Mathematical Thinking.pdf

CLAST Preparation Guide
To accompany
Mathematical Thinking
and Quantitative Reasoning
Richard N. Aufmann
Palomar College
Joanne S. Lockwood
New Hampshire Community Technical College
Richard D. Nation
Palomar College
Daniel K. Clegg
Palomar College
____________________________________
Rosalie Abraham
Florida Community College at Jacksonville
Houghton Mifflin Company Boston New York

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Printed in the U.S.A.
ISBN 13: 978-0-618-77740-2
ISBN 10: 0-618-77740-7
1 2 3 4 5 6 7 8 9- XX -10 09 08 07 06

TABLE OF CONTENTS
Preface iii
Diagnostic Tests and Answers 1
Chapter 1: Sets and Logic
1.1 Sets 21
1.2 Symbolic Transformation 26
1.3 Negations 28
1.4 Equivalent Statements 31
1.5 Valid and Invalid Arguments 34
1.6 CLAST-Like Questions (includes answers and solutions) 39
Chapter 2: Arithmetic
2.1 Basic Operations on Rationals 51
2.2 Interchanging Fractions, Decimals, and Percents 56
2.3 Percents 59
2.4 Place Value, Rounding, and Estimations 63
2.5 Exponents, Comparisons, and Sequences 68
2.6 Word Problems 74
Chapter 3: Algebra
3.1 Operations with Real Numbers 89
3.2 Real Number Properties and Scientific Notation 94
3.3 Linear Equations and Inequalities 98
3.4 Substitutions 103
3.5 Graphing Equations and Inequalities 106
3.6 Factoring and the Quadratic Equation 118
3.7 Proportions and Variations 125
3.8 Word Problems 130
Chapter 4: Geometry
4.1 Linear Measurement 145
4.2 Perimeter 148
4.3 Area 153
4.4 Volume and Surface Area 160
4.5 Appropriate Measurement 167
4.6 Angle Measurement 169
4.7 Triangles 175
© Houghton Mifflin Company. All rights reserved.

x CONTENTS
4.8 Pythagorean Theorem 182
4.9 Polygons 187
Chapter 5: Probability and Statistics
5.1 Counting Techniques 201
5.2 Probability and Odds 204
5.3 Addition Rule and Multiplication Rule 207
5.4 Unbiased Samples 213
5.5 Statistical Graphs 215
5.6 Mean, Median, Mode 221
Practice Examinations
Practice Examination A (includes answers and solutions) 237
Practice Examination B (includes answers and solutions) 253
Conversion Score Table 267
Answers to Check Your Progress and See If You Remember 271
Glossary 285

PREFACE
About The Study Guide
This guide is designed to give CLAST (College Level Academic Skills Test) students a
review of basic mathematics skills. It will augment the ability of students who have been
exposed to sets, logic, arithmetic, algebra, geometry, probability, and statistics. Students who
are not proficient in the topics mentioned above need to take a liberal arts mathematics
course, or read related texts such as Mathematical Thinking and Quantitative Reasoning by
Aufmann, Nation, Lockwood, and Clegg, which is the text associated with this study guide.
General information in the CLAST, with emphasis on mathematics, will be given in this
section of the guide, as well as a diagnostic test. The guide also includes five chapters for
each topic in the math CLAST, two practice examinations, an answer key, and a glossary.
CLAST information will be given in question and answer format.
The diagnostic test has five sections: sets and logic; arithmetic; algebra; geometry;
probability and statistics. Each section has fifteen questions. If a student answers twelve or
more questions correctly in a certain topic, he may review the corresponding chapter at
leisure. On the other hand, if a student answers less than twelve questions correctly, he
should commence review of the corresponding chapter immediately. The recommended
order of chapter review is from the lowest topic score to the highest topic score.
Each chapter of the guide covers one of the five topics for the mathematics portion of the
CLAST. Each chapter is divided into sections, which address particular objectives for the
math sub-test. These objectives are stated at the start of the section. Definitions, rules,
formulas, or explanations are given next, followed by examples and their solutions.
Subsequently, the student is given an opportunity to “check his progress” by way of
questions. Towards the end of a section there are review questions to “see if he remembers”.
At the end of each chapter there are twenty-five multiple-choice questions with answers and
explanations.
The last section of the guide has two practice examinations, each containing fifty multiple-
choice questions, followed by the answers and their solutions. In addition, there is a score
converter sheet that would allow students to convert their raw score to the reported score as
given by the examination board. Students are encouraged to take at least one of these practice
examinations so as to determine their level of competency before taking the actual
mathematics sub-test.
Acknowledgements
I wish to thank my dear friend, Deborah Boykin, and my loving husband, Wayne Abraham,
for their support, encouragement, expertise, suggestions, and criticisms in the production of
this study guide.
iii

iv PREFACE
What is the College Level Academic Skills Test (CLAST)?
The CLAST is an achievement test that measures selected communication and mathematics
skills. It includes four sub-tests: essay, English language skills, reading, and mathematics.
The mathematics sub-test is multiple-choice.
Is the CLAST a required test?
Demonstrating proficiency of basic college-level communications and mathematics skills is
required for
• The award of the associate in arts degree from a community college or state
university
• Admission to upper-division status in a state university
• Receipt of a baccalaureate degree from a state university
The CLAST is one measure of students’ attainment. However, there are alternative ways for
students to display proficiency. For further information on these alternatives, contact the
department responsible for the CLAST at your college or university.
How many types of CLAST are there?
There is the traditional paper-and-pencil CLAST, which is administered on a Saturday three
times during the year: once in the spring, once in the summer, and once in the winter. In
addition, there is the Computer Adaptive Testing CLAST (CAT-CLAST), which is
administered on weekdays during regular working hours. Both tests measure the same level
of competency.
Do teachers take the CLAST?
Prior to July 1, 2002, the required basic skills test for teacher certification had been the
CLAST. Beginning July 1, 2002, the required basic skills test for teacher certification is the
General Knowledge Test, which is very similar to the CLAST. For further information on the
General Knowledge Test, contact your local school board.
What is a passing score on the mathematics sub-test of the CLAST?
The passing mathematics score is at least 295 points, which is equivalent to answering
approximately 35 or more questions correctly.

