Peterson's New SAT Math Workbook (Academic Test Preparation Series)

354 Pages·2004·3.6 MB·English

by  Peterson's

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Contents
About the SAT.............................................................................................................................vii
1. Operations with Whole Numbers and Decimals.................................................................1
Diagnostic Test • Addition of Whole Numbers • Subtraction of Whole Numbers •
Multiplication of Whole Numbers • Division of Whole Numbers • Addition or
Subtraction of Decimals • Multiplication of Decimals • Division of Decimals • The
Laws of Arithmetic • Estimating Answers • Retest • Solutions to Practice Exercises
2. Operations with Fractions...................................................................................................19
Diagnostic Test • Addition and Subtraction • Multiplication and Division • Simplifying
Fractions • Operations with Mixed Numbers • Comparing Fractions • Retest • Solutions
to Practice Exercises
3. Verbal Problems Involving Fractions..................................................................................39
Diagnostic Test • Part of a Whole • Finding Fractions of Fractions • Finding Whole
Numbers • Solving with Letters • Retest • Solutions to Practice Exercises
4. Variation................................................................................................................................53
Diagnostic Test • Ratio and Proportion • Direct Variation • Inverse Variation • Retest •
Solutions to Practice Exercises
5. Percent...................................................................................................................................69
Diagnostic Test • Fractional and Decimal Equivalents of Percents • Finding a Percent of
a Number • Finding a Number When a Percent Is Given • To Find What Percent One
Number Is of Another • Percents Greater Than 100 • Retest • Solutions to Practice
Exercises
6. Verbal Problems Involving Percent ....................................................................................85
Diagnostic Test • Percent of Increase or Decrease • Discount • Commission • Profit and
Loss • Taxes • Retest • Solutions to Practice Exercises
7. Averages..............................................................................................................................103
Diagnostic Test • Simple Average • To Find a Missing Number When an Average Is
Given • Weighted Average • Retest • Solutions to Practice Exercises
8. Concepts of Algebra—Signed Numbers and Equations.................................................115
Diagnostic Test • Signed Numbers • Solution of Linear Equations • Simultaneous
Equations in Two Unknowns • Quadratic Equations • Equations Containing Radicals •
Retest • Solutions to Practice Exercises
9. Literal Expressions .............................................................................................................133
Diagnostic Test • Communication with Letters • Retest • Solutions to Practice Exercises
v
vi Contents
10. Roots and Radicals.............................................................................................................143
Diagnostic Test • Addition and Subtraction of Radicals • Multiplication and Division of
Radicals • Simplifying Radicals Containing a Sum or Difference • Finding the Square
Root of a Number • Retest • Solutions to Practice Exercises
11. Factoring and Algebraic Fractions....................................................................................155
Diagnostic Test • Simplifying Fractions • Addition or Subtraction of Fractions •
Multiplication or Division of Fractions • Complex Algebraic Fractions • Using
Factoring to Find Missing Values • Retest • Solutions to Practice Exercises
12. Problem Solving in Algebra...............................................................................................171
Diagnostic Test • Coin Problems • Consecutive Integer Problems • Age Problems •
Investment Problems • Fraction Problems • Mixture Problems • Motion Problems •
Work Problems • Retest • Solutions to Practice Exercises
13. Geometry ............................................................................................................................197
Diagnostic Test • Areas • Perimeter • Right Triangles • Coordinate Geometry • Parallel
Lines • Triangles • Polygons • Circles • Volumes • Similar Polygons • Retest • Solutions
to Practice Exercises
14. Inequalities..........................................................................................................................231
Diagnostic Test • Algebraic Inequalities • Geometric Inequalities • Retest • Solutions to
Practice Exercises
15. Numbers and Operations, Algebra, and Fractions..........................................................243
16. Additional Geometry Topics, Data Analysis, and Probability........................................273
Practice Test A....................................................................................................................313
Practice Test B ....................................................................................................................319
Practice Test C ....................................................................................................................327
Solutions to Practice Tests................................................................................................333
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About the SAT
PURPOSE OF THE SAT
The SAT is a standardized exam used by many colleges and universities in the
United States and Canada to help them make their admissions decisions. The test
is developed and administered by Educational Testing Service (ETS) for the Col-
lege Entrance Examination Board.
The SAT consists of two different types of exams designated SAT and SAT II.
The SAT tests verbal and mathematical reasoning skills — your ability to under-
stand what you read, to use language effectively, to reason clearly, and to apply
