155 Diagnostic Test • Simplifying Fractions • Addition or Subtraction of Fractions • Multiplication or Division of Fractions • Complex Algebraic Fractions • Using Factoring to Find Missi
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Trang 5Contents
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
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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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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
(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
Standard Multiple Choice (6) Mathematics 25 min Standard Multiple-choice 8
(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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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 14 inch = 20 miles, how many inches should
there be between towns that are 70 miles apart?
Solution:
The correct answer is 78 or 875, depending upon whether you choose
to solve the problem using fractions or decimals
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Using fractions Using decimals
1 4
20 70
4 70 4
1 20
7 8
=
=
=
=
x
x x
25
875
=
=
=
x x x
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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3 Start your answer in any column, if space permits.
Unused columns should be left blank Don’t put in zeroes Look at this
example:
Here are two ways to enter an answer of “150.”
4 Work with decimals or fractions.
An answer can be expressed as 34 or as 75 Do not put a zero in front of a
decimal that is less than 1 Just remember that you have only four spaces to work
with and that a decimal point or a fraction slash uses up one of the spaces
For decimal answers, be as accurate as possible but keep within the four spaces
Say you get an answer of 1777 Here are your options:
Fractions do not have to be simplified to simplest form unless they don’t fit in
the answer grid For example, you can grid 104, but you can’t grid 1216 because
you’d need five spaces So, you would simplify it and grid 34
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5 Express a mixed number as a decimal or as an improper fraction.
If you tried to grid 134, it would be read as 134 , which would give you a wrong answer Instead you should grid this answer as 1.75 or as 74
6 If more than one answer is possible, grid any one.
Sometimes the problems in this section will have more than one correct answer
In such cases, choose one answer and grid it For example, if a question asks for a prime number between 5 and 13, the answer could be 7 or 11 Grid 7 or grid 11, but don’t put in both answers
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CALCULATORS AND THE SAT
Calculators are allowed on the SAT You may bring to your exam any of the
fol-lowing types of calculators:
• four-function
• scientific
• graphing
You may not bring calculators of the following types:
• calculators with paper tape or printers
• laptop computers
• telephones with calculators
• “hand-held” microcomputers
Make sure that the calculator you bring is one you are thoroughly familiar with
WHEN TO USE A CALCULATOR
No question requires the use of a calculator For some questions a calculator may
be helpful; for others it may be inappropriate In general, the calculator may be
useful for any question that involves arithmetic computations Remember, though,
that the calculator is only a tool It can help you avoid inaccuracies in computation,
but it cannot take the place of understanding how to set up and solve a
mathemati-cal problem
Here is a sample problem for which a calculator would be useful:
Example:
The cost of two dozen apples is $3.60 At this rate, what is the cost of 10
apples?
(A) $1.75
(B) $1.60
(C) $1.55
(D) $1.50
(E) $1.25
Solution:
The correct answer is (D)
Make a ratio of apples to dollars:
apples dollars:
$
24
3 60 10
24 36 36
24 1 50
=
=
x x
x
A calculator would be useful in solving this problem Although the calculations
are fairly simple, the calculator can improve your speed and accuracy
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Here is a problem for which a calculator would not be useful:
Example:
Joshua travels a distance of d miles in t - 6 hours At this rate, how many miles will he travel in t2 - 36 hours?
(A) d(t + 6)
(B) d(t - 6)
(C) d
t+ 6
(D) d
t− 6
(E) t
d
+ 6
Solution:
The correct answer is (A)
rate = distance time Joshua's rate = d
t− 6
To calculate his new distance, use distance = rate × time
Distance = d
d
−
−
=
−
+
2
( ))( )
t
d t
−
6 6
This is an algebra problem Using a calculator would not be helpful
SCORING THE SAT
Every correct answer is worth one point If you leave an answer blank, you score
no point For incorrect answers to all verbal questions and to regular mathematics questions, you lose one-fourth of a point For incorrect answers to quantitative comparisons, you lose one-third of a point For incorrect answers to student-pro-duced responses, there is no penalty The penalties for wrong answers are intended
to discourage random guessing
Regardless of the number of questions on the test, all SAT scores are reported
on a scale of 200 to 800 The scores are based on the nonexperimental sections and are broken down into separate math and verbal scores
Five or six weeks after the exam, your scores will be sent to the colleges you have named on your registration form, to your high school, and to you
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HOW TO USE THIS BOOK
The math review that follows is designed as a self-teaching text to help you
pre-pare for the mathematics sections of the SAT At the beginning of each chapter,
you will find a ten-question diagnostic test Try this test before you read the
chap-ter Check your answers with the solutions provided at the end of the chapchap-ter If
you get eight to ten questions right, you may skip that chapter and go right on to
the next diagnostic test at the beginning of the following chapter Or you may
prefer to skim the instructional material anyway, just for review, but not bother
with the practice exercises If you get five to seven questions right, you might do
the practice exercises only in the sections dealing with problems you missed If
you get fewer than five questions right, you should work carefully through the
entire chapter
At the end of each chapter you will find a retest that is similar to the diagnostic
test After working through the chapter, you should do better on the retest If not,
go back and review any instructional material dealing with errors you made before
proceeding to the next chapter
Working diligently through each chapter in this manner will strengthen your
weaknesses and prepare you to get your best score on the three Practice new SAT
Math Tests at the end of this book—and on your actual SAT
Good luck