PREFACE v
How many questions are there on the mathematics sub-test?
The number of questions on the paper-and-pencil mathematics sub-test can range from 55 –
61. However, only 50 questions are used to calculate a student’s score. The other questions
are experimental to determine use in a future test.
The number of questions on the CAT-CLAST varies dependent upon the student’s
performance during the test. However, 30 – 32 questions are usual.
May students retake the CLAST?
Students who have not passed a sub-test of the CLAST may retake it during another
administering for which they are properly registered. Students may retake as many sub-tests
as necessary to meet the CLAST requirements. However, the waiting period to retake a sub-
test is thirty-one days. If a student takes the sub-test of the CLAST prior to the thirty-one day
time frame, the scores will be invalid and no score report will be mailed.
Students may not retake any CLAST subtest for which they already have a passing score.
What is the total testing time for the CLAST?
The total testing time for the CLAST is five hours, including time for instructions, and a
break. The time allotted for the mathematics sub-test is 90 minutes.
Retake examinees are allowed double time for each sub-test. Thus, there are 3 hours for a
mathematics sub-test retake.
When will results be mailed?
Test results will be mailed to students within six weeks of taking the test.
How do students register for the CLAST?
Students may register for the CLAST at any institution that can determine the eligibility to
take the test. Normally, this will be the institution in which the students are enrolled.
Where is the CLAST administered?
The CLAST is administered in all community colleges, state universities, and many private
institutions in Florida.

vi PREFACE
How do students prepare to take the CLAST?
Community colleges and state universities in the state of Florida are required to afford
students the opportunity to acquire the skills that are measured in the CLAST.
Each community college and state university has resources and staff to assist students in
preparing for the CLAST.
Students may choose to individually prepare by using study guides such as this one.
What do students need for the CLAST?
Students need a rested brain and a well-nourished body. In addition, students need an
admission ticket, two forms of identification with at least one showing a picture, several soft
leaded pencils with erasers, and blue or black ball-point pens.
No calculators are currently allowed. However, you may verify the policy on calculators
with your testing center prior to the examination.
What are the advantages of the CAT-CLAST?
You may answer fewer questions.
You may take it at your convenience.
The test is offered more often (everyday at some institutions).
You receive an unofficial score immediately.
What are the disadvantages of the CAT-CLAST?
You may have “computer-phobia.”
You have to answer the questions in the order presented. Once answered, a question cannot
be retrieved. There is no skipping or returning to questions to change answers.

PREFACE vii
CLAST Mathematics Skills
Sets and Logic
• Deduce facts of set inclusion or set non-inclusion from a diagram
• Identify negations of simple and compound statements
• Determine equivalence and nonequivalence of statements
• Draw logical conclusions from data
• Recognize invalid arguments with true conclusions
• Recognize valid reasoning patterns shown in everyday life
• Select applicable rules for transforming statements without affecting their meaning
• Draw logical conclusions when facts warrant them
Arithmetic
• Add, subtract, multiply, and divide rational numbers in fractional form
• Add, subtract, multiply, and divide rational numbers in decimal form
• Calculate percent increase and percent decrease
• Solve “a % of b is c,” where two of the variables are given
• Recognize the meaning of exponents
• Recognize the role of the base number in the base-ten numeration system
• Identify equivalent forms of decimals, percents, and fractions
• Determine the order relations between real numbers
• Identify a reasonable estimate of a sum, average, or product
• Infer relations between numbers in general by examining number pairs
• Solve real-world problems that do not involve the use of percent
• Solve real-world problems that involve the use of percent
• Solve problems that involve the structure and logic of arithmetic
Algebra
• Add, subtract, multiply, and divide real numbers
• Apply the order-of-operations agreement
• Use scientific notation
• Solve linear equations and inequalities
• Use formulas to compute results
• Find particular values of a function
• Factor a quadratic expression
• Find the roots of a quadratic equation
• Solve a system of two linear equations in two unknowns
• Use properties of operations correctly
• Determine whether a number is among the solutions of a given equation or inequality
• Recognize statements and conditions of proportionality and variation
• Identify regions of the coordinate plane that correspond to specific conditions and
vice versa
• Use applicable properties to select equivalent equations or inequalities
• Solve real-world problems involving the use of variables
• Solve problems that involve the structure and logic of algebra

viii PREFACE
Geometry and Measurement
• Round measurements
• Calculate distance, area, and volume
• Identify relationships between angle measures
• Classify simple plane figures by recognizing their properties
• Recognize similar triangles and their properties
• Identify units of measurement for geometric objects
• Infer formulas for measuring geometric figures
• Select applicable formulas for computing measures of geometric figures
• Solve real-world problems involving perimeters, areas, and volumes of geometric
figures
• Solve real-world problems involving the Pythagorean property
Probability and Statistics
• Identify information contained in graphs
• Determine the mean, median, and mode
• Use the fundamental counting principle
• Recognize properties and interrelationships among the mean, median, mode
• Choose the most appropriate procedures for selecting an unbiased sample
• Identify the probability of a specified outcome
• Infer relations and make accurate predictions from studying statistical data
• Interpret real-world data involving frequency and cumulative frequency tables
• Solve real-world problems involving probabilities

Diagnostic Test
This test is designed to determine a student’s strengths and weaknesses for the purpose of
planning an effective review, thereby using this study guide efficiently.
The diagnostic test is issued in five sections:
• Sets and Logic
• Arithmetic
• Algebra
• Geometry
• Probability and Statistics
Each section of the test has fifteen questions with an allotted time of 25 minutes per section.
One may take a break between sections, but not during a section test.
Use your time wisely. Do not linger on a question. If you are unsure about the working or the
solution to a question, place a star next to it and move onto another question. Give priority to
the questions that come easy to you. You can always go back to your starred questions later,
if time permits. However, do not leave any questions unanswered; by a process of
elimination make a logical choice.
The answers to the diagnostic test are provided at the end of the test. The interpretation for
the score in each section is:
14 – 15 correct: Review this chapter at your leisure.
11 – 13 correct: Review this chapter, but not necessarily first.
Below 11 correct: Review this chapter immediately.
The recommended order of review for the chapters in this study guide depends on the scores
of the diagnostic test. Review the chapters in order of least score to highest score.
DO NOT USE A CALCULATOR
1

2 DIAGNOSTIC TESTS
Sets and Logic – Diagnostic Test 25 minutes
1. Consider the diagram. Which statement is true if all regions are occupied?
a) Any element of A is an element of B.
b) Any element of B is an element of A.
c) Some element of A is an element of C.
d) None of the above.
2. Select the rule that directly transforms statement i) into statement ii).
i) Not all pens are blue.
ii) Some pens are not blue.
a) “ All are not p ” into “ Some are not p ”.
b) “ Not (not p) ” into “ p ”.
c) “ All are not p ” into “ None are p ”.
d) “ If p then q ” into “ If not q, then not p ”.
A
B
C
i) It is not true that Jim speaks Spanish or English.
ii) Jim does not speak Spanish and does not speak English
a) “ Not (p or q) ” into “ Not p and not q ”.
b) “ Not (p and q) ” into “ Not p or not q ”.
c) “ None are p ” into “ All are not p ”.