fundamental mathematical principles to unfamiliar problems. SAT II tests mastery
of specific subjects such as Chemistry or French or World History.
TAKING THE SAT
The SAT is offered on one Saturday morning in October, November, December,
January, March, May, and June. When you apply to a college, find out whether it
requires you to take the SAT and if so when scores are due. To make sure your
scores arrive in time, sign up for a test date that’s at least six weeks before the
school’s deadline for test scores.
Registration forms for the SAT are available in most high school guidance of-
fices. You can also get registration forms and any other SAT information from:
College Board SAT Program
P.O. Box 6200
Princeton, NJ 08541-6200
609-771-7600
Monday through Friday, 8:30 a.m. to 9:30 p.m. Eastern Time
www.collegeboard.com
Along with your registration form you will receive a current SAT Student Bulletin.
The bulletin includes all necessary information on procedures, exceptions and spe-
cial arrangements, times and places, and fees.
vii
viii About the SAT
FORMAT OF THE NEW SAT
The new SAT is a three-hour, mostly multiple-choice examination divided into
sections as shown in the chart below. One of the sections is experimental. Your
score on the six nonexperimental sections is the score colleges use to evaluate your
application.
The critical reading sections of the SAT use Sentence Completions to measure
your knowledge of the meanings of words and your understanding of how parts of
sentences go together, and Critical Reading questions (short and long passages) to
measure your ability to read and think carefully about the information presented in
passages.
The mathematical sections use Standard Multiple-Choice Math, Quantitative
Comparisons, and Student-Produced Response Questions to test your knowledge
of arithmetic, algebra, and geometry. Many of the formulas that you need will be
given in the test instructions. You are not required to memorize them. SAT math
questions are designed to test your skill in applying basic math principles you
already know to unfamiliar situations.
The experimental section of SAT may test critical reading or mathematical rea-
soning, and it can occur at any point during the test. This section is used solely
by the testmakers to try out questions for use in future tests. You won’t know which
section it is. So you’ll have to do your best on all of the sections.
FORMAT OF A TYPICAL SAT
Number of
Section #/Content Questions Time
(1) Writing 1 essay 25 min.
(2) Mathematics 20 25 min.
Standard Multiple Choice
(3)* “Wild Card” an Experimental Section varies 30 min.
(Varies with test)
(4) Critical Reading 16 25 min.
Sentence Completions 8
(5) Writing 35 25 min.
Standard Multiple Choice
(6) Mathematics 25 min.
Standard Multiple-choice 8
Grid-Ins 10
(7) Critical Reading 19 25 min.
Sentence Completions 5
(8) Mathematics 20 min.
Standard Multiple Choice 16
(9) Critical Reading 13 20 min.
Sentence Completions 6
* Can occur in any section
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About the SAT ix
THE NEW SAT MATH QUESTIONS
The mathematical reasoning sections of the SAT test problem solving in numbers
and operations, algebra I and II, geometry, statistics, probability, and data analysis
using two question types:
• Standard multiple-choice questions give you a problem in arithmetic,
algebra, or geometry. Then you choose the correct answer from the five
choices.
• Grid-Ins do not give you answer choices. You have to compute the answer
and then use the ovals on the answer sheet to fill in your solution.
Although calculators are not required to answer any SAT math questions, students
are encouraged to bring a calculator to the test and to use it wherever it is helpful.
Mathematics tests your knowledge of arithmetic, algebra, and geometry. You
are to select the correct solution to the problem from the five choices given.
Example:
If (x + y)2 = 17, and xy = 3, then x2 + y2 =
(A) 11
(B) 14
(C) 17
(D) 20
(E) 23
Solution:
The correct answer is (A).
(x + y)2 = 17
(x + y)(x + y) = 17
x2 + 2xy + y2 = 17
Since xy = 3,
x2 + 2(3) + y2 = 17
x2 + 6 + y2 = 17
x2 + y2 = 11
Student-Produced Responses test your ability to solve mathematical problems when
no choices are offered.
Example:
On a map having a scale of 1 inch = 20 miles, how many inches should
4
there be between towns that are 70 miles apart?
Solution:
The correct answer is 7 or .875, depending upon whether you choose
8
to solve the problem using fractions or decimals.
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x About the SAT
Using fractions Using decimals
1 x .25 x
4 = =
20 70 20 70
70 20x=17.5
20x=
4 x=.875
70 1  7
x=   =
 4 20 8
HOW TO USE THE ANSWER GRID
The answer grid for student-produced response (grid-ins) questions is similar to
the grid used for your zip code on the personal information section of your answer
sheet. An example of the answer grid is shown below.
The open spaces above the grid are for you to write in the numerical value of your
answer. The first row of ovals has only two ovals in the middle with a “/”. These
allow you to enter numbers in fractional form. Since a fraction must have both a
numerator and a denominator, it is not possible that the leftmost or rightmost posi-
tions could have a “/”. To protect you from yourself, there are no “/s” in those
positions. The next row has decimal points. The horizontal bar separates the frac-
tion lines and decimal points from the digits 0 to 9. Record your answers to grid-in
questions according to the rules that follow.
GRID RULES
1. Write your answer in the boxes at the top of the grid.
Technically this isn’t required by the SAT. Realistically, it gives you something
to follow as you fill in the ovals. Do it—it will help you.
2. Mark the bubbles that correspond to the answer you entered in the boxes.
Mark one bubble per column. The machine that scores the test can only read the
bubbles, so if you don’t fill them in, you won’t get credit. Just entering your
answer in the boxes is not enough.
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