CHAPTER 6 / WHAT THE SAT MATH IS REALLYTESTING 263
SAT Practice 7
- A Since ais greater than b, b– amust be a negative
number. Since b(b – a) must be positive, but b– ais
negative, balso must be negative because negative×
negative = positive, but positive×negative = negative.
This proves that statement I must be true. However,
statement II does not have to be true because a coun-
terexample is a= 1 and b= −1. Notice that this satis-
fies the conditions that a> band that b(b – a) > 0.
Statement III also isn’t necessarily true because a
counterexample is a= −1 and b= −2. Notice that this
also satisfies the conditions that a> band b(b – a) > 0
but contradicts the statement that ab< 0. - 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 (ABand DC)
in a plane so that same-side interior angles are
supplementary, then the two lines are parallel.” - C First, translate each choice according to the
definition of the bizarre new symbol. This gives us
(A) 3/5 > 5/3, (B) 5/5 > 3/3, (C) 4/5 > 2/3, (D) 6/5 > 4/3,
and (E) 7/5 > 5/3. The only true statement among
these is (C).
4. E The question asks whether the statements can
be true, not whether they mustbe true. The equa-
tion says that two numbers have a product of 1.
You might remember that such numbers are reci-
procals, so we want to find values such that m+ 1
and n+ 1 are reciprocals of each other. One pair
of reciprocals is 2 and^1 / 2 , which we can get if m= 1
and n= −^1 / 2. Therefore, statement III can be true,
and we can eliminate choices (A) and (C). Next,
think of negative reciprocals, such as −2 and −^1 / 2 ,
which we can get if m= −3 and n= −^1 / 2. 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 mand
nare 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
pand qthat satisfy the conditions, such as p= 7
and q= 9. When you plug these values in, all three
statements are true. Bummer, because this nei-
ther proves any statement true nor proves any
statement false. Are there any interestingpossible
values for pand qthat might disprove one or more
of the statements? Notice that nothing says that
pand qmust be different, so choose p= 7 and q= 7.
Now pq= 49, which only has 1, 7, and 49 as fac-
tors. Therefore, it does nothave at least three pos-
itive integer factors greater than 1, and statement
II is not necessarily true. Also, q/p= 1, which is an
integer, so statement III is not necessarily true. So
we can eliminate any choices with II or III, leav-
ing only choice (A).
Concept Review 7
- 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. - 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. - 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 numeri-
cal 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 parallellines
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 dis-
cussed in Chapter 9, Lesson 3, and Chapter 7, Les-
son 7. They include “odd×odd = odd” and “positive
×negative = negative.”
Answer Key 7: Thinking Logically