DIAGNOSTIC TESTS 3
4. Negate: Jack will study and he will pass the test.
a) Jack will not study and he will not pass the test.
b) Jack will not study or he will not pass the test.
c) If Jack studies, then he will not pass the test.
d) Jack will study or he will pass the test.
5. Find the negation of “ If loving you is wrong, then I don’t want to be right.”
a) Loving you is not wrong or I want to be right.
b) If loving you is not wrong, then I want to be right.
c) If I want to be right, then loving you is not wrong.
d) Loving you is wrong and I want to be right.
6. Negate: All teachers are intelligent.
a) Some teachers are intelligent.
b) No teacher is intelligent.
c) Some teachers are not intelligent.
d) Every teacher is intelligent.
7. Negate: Some students are taking Algebra I.
a) All students are taking Algebra I.
b) Some students are not taking Algebra I.
c) Not all students are not taking Algebra I.
d) No students are taking Algebra I.
8. Select a statement that is logically equivalent to “ If Jerry is tall, then Betty is
short.”
a) If Betty is not short, then Jerry is not tall.
b) Jerry is tall and Betty is short.
c) If Betty is short, then Jerry is tall.
d) If Jerry is not tall, then Betty is short.

4 DIAGNOSTIC TESTS
9. Select the logical equivalent to “ It is not true that I like mathematics and you like
English.”
a) I do not like mathematics and you do not like English.
b) I like mathematics or you like English.
c) If I like mathematics, then you like English.
d) I do not like mathematics or you do not like English.
10. Select the statement that is logically equivalent to “ It is not true that if it rains, it
pours.”
a) If it does not rain, then it does not pour.
b) It rains and it does not pour.
c) It does not rain and it does not pour.
d) If it does not rain, then it pours.
11. Consider the following pair of statements and find a valid conclusion, if possible.
i) No student likes homework.
ii) All teaches like homework.
a) Some students are teachers.
b) No teachers are students.
c) Some students like homework.
12. Consider the following pair of statements and find a valid conclusion.
i) All drivers have a license.
ii) Some drivers own a car.
a) All drivers own a car.
b) All licensed people drive.
c) Some licensed people who are not drivers own a car.

DIAGNOSTIC TESTS 5
13. Select the conclusion to make the argument valid.
If I take the bus, then I will arrive late.
If I arrive late, then I will work more hours.
a) If I work more hours, then I will take the bus.
b) I take the bus or I will work more hours.
c) If I take the bus, then I will work more hours.
d) I take the bus and I arrive late.
i) If you practice, then you will succeed.
ii) You will not succeed.
a) You practice.
b) You do not practice.
c) You practice and you will succeed.
d) You do not practice or you succeed.
15. To qualify for a loan of $100,000, an applicant must have a gross income of $40,000
if single or $50,000 if married and the applicant(s) must have assets of at least
$20,000. Read the requirements and each applicant’s qualifications for obtaining a
$100,000 loan. Select the qualified applicant.
Mr. A and his wife have assets of $25,000. He makes $32,000 and she makes
$23,000.
Mr. B is married with five children and makes $53,000. His wife is unemployed.
Miss C is single and works two jobs. She makes $28,000 on her day job and $13,000
on her night job. She has assets of $17,000.
a) Mr. A
b) Mr. B
c) Miss C
d) No one

6 DIAGNOSTIC TESTS
Arithmetic – Diagnostic Test 25 minutes
1. 82.965 + 3.47 + 108.2
a) 192.035 b) 194.635 c) 843.94 d) 84.394
2. 2.28 ÷ 0.012
a) 19 b) 190 c) 1.9 d) 0.19
3. ⎟
⎠
⎞
⎜
⎝
⎛
−
−
5
1
2
4
3
7
a)
20
19
5 b)
20
11
9 c)
20
11
5 d)
20
19
9
4. 407 % is equivalent to
a) 4.07 b) 407 c) 40.7 d) 0.407
5.
25
11
is equivalent to
a) 4.4 % b) 4.4 c) 0.44 % d) 0.44
6. 0.28 is equivalent to
a)
7
25
b) 2.8 % c)
25
7
d)
25
7
%
7. 60 is what percent of 40?
a) 50 % b) 150 % c) 6
.
66 % d)
2
3
%
8. An item that regularly sells for $80 is on sale for $60. What is the percent decrease?
a) 250 % b) 25 % c) 2.5 % d) 0.25 %

DIAGNOSTIC TESTS 7
9. Round to the nearest hundredth: 547.3951
a) 547 b) 547.395 c) 500 d) 547.4
10. Select the expanded form for 27.09.
a) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
⎟
⎠
⎞
⎜
⎝
⎛
×
+
× 2
3
10
1
9
10
1
7
10
2 b) ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
× 2
2
10
1
9
10
7
10
2
c) ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
× 2
0
10
1
9
10
7
10
2 d) ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
×
10
1
9
10
7
10
2 0
11. An investor buys 203 shares of stock. Each share costs $52.50. What is a reasonable
estimate of the purchase?
a) $8,000 b) $10,000 c) $12,000 d) $14,000
12. 3 3
+ 5 3
is equivalent to
a) ( 3 + 5 ) 3
b) ( 3 • 3 ) + ( 5 • 3 ) c) ( 3 + 5 ) 6
d) ( 3 • 3 • 3 ) + ( 5 •5 • 5 )
13. Place = , < , or > between the numbers: 17
.
9
?
7
1
.
9
a) > b) < c) =
14. A checking account charges $6 per month plus $0.20 per check. How many checks
were written in a month in which the total charges amounted to $22.00?
a) 6 b) 140 c) 80 d) 110
15. Find the largest positive integer which is a factor of 24 and 40 and is also a factor of
the sum of 24 and 40.
a) 4 b) 8 c) 16 d) 32

