SAT 8

34

48. D The largest angle of a triangle is opposite its longest side. Let C = the triangle’s largest angle. The side opposite C is side AB, and it measures 16 inches.

Recall that the Law of Cosines states: c2 = a2 + b2 − 2ab cos C,

where a, b, and c are the lengths of the sides of the triangle.

162 = 42 + 132 − 2(4)(13)cos C. 162 = 16 + 169 − 104 cos C.

71 = −104 cos C. cos C = − 10471 .

C= cos−1 − 10714 ≈ 133.1º.

49.C Recall that the absolute value of a complex

number is given by: a + bi = (a2 + b2 ). 4 + 2i = (42 + 22 ) = 20 = 2 5.

PART I / ABOUT THE SAT MATH LEVEL 2 TEST

50. E Using right triangle trigonometric ratios:

cos B =

a

.

c

sin A =

a

.

c

sec A =

1

=

c

.

cos A

b

a a

cos B sin A

c

c

=

.

sec A

c

b

Multiply the numerator and denominator by the LCD, c2b, to get:

a2b . c3

CHAPTER 3 / DIAGNOSTIC TEST

35

DIAGNOSE YOUR STRENGTHS AND WEAKNESSES

Check the number of each question answered correctly and “X” the number of each question answered incorrectly.

Algebra	1	2	3	4	24	25	26	28	34		39		Total Number Correct
10 questions
Solid Geometry	8	40	Total Number Correct
2 questions
Coordinate
Geometry	5	7	12	21	38	46	Total Number Correct
6 questions
Trigonometry	6	9	14	30	45	48	50	Total Number Correct
7 questions
Functions	10	11	13	15	16	17	18	20	27		32	33		36	37		41		42	Total Number Correct
15 questions
Data Analysis,
Statistics,
and Probability	35	43	44	47	Total Number Correct
4 questions
Numbers
and Operations	19	22	23	29	31	49	Total Number Correct
6 questions

Number of correct answers − 14 (Number of incorrect answers) = Your raw score

___________________________ − 14 (_____________________________) =

36	PART I / ABOUT THE SAT MATH LEVEL 2 TEST

Compare your raw score with the approximate SAT Math Test score below:

		SAT Math
	Raw Score	Approximate Score
Excellent	43–50	770–800
Very Good	33–43	670–770
Good	27–33	620–670
Above Average	21–27	570–620
Average	11–21	500–570
Below Average	< 11	<500

PART II

MATH REVIEW

This page intentionally left blank

CHAPTER 4

ALGEBRA

This chapter provides a review of basic algebraic principles. On the Level 2 test, 48–52% of the questions relate to algebra and functions. That translates to about 20% of the test questions relating specifically to algebra and about 30% to functions. In reality, however, algebra is needed to answer nearly all of the questions on the test including coordinate geometry, solid geometry, and, especially, functions. The pie chart shows approximately how much of the Level 2 test is related directly to algebra.

Numbers and

Operations

12% Algebra

20%

Data Analysis,

Statistics, and

Probability

8%

Solid

Geometry

4%

Coordinate

Geometry

12%

Functions

30%

Trigonometry

14%

A wide variety of algebraic topics are covered in this chapter as an overall review of basic algebraic concepts. The topics are:

1.Evaluating Expressions

2.Fractions

a.Simplifying Fractions

b.Least Common Denominator

c.Multiplying Fractions

d.Using Mixed Numbers and Improper Fractions

e.Variables in the Denominator

39

40	PART II / MATH REVIEW

3.Percentages

a.Converting Percentages to Decimals

b.Converting Fractions to Percentages

c.Percentage Problems

4.Exponents

a.Properties of Exponents

i.Rules of Exponents

b.Common Mistakes with Exponents

c.Rational Exponents

d.Negative Exponents

e.Variables in an Exponent

5.Real Numbers

a.Vocabulary

b.Properties of Real Numbers

i.Properties of Addition

ii.Properties of Multiplication

iii.Distributive Property

iv.Properties of Positive and Negative Numbers

6.Absolute Value

7.Radical Expressions

a.Roots of Real Numbers

b.Simplest Radical Form

c.Rationalizing the Denominator

d.Conjugates

8.Polynomials

a.Vocabulary

b.Adding and Subtracting Polynomials

c.Multiplying Polynomials

d.Factoring

i.Trinomials

ii.Difference of Perfect Squares

iii.Sum and Difference of Cubes

9.Quadratic Equations

a.Factoring

b.Quadratic Formula

c.Solving by Substitution

d.The Discriminant

e.Equations with Radicals

10.Inequalities

a.Transitive Property of Inequality

b.Addition and Multiplication Properties

c.“And” vs. “Or”

