofs?

4. Name two geometric theorems that are useful on the SAT.

5. Name two algebraic theorems that are useful on the SAT.

6. Name two numerical theorems that are useful on the SAT.

SAT Practice 7: Thinking Logically

Use logical methods to solve each of the following SAT questions.

1. If and , which of the following must be true?

(A) I only

(B) II only

(C) I and II only

(D) I and III only

(E) I, II, and III

A, B, C, and D are the consecutive vertices of a quadrilateral.

∠ABC and a ∠BCD are right angles.

2. If the two statements above are true, then which of the following also must be true?

(A) ABCD is a rectangle.

(B) is parallel to .

(C) is parallel to .

(D) Triangle ACD is a right triangle.

(E) Triangle ABD is a right triangle.

3. The statement is defined to be true if and only if Which of the following is true?

4. If , which of the following can be true?

I. m and n are both positive.

II. m and n are both negative.

III. m is positive and n is negative.

(A) II only

(B) III only

(C) I and II only

(D) I and III only

(E) II and III only

5. If p is a prime number greater than 5 and q is an odd number greater than 5, which of the following must be true?

I. is not a prime number.

II. pq has at least three positive integer factors greater than 1.

III. is not an integer.

(A) I only

(B) I and II only

(C) I and III only

(D) I and III only

(E) I, II, and III

Answer Key 7: Thinking Logically

Concept Review 7

1. A proof is a sequence of logical statements that begins with a set of assumptions and proceeds to a desired conclusion. You construct a logical proof every time you solve an equation or determine a geometric or arithmetic fact.

2. The process of elimination (POE) is the process of eliminating wrong answers. Sometimes it is easier to show that one choice is wrong than it is to show that another is right, so POE may provide a quicker path to the right answer.

3. Geometric proofs depend on geometric facts such as “angles in a triangle have a sum of 180°,” algebraic proofs use laws of equality such as “any number can be added to both sides of an equation,” and numerical proofs use facts such as “an odd number plus an odd number always equals an even number.”

4. The most important geometric theorems for the SAT are given in Chapter 10. They include parallel lines theorems such as “if two parallel lines are cut by a transversal, then alternate interior angles are congruent” and triangle theorems such as “if two sides of a triangle are congruent, then the angles opposite those sides are also congruent.”

5. The most important algebraic theorems are the laws of equality, such as “you can subtract any number from both sides of an equation.”

6. The most important numerical theorems are discussed in Chapter 9, Lesson 3, and Chapter 7, Lesson 7. They include “odd× odd = odd” and “positive × negative = negative.”

SAT Practice 7

1. A Since a is greater than b, b – a must be a negative number. Since must be positive, but is negative, b also must be negative because negativ = negative = positive, but positiv +× negative = negative. This proves that statement I must be true. However, statement II does not have to be true because a counterexample is a = 1 and . Notice that this satisfies the conditions that . Statement III also isn”t necessarily true because a counterexample is and . Notice that this also satisfies the conditions that a > b and but contradicts the statement that .

2. B First draw a diagram that illustrates the given conditions, such as the one above. This diagram shows that the only true statement among the choices is (B). This fact follows from the fact that “if a line (BC), crosses two other lines (AB and DC) in a plane so that same-side interior angles are supplementary, then the two lines are parallel.”

3. C First, translate each choice according to the definition of the bizarre new symbol. This gives us , ,,, and (E) . The only true statement among these is (C).

4. E The question asks whether the statements can be true, not whether they must be true. The equation says that two numbers have a product of 1. You might remember that such numbers are reciprocals, so we want to find values such that and are reciprocals of each other. One pair of reciprocals is 2 and ½, which we can get if and . Therefore, statement III can be true, and we can eliminate choices (A) and (C). Next, think of negative reciprocals, such as –2 and –½, which we can get if and Therefore, statement II can be true, and we can eliminate choices (B) and (D), leaving only (E), the correct answer. Statement I can”t be true because if m and n are both positive, then both m + 1 and n + 1 are greater than 1. But, if a number is greater than 1, its reciprocal must be less than 1.

5. A You might start by just choosing values for p and q that satisfy the conditions, such as and . When you plug these values in, all three statements are true. Bummer, because this neither proves any statement true nor proves any statement false. Are there any interestingpossible values for p and q that might disprove one or more of the statements? Notice that nothing says that p and q must be different, so choose and . Now , which only has 1, 7, and 49 as factors. Therefore, it does not have at least three positive integer factors greater than 1, and statement II is not necessarily true. Also, , which is an integer, so statement III is not necessarily true. So we can eliminate any choices with II or III, leaving only choice (A).