8 DIAGNOSTIC TESTS
Algebra – Diagnostic Test 25 minutes
1. Simplify: )
7
(
4
3
36
8 2
2
−
−
•
÷
−
a) 23 b) 55 c) 70 d) 82
2. Simplify: 8
3
18
5 −
a) 2
9 b) 10
2 c) 26
8 d) 2
21
3. Name the property: - 4(9x) + (- 4)(3) = - 4( 9x + 3)
a) associative property c) distributive property
b) commutative property d) additive identity
4. ( ) ( )
6
13
10
3
10
2
.
4 −
×
×
a) 8
10
26
.
1 × b) 8
10
26
.
1 −
× c) 19
10
4
.
1 × d) 8
10
6
.
12 −
×
5. Solve: – 4(x + 2) > 3x + 20
a) x ≥ – 4 b) x = – 4 c) x > – 4 d) x < – 4
6. Find the solution:
1
5
4
2
3
=
+
=
+
y
x
y
x
a) (-1, 1) b) (1, -1) c) (2, -1) d) (-1, 2)
7. If f (x) = x 2
+ 4x – 5, find f (-2).
a) –1 b) 7 c) – 9 d) –17
8. For which statements is –1 a solution?
i) | t – 1| = 0 ii) (x – 2)(x – 5) ≤ 6 iii) r 2
+ 4r + 16 = 13
a) i only b) iii only c) none d) all

DIAGNOSTIC TESTS 9
9. Which graph represents 2x + y = -3 ?
a) b)
c) d)
10. Identify the conditions that correspond to the shaded region.
a) 2x – 3y > 6 b) 2x – 3y ≥ 6 c) 2x – 3y ≤ 6 d) 2x – 3y < 6
11. Which is a factor of 3x 2
+ 7x – 6 ?
a) (x – 3) b) (3x – 2) c) (x – 2) d) (3x + 3)
4
3
2
1
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4
4
3
2
1
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4
4
3
2
1
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4
4
3
2
1
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4
4
3
2
1
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4

10 DIAGNOSTIC TESTS
12. Find the real roots: 2x 2
+ 3x = 1
a)
4
17
3 ±
− b) 1
,
2
1
−
− c)
4
17
3 ± d) 1
,
2
1
13. A machine can do 7 jobs in 4 hours. Which equation can be used to find the number of
hours (x) the machine takes to do 11 jobs?
a)
x
11
4
7
= b)
11
4
7
=
x
c)
4
11
7
=
x
d)
11
4
7 x
=
14. The tension (t) in a spring varies directly with the distance (d) the spring is stretched. If
the tension is 42 pounds when the spring is stretched 2 inches, find the tension when
the spring is stretched twice as far.
a) 4 inches b) 84 inches c) 21 inches d) 44 inches
15. The difference between a number and five more than three times the number is five.
What equation should be used to find the number?
a) x + (3x – 5) = 5 b) x – (3x + 5) = 5 c) b) x – (5x + 3) = 5 d) x – (5x – 3) = 5

DIAGNOSTIC TESTS 11
s reserved.
Geometry – Diagnostic Test 25 minutes
1.Convert 752.4 mm to cm.
a) 7524 cm b) 75.24 cm c) 7.524 cm d) 0.7524 cm
2. Round measurement A to the nearest ½ inch.
a) 2 inches b) 2
2
1
inches c) 2
4
3
inches d) 3 inches
3. Find the distance around a circular table of radius 1.3 m.
a) 1.3 π m b) 1.69 π m c) 1.96 π m 2
d) 2.6 π m
4. The length of a rectangle is 5 feet more than twice the width. If the perimeter is 22
feet. Find the length of the rectangle.
a) 2 ft b) 4 ft c) 9 ft d) 18 ft
5. Find the shaded area.
5 m
a) 25 m 2
b) 50 m 2
c) 75 m 2
d) 50 m
6. Find the volume of the cone.
2 cm
9 cm
a) 6 π cm 3
b) 36 π cm 3
c) 12 π cm 3
d) 18 π cm 3
A
1 2 3 4 5
10 m
© Houghton Mifflin Company. All right

12 DIAGNOSTIC TESTS
7. Jan is wrapping a cube shaped gift box. The side of the cube is 10 cm. How much
paper, in cm 2
, does Jan need to cover the gift box?
a) 10 cm 2
b) 60 cm 2
c) 100 cm 2
d) 600 cm 2
8. What is the appropriate measure for the amount of water in a pool?
a) meters b) square inches c) miles d) liters
9. Find the angle measurements if lines L1 and L2 are parallel.
L1 3x
L2 5x-20
a) 25 º b) 25 º, 75 º c) 75 º, 105 º d) 105 º
10. Find the angle measure of x.
110°
x
a) 90 º b) 70 º c) 110 º d) 180 º
11. Find the height of a flagpole that casts a shadow of 5 feet while a person that is 6 feet
tall casts a shadow of 3 feet.
a) 2
2
1
feet b) 4 ft c) 8 ft d) 10 ft
12. Find the missing leg of the triangle.
5 cm
13 cm
a) 12 cm b) 8 cm c) 144 cm d) 194 cm

DIAGNOSTIC TESTS 13
13. How high up a wall does a 50-foot ladder reach if the foot of the ladder is 30 feet
from the wall?
a) 30 feet b) 40 feet c) 1600 feet d) 3400 feet
14. What quadrilateral has all sides equal, but no right angles?
a) square b) rectangle c) rhombus d) trapezoid
15. Find the value of x.
2x
─ ─
3x + 10
a) 34 b) 68 c) 112 d) 180
″
″