d.Inequalities with Absolute Value

11.Rational Expressions

a.Simplifying Rational Expressions

b.Multiplying and Dividing Rational Expressions

c.Adding and Subtracting Rational Expressions

d.Solving Equations with Rational Expressions

CHAPTER 4 / ALGEBRA

41

12.Systems

a.Solving by Substitution

b.Solving by Linear Combination

c.No Solution vs. Infinite Solutions

d.Word Problems with Systems

13.*Binomial Theorem

EVALUATING EXPRESSIONS

Problems asking you to evaluate an expression represent the easiest of the algebra questions on the Math Level 2 test. To answer this type of question, simply substitute the value given for the variable.

E X A M P L E :

If x = n3 and n = 1 y, then find the value of x when y = 4.

2

Substitute y = 4 into the second equation to get n = 1 (4) = 2.

2

Now substitute n = 2 into the first equation to get x = 23 = 8.

The correct answer is 8.

FRACTIONS

Simplifying Fractions

Fractions are in simplest form when the numerator and denominator have no common factor other than 1. To simplify a fraction, factor both the numerator and denominator.

Don’t cancel terms that are not common factors. Below is a common mistake:

x2 − 4 ≠ x2 .

82

x2 − 4 and 8 do not have a common factor, so this expression is already in simplest form.

E X A M P L E :

Simplify (3x + 12) .

(3x + 3y)

=3(x + 4) 3(x + y)

=(x + 4) x ≠ − y (x + y)

Factor the numerator and denominator. 3 is a common factor.

(This restriction is important because you cannot divide by zero!) (Answer)

*Denotes concepts that are on the Level 2 test only.

42	PART II / MATH REVIEW

Least Common Denominator

The least common denominator (LCD) of two or more fractions is the least common multiple (LCM) of their denominators. To find the LCD:

1.Factor each denominator completely and write as the product of prime factors. (Factor trees are usually used for this.)

2.Take the greatest power of each prime factor.

3.Find the product of these factors.

E X A M P L E :

Find the LCD of 1 and 7 .

430

4 = 2 × 2 = 22. 30 = 2 × 3 × 5.

The greatest power of 2 is 22. The greatest power of 3 is 3, and the greatest power of 5 is 5.

22 × 3 × 5 = 60.
60 is the LCD.	(Answer)

The least common denominator is helpful when adding and subtracting

	fractions.
1 + 7		= 15	+ 14 = 29 .
4	30	60	60	60

The least common denominator is also used when simplifying complex fractions. A complex fraction is a fraction whose numerator or denominator contains one or more fractions. Find the LCD of the simple fractions and multiply the numerator and denominator of the complex fraction by it.

E X A M P L E :
			5
Simplify				x		.
		1	−
					5x
	x

We first start by finding a common denominator for the expression in the

numerator, which is x. This gives us:

5

5

=

x

=

x

.

1

−

5x2

1

− 5x2

x

x

x

5

x

=

×

x

1 − 5x2

=

5

.

(1

− 5x2 )

CHAPTER 4 / ALGEBRA

43

Multiplying Fractions

a × c = ac

To multiply fractions, simply multiply straight across. b d bd .

E X A M P L E :

Simplify 2 × 3 × 4 .

5 6 7

=2 × 3 × 4 .

5 × 6 × 7

=. Divide by a common 210 factor to simplify.

=4 . (Answer)24

35

You can also simplify the fractions before multiplying to save time.

E X A M P L E :
	2	×	3	×	4	=	1	×	1	×	4	. Remove the common factors of 2 and 3.
5
		6		7		5		1		7

=1 × 1 × 4 .

5 × 1 × 7

=4 .

35

To divide by a fraction, multiply by its reciprocal. a ÷ c = a × d . This is b d b c

known as the Division Rule for Fractions. Of course, b, c, and d cannot equal zero because you cannot divide by zero.

E X A M P L E :

Simplify 18 ÷ 6 .

11

=18 × 11.

6

=3 × 11. Divide through by a common factor of 6. 1

= 3 × 11 = 33.	(Answer)
1

Using Mixed Numbers and Improper Fractions

A mixed number represents the sum of an integer and a fraction.

3 1 = 3 + 1

44