14 DIAGNOSTIC TESTS
Probability and Statistics – Diagnostic Test 25 minutes
1. Jane can have her potato baked or fried. She can have her chicken baked, fried, or
barbequed. She can drink soda or water. How many choices does Jane have for a meal?
a) 3 b) 12 c) 7 d) 21
2. A student is asked to rank his 5 teachers from best to worst. How many ways can the
teachers be ranked?
a) 5 b) 25 c) 60 d) 120
3. A committee must consist of two men and two women. How many different
committees can be formed if there are four men and three women?
a) 18 b) 6 c) 9 d) 3
4. A box has 3 red pens, 2 black pens, and 4 green pens. What is the probability of
randomly selecting a black pen?
a)
9
3
b)
9
2
c)
9
4
d)
2
1
5. The probability that it will rain tomorrow is 47%. What is the probability that it will
not rain tomorrow?
a) 0.53 b) 0.47 c) 0.92 d) 1
6. A box has 3 red pens, 2 black pens, and 4 green pens. What are the odds in favor of
selecting a green pen?
a) 5 : 4 b) 4 : 9 c) 4 : 5 d) 9 : 4
7. Seventy percent of the students at a certain community college are taking Math.
Eighty percent are taking English, and sixty percent are taking both Math and English.
What percent of students are taking either Math or English?
a) 70 % b) 80 % c) 90 % d) 150 %

DIAGNOSTIC TESTS 15
8. Using the table below, what is the probability that a nurse received a good evaluation
given that she is part-time? (Table shows results from a survey of 200 nurses.)
Good Evaluations Bad Evaluations
Part-time 38 42
Full-time 84 36
a)
100
19
b)
40
19
c)
200
112
d)
100
61
9. A box has six good apples and four bad apples. If two apples are chosen at random
with replacement, what is the probability that the first one will be good, and the second
one will be bad?
a)
5
2
b)
5
3
c)
25
6
d)
5
1
10. A toy-store owner wants to determine which electronic game system is the most
popular among students in his county. Which procedure is best for obtaining an unbiased
sample?
a) Survey a sample of students in Art class.
b) Survey the first two hundred students who enter his store.
c) Survey a random sample of students from the entire student body.
d) Survey a random sample of student responses from Pre-K through 6th
graders.
For Question 11, use the graph below.
Class-Skipping Frequency
11. What percentage of students never skip class?
a) 25 % b) 75 % c) 37.5 % d) 125 %
Bi-weekly
25
Weekly
30
Monthly 70
75 Never

16 DIAGNOSTIC TESTS
For Question 12, use the graph below.
Yearly Rainfall
0
20
40
60
80
1991 1992 1993 1994 1995 1996
Year
Rainfall
(in
inches)
12. Which year had the greatest rainfall?
a) 1993 b) 1994 c) 1995 d) 1996
13. Consider a student’s scores for one semester in an English class: {85, 72, 93, 96, 88}.
What is the mean score?
a) 93 b) 87 c) 88 d) 89
14. More than half of the shirts in the store cost $15.00. Most of the other shirts cost
$25.00, and the remaining few cost $35.00. Which of the statements below is true?
a) The mean is greater than the mode.
b) The median is equal to the mean.
c) The median is less than the mode.
d) The mean is less than the median.
15. The scores on a placement examination are as follows:
Score Percentile Rank
25 99
20 82
15 74
10 45
5 22
What percentage of students taking the examination scored between 10 and 20?
a) 37 % b) 45 % c) 82 % d) 127 %

DIAGNOSTIC TESTS 17
Answers to Diagnostic Test
Sets and Logic
1 b 4 b 7 d 10 b 13 c
2 a 5 d 8 a 11 b 14 b
3 a 6 c 9 d 12 c 15 a
Score: _____
Arithmetic
1 b 4 a 7 b 10 c 13 a
2 b 5 d 8 b 11 b 14 c
3 d 6 c 9 d 12 d 15 b
Score: _____
Algebra
1 b 4 a 7 c 10 d 13 a
2 a 5 d 8 b 11 b 14 b
3 c 6 a 9 c 12 a 15 b
Score: _____
Geometry
1 b 4 c 7 d 10 b 13 b
2 b 5 a 8 d 11 d 14 c
3 d 6 c 9 c 12 a 15 a
Score: _____

18 DIAGNOSTIC TESTS
Probability and Statistics
1 b 4 b 7 c 10 c 13 b
2 d 5 a 8 b 11 c 14 a
3 a 6 c 9 c 12 d 15 a
Score: _____

q → r

21
1.1 SETS Textbook Reference Section 1.3
CLAST OBJECTIVE
" Deduce facts of set inclusion or set non-inclusion from a diagram
A set is a collection of objects and is usually denoted with a capital letter.
The objects in a set are called members or elements.
Symbols
∈ - “ is a member of ”
∉ - “ is not a member of ”
Examples Solutions
Consider the two sets A = { 1, 2, 3, 4, 5, 6 }
and B = { 2, 4, 6, 8 }.
Answer True or False.
a) 3 ∈ A
b) 1 ∉ B
c) 8 ∈ A
The first statement is read “ Three is a
member of set A.” The statement is true.
The second statement is read “ One is not a
member of set B.” The statement is true.
The last statement is read “ Eight is a
member of set A.” The statement is false.
Union
The union of two sets, denoted A B, is the set of all elements from sets A, B, or both.
∪
Intersection
The intersection of two sets, denoted by A ∩ B, is the set of elements that are common to
both sets A and B.
Example Solution
Let A = { 1, 2, 3, 4, 5, 6 } and
B = { 2, 4, 6, 8 }
d) Find A ∪ B.
A ∪ B = { 1, 2, 3, 4, 5, 6, 8 }
1
3
5
A B
8
2
4
6

22 CHAPTER 1 Sets and Logic
Example Solution
e) Find A B.
∩ A ∩ B = { 2, 4, 6 }
1
3
5
2
4
6
A B
8
Special Sets
The empty set contains no members. It is denoted by { } or Ø.
The universal set contains all possible elements. It is denoted by U.
The complement of a set contains elements in the universal set that do not belong to the set
under consideration. It is denoted by A '.
Example Solution
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
and A = {2, 4, 6, 8, 9, 10, 11}
f) Find A'. A ' = {1, 3, 5, 7, 12 }
Venn Diagrams
Venn Diagrams are used to give a visual representation of sets and their relationships. The
universal set is shown by a rectangle. All other sets are shown by circles inside the universal
set.
Consider the shaded regions for the sets mentioned.
Set A
Set A'
U
A
U
A'

SECTION 1.1 Sets 23
Set A ∪ B
A union B
B union A
Set A B
∩
A intersect B
B intersect A
Disjoint sets A and B have no common elements.
The intersection of
disjoint sets is the
empty set.
A ∩ B = Ø
A is a subset of B if all members of A are contained in B.
Note that B is not a
subset of A.
A
U
B
U
B
A
B
A
U
U
B
A
Example Solution
g) Put the following information in a Venn
Diagram.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8}
C = {10} 9
7
1
3
5
2
4
6
8
10
C
U
A B

Examples Solutions
h) Consider the Venn Diagram and find U, A,
B, C, A B, A ∪ B, and B C.
∩ ∩
1
6
U = {1, 2, 3, 4, 5, 6}
A = {2, 3, 4}; B = {4, 5}; C = {3}
A ∩ B = {4}
A ∪ B = {2, 3, 4, 5}
B C = {}
∩
i) Consider the Venn Diagram below.
Note: No regions are empty.
Describe the relationship between sets A and
B, and the relationship between sets B and C.
Set B is a subset of set A because the set
B is contained in the set A.
Sets B and C are disjoint: the two sets
have no common elements.
U
2
4
5
3
A B
C
A
C
B
U
Check Your Progress 1.1
For Questions 1 – 4, use the Venn Diagram below.
1. Find U. 2. Find B.
3. Find A ∩ B. 4. Find B C.
∪
2 5 6
8
3
7
4 9
1
A
B
C
U

SECTION 1.1 Sets 25
For Questions 5 – 8, use the Venn Diagram below.
5. Find U. 6. Find B.
7. Find A ∩ B. 8. Find C ∪ A.
For Questions 9 and 10, use the Venn Diagram below.
9. Name two disjoint sets.
10. Name two intersecting sets.
For Questions 11 - 15, use the Venn Diagram below to fill in the blanks.
11. A is a subset of ______________. 12. C is a subset of ______________.
13. B is a subset of ______________. 14. Sets B and C intersect at ______.
15. Sets A and B intersect at _____________.
2
3 4
5
1
A
B C
U
U
A
B
C
A
B
C
U
No region is empty.

1.2 SYMBOLIC TRANSFORMATION Textbook Reference Section 2.1
CLAST OBJECTIVES
" Identify negations of simple and compound statements
" Select applicable rules for transforming statements without affecting their
meaning
A simple statement is a sentence that can be classified as true or false.
“Today is Friday.” is a statement.
“Red is the best color.” is not a statement.
“It is raining.” is a statement.
A statement can be denoted by a letter, usually p , q , or r.
Two statements can be connected by “ or ” , “ and ” , or “ if…then…”.
A statement can be negated or made completely opposite by introducing the word “ not ”.
Statement Symbolic Form
a) Today is Friday. p
b) It is raining. q
c) Today is Friday and it is raining. p and q
d) Today is Friday or it is raining. p or q
e) If today is Friday, then it is raining. if p then q
f) Today is not Friday. not p
g) Today is not Friday and it is raining. not p and q
h) Today is not Friday and it is not raining. not p and not q
i) It is not true that today is Friday or it is not raining. ~ (p or not q)
Note: “ It is not true that ” is a negation sign , ~ , followed by parentheses.
For Questions 1 – 5, use the following statements and write in symbolic form.
p: I live in Florida. q: The sun is shining. r: Bill Clinton is the president.
1. The sun is not shining.
2. I live in Florida and Bill Clinton is president.
3. If the sun is shining, then I live in Florida.
4. It is not true that I live in Florida and the sun is not shining.
5. If Bill Clinton is not the president, then the sun is shining.

SECTION 1.2 Symbolic Transformation 27
Transforming One Statement to Another
Examples Solutions
j) Write the rule that transforms statement “ i ” into
statement “ ii ”.
Statement - i statement - ii
i) If today is Friday, then it is raining.
ii) Today is not Friday or it is raining.
If p, then q not p or q
k) Write the rule that transforms statement “ i ” into
i) It is not true that it is Friday and it is
raining.
ii) It is not Friday or it is not raining.
not (p and q ) not p or not q
l) Write the rule that transforms statement “ i ” into
i) It is not true that today is Friday or it is
not raining.
ii) Today is not Friday and it is raining.
not (p or not q ) not p and q
m) Write the rule that transforms statement “ i ” into
i) Not all dogs are bad.
ii) Some dogs are not bad.
not all are p some are not p
For Questions 6 – 10, write the rule that transforms statement i) into statement ii).
6.
i) Not all teachers are smart.
ii) Some teachers are not smart.
7.
i) If the snow is falling, then it must be Christmas.
ii) The snow is not falling or it must be Christmas.
8.
i) It is not true that Mary eats chocolate and peanuts.
ii) Mary does not eat chocolate or she does not eat peanuts.
9.
i) All tests are not easy.
ii) No test is easy.
10.
i) Not all musicians are guitarists.
ii) Some musicians are not guitarists.

1.3 NEGATIONS Textbook Reference Section 2.1
CLAST OBJECTIVE
" Identify negations of simple and compound statements
Case 1: Negating Statements
Statement Negation
p not p
not p p
p and q not p or not q
p or q not p and not q
if p then q p and not q
Examples
Statement Negation
a) It is Friday. It is not Friday.
b) It is not raining. It is raining.
c) It is Friday and it is raining. It is not Friday or it is not raining.
d) It is Friday or it is raining. It is not Friday and it is not raining.
e) If it is Friday, then it is raining. It is Friday and it is not raining.
Write the negation of each of the statements below.
1. Mary is not at home. 2. The fish weighs five pounds.
3. It is January and the sun is shining. 4. The sun is not shining or it is July.
5. If the fish weighs five pounds, then Lilly will cook dinner.
Case 2: Negating Universal Quantifiers: All, Some, None
Statement Negation
All…are… Some…are not…
Some…are not… All…are…
Some…are… No…are…
No…are… Some…are…

SECTION 1.3 Negations 29
The Negation “X” below may be helpful.
Every…are…
All…are… Some...are …
No…are…
None of the…are…
Some...are not…
Examples
Statement Negation
f) All students have pencils. Some students do not have pencils.
g) Some students will pass the test. No student will pass the test.
h) Some dogs are not brown. All dogs are brown.
i) No cat drinks milk. Some cats drink milk.
Write the negation of each of the statements below.
6. Some children eat vegetables. 7. Some cats do not drink milk.
8. No dog plays the piano. 9. All dogs can jump rope.
10. Every boy plays football. 11. None of the babies cry.
12. Some babies drink juice.

See If You
Remember
SECTIONS 1.1 & 1.2
For Questions 1 – 3, write the rule that transforms statement i) into statement ii).
1.
i) All singers are not talented.
ii) No singer is talented.
2.
i) If it is Tuesday, then I am studying math.
ii) It is not Tuesday or I am studying math.
3.
i) Not all elephants are gray.
ii) Some elephants are not gray.
For Questions 4 – 6, use the following Venn Diagram.
4. Is there an element that is in sets A, B, and C?
5. What is the relationship between sets B and C?
6. What is the relationship between sets A and U?
U
A B
C

SECTION 1.4 Equivalent Statements 31
1.4 EQUIVALENT STATEMENTS Textbook Reference Section 2.1, 2.3
CLAST OBJECTIVE
" Determine equivalent and non-equivalent statements
Equivalent Statements are statements that are written differently, but hold the same
logical equivalence.
Case 1: “ If p then q ” has three equivalent statements.
RULE
Statement Equivalent Statement
1) If p then q not p or q
2) If p then q q or not p
3) If p then q If not q then not p
Examples
a) If it is blue, then it is the sky. It is not blue or it is the sky.
b) If it is blue, then it is the sky. It is the sky or it is not blue.
c) If it is blue, then it is the sky. If it is not the sky, then it is not blue.
Case 2: Statements beginning with “ It is not true that… ”
RULE
1) ~ (p and q) not p or not q
2) ~ (p or q) not p and not q
3) ~ (if p then q) p and not q
4) ~ (All p are q) Some p are not q
5) ~ (No p are q) Some p are q
6) ~ (Some p are q) No p are q
7) ~ (Some p are not q) All p are q
Examples
d) It is not true that Peter has an A and
Mary has a C.
Peter does not have a A or Mary does not
have a C.
e) It is not true that the car is red or the
pole is not blue.
The car is not red and the pole is blue.

Examples
f) It is not true that if Peter has an A, then
Mary has a C.
Peter has an A and Mary does not have a C.
g) It is not true that all apples are red. Some apples are not red.
h) It is not true that no banana is yellow. Some bananas are yellow.
i) It is not true that some grapes are not
green.
All grapes are green.
For Questions 1- 4, answer “ yes ” if the statements are equivalent. Answer “ no ” if the
statements are not equivalent.
1. If it is heavy, then I am strong.
If I am not strong, then it is not heavy.
2. If it is winter, then Tom is cold.
It is winter and Tom is cold.
3. If Jack wins, then Mary loses.
Jack does not win or Mary loses.
4. If cats like fish, the dogs like beef.
Dogs like beef or cats do not like fish.
For Questions 5 – 9, write an equivalent statement.
5. It is not true that Ben plays the flute or Jan plays the piano.
6. It is not true that chocolate is white and milk is green.
7. It is not true that all puppies are playful.
8. It is not true that some elephants are grey.
9. It is not true that no berries are red.

SECTION 1.4 Equivalent Statements 33
See If You
Remember
SECTIONS 1.1 – 1.3
For Questions 1 and 2, write the rule that directly transforms statement i) into statement
ii).
1.
i) Not all clowns are funny.
ii) Some clowns are not funny.
2.
i) If Mary has a B, then Kyle has a D.
ii) If Kyle does not have a D, then Mary does not have a B.
3.
i) It is not true that you study French or English.
ii) You do not study French and you do not study English.
For Questions 4 – 8, negate the statement.
4. No cats are furry.
5. All dogs are fluffy.
6. If Jim wins, then he flies to Vegas.
7. Tom is in French class and Ben is in Spanish class.
8. I am a student or Jean is a teacher.

1.5 VALID AND INVALID ARGUMENTS Textbook Reference Section 2.4, 2.5
CLAST OBJECTIVES
" Draw logical conclusions from data
" Draw logical conclusions when facts warrant them
" Recognize invalid arguments with true conclusions
" Recognize valid reasoning patterns shown in everyday language
An argument is made up of premises and a conclusion. Premises are statements that must
be accepted as true. The conclusion given may be invalid or valid. A valid conclusion is
logically deduced from the premises and thus the argument is valid.
Case 1: Arguments using universal quantifiers: all , some , none , no.
(Venn Diagrams aid in determining a valid conclusion for these types of arguments.)
Premise Diagram
All A’s are B’s
Some A’s are B’s
No A’s are B’s
B
A
B
A
B
A

SECTION 1.5 Valid and Invalid Arguments 35
Examples Solutions
a) Given the following:
i) No persons who
grade work are
intelligent.
ii) All teachers grade
work.
Find a valid conclusion.
For premise i, we need two circles: G for grade and I for
intelligent.
For premise ii, we add another circle, T, for teachers.
Conclusion: No teacher is intelligent.
b) Given the following:
i) All parents make
promises.
ii) Some parents are
liars.
For premise i, we need two circles: P a for parents and P r
for premises.
For premise ii, we add another circle, L, for liars.
Conclusion: Some people who make promises are liars.
c) Given the following:
i) All dogs are
playful.
ii) Rover is a dog.
For premise i, we need two circles: D for dogs and P for
playful.
For premise ii, we need to add a dot, R, for Rover.
Conclusion: Rover is playful.
I
G
T
I
G
P a
P r
L
P a
P r
P
D
R•
D
P

For Questions 1 – 6, read each pair of statements and find a valid conclusion, if possible.
1.
i) No people who assign work are rich.
ii) All teachers assign work.
2.
i) Some students are happy.
ii) All happy people are irritating.
3.
i) All politicians are liars.
ii) No liar is intelligent.
4.
i) Some dogs have fleas.
ii) Spot is a dog.
5.
i) All horses eat hay.
ii) Harry eats hay.
6.
i) All birds have wings.
ii) Robin is a bird.
Case 2: Arguments without universal qualifiers.
Five valid argument forms are symbolized below. Arguments outside these will be
considered invalid.
Valid Forms
1. i) If p then q
ii) p
2. i) If p then q
ii) not q
Therefore, q. Therefore, not p.
3. i) p or q
ii) not p
4. i) p or q
ii) not q
Therefore, q. Therefore, p.
5. i) If p then q
ii) If q then r
Therefore, if p then r.

SECTION 1.5 Valid and Invalid Arguments 37
Examples Solutions
d) Given the following:
i) If you wear a ring, then you are
married.
ii) You wear a ring.
i) If p then q
ii) p
Form 1 indicates that the conclusion is q:
You are married.
e) Given the following:
i) You study French or Spanish.
ii) You do not study Spanish.
i) p or q
ii) not q
Form 4 indicates that the conclusion is p:
You study French.
f) Given the following:
i) If you study, you will get a job.
ii) If you get a job, you can buy a car.
i) If p then q
ii) If q then r
Form 5 indicates that the conclusion is “ if
p then r ”: If you study, you can buy a car.
For Questions 7 – 11, consider each pair of statements and find a valid conclusion, if
possible.
7.
i) You play the piano or guitar.
ii) You do not play the piano.
8.
i) If you speed, you will get a ticket.
ii) If you get a ticket, you lose your license.
9.
i) If you water the plant, it will grow.
ii) You water the plant.
10.
i) You sing or dance.
ii) You do not dance.
11. i) If you run, then you will win.
ii) You do not win.

See If You
Remember
1. Consider the diagram below, in which no regions are empty.
What is the relationship between sets B and C?
For Questions 2 and 3, write the rule that directly transforms statement i) into statement
ii).
2.
i) Not all babies cry.
ii) Some babies do not cry.
3.
i) If today is Tuesday, then I will go to school.
ii) Today is not Tuesday or I will go to school.
For Questions 4 and 5, negate the statement.
4. If it does not rain, then I will go shopping.
5. Some dancers are in good shape.
Are the following pairs of statements equivalent?
6.
i) If the water is warm, Jan will go swimming.
ii) The water is not warm or Jan will go swimming.
7.
i) It is not true that you smoke and drink.
ii) You do not smoke and you do not drink.
8.
i) Not all students are failing the course.
ii) Some students are not failing the course.
A
B
C

SECTION 1.6 CLAST – Like Questions 39
1.6 CLAST-LIKE QUESTIONS No Calculator Allowed
1. Consider the diagram. Which statement is true if all regions are occupied?
a) Any element of A is an element of B.
b) Any element of B is an element of A.
c) Some element of A is an element of C.
2. Consider the diagram. Which statement is true if no region is empty?
a) No members of B are members of C.
b) No members of A are members of C.
c) Some members of B are members of C.
i) Not all pens are blue.
ii) Some pens are not blue.
a) “ All are not p ” into “ Some are not p ”.
b) “ Not (not p) ” into “ p “.
d) “ If p then q ” into “ If not q, then not p ”.
A
B
C
A
B
C

i) If it is a diamond, then it is small.
ii) If it is not small, then it is not a diamond.
a) “ If p then q ” into “ If not p, then not q ”.
b) “ If p then q ” into “ If q then p ”.
c) “ If p then q ” into “ If not q, then not p ”.
i) All grapes are not green.
ii) No grape is green.
a) “ Not (not p) ” into “ p ”.
b) “ Not some p ” into “ All are not p” .
i) It is not true that Jim speaks Spanish or English.
ii) Jim does not speak Spanish and does not speak English.
a) “ Not (p or q) ” into “ Not p and not q ”.
b) “ Not (p and q) ” into “ Not p or not q ”.
c) “ None are p ” into “ All are not p ”.
7. Negate: It is sunny or I am blind.
a) It is not sunny or I am not blind.
b) It is not sunny and I am not blind.
c) If it is sunny, then I am blind.
d) It is not sunny or I am blind.
8. Negate: Jack will study and he will pass the test.
a) Jack will not study and he will not pass the test.
b) Jack will study or he will pass the test.
c) If Jack studies, then he will not pass the test.
d) Jack will not study or he will not pass the test.

9. Find the negation of “ If loving you is wrong, then I don’t want to be right .”
a) Loving you is wrong and I want to be right.
b) If loving you is not wrong, then I want to be right.
c) If I want to be right, then loving you is not wrong.
d) Loving you is not wrong or I want to be right.
10. Negate: All teachers are intelligent.
a) Some teachers are intelligent.
b) No teacher is intelligent.
c) Some teachers are not intelligent.
d) Every teacher is intelligent.
11. Negate: Some students are taking Algebra I.
a) All students are taking Algebra I.
b) Some students are not taking Algebra I.
c) Not all students are not taking Algebra I.
d) No students are taking Algebra I.
12. Find the negation of “ No dogs have three legs. ”
a) Every dog has three legs.
b) All dogs have three legs.
c) Some dogs do not have three legs.
d) Some dogs have three legs.
13. Negate: Some horses are not in the race.
a) Some horses are in the race.
b) All horses are in the race.
c) No horse is in the race.
d) None of the horses are in the race.
14. Select a statement that is logically equivalent to “ If Jerry is tall, then Betty is short.”
a) If Betty is short, then Jerry is tall.
b) Jerry is tall and Betty is short.
c) If Betty is not short, then Jerry is not tall.
d) If Jerry is not tall, then Betty is short.

15. Select a statement that is logically equivalent to “ If Florida is sunny, then Chicago is
windy.”
a) Florida is not sunny or Chicago is windy.
b) If Chicago is windy, then Florida is sunny.
c) If Florida is not sunny, then Chicago is not windy.
d) Florida is not sunny and Chicago is windy.
16. Select the logical equivalent to “ It is not true that I like mathematics and you like
English.”
a) I do not like mathematics or you do not like English.
b) I like mathematics or you like English.
c) If I like mathematics, then you like English.
d) I do not like mathematics and you do not like English.
17. Select the logical equivalent to “ It is not true that Bob sings or Jane dances. ”
a) Bob does not sing or Jane does not dance.
b) Bob sings and Jane dances.
c) If Jane dances, then Bob sings.
d) Bob does not sing and Jane does not dance.
18. Select the statement that is logically equivalent to “ It is not true that if it rains, it
pours. ”
a) If it does not rain, then it does not pour.
b) It rains and it does not pour.
c) It does not rain and it does not pour.
d) If it does not rain, then it pours.
19. Consider the following pair of statements and find a valid conclusion, if possible.
i) No student likes homework.
ii) All teaches like homework.
a) Some students are teachers.
b) No teachers are students.
c) Some students like homework.

i) All dogs have tails.
ii) Bruno has a tail.
a) Some Brunos have a tail.
b) Bruno is a dog.
c) Some dogs have tails.
i) All drivers have a license.
ii) Some drivers own a car.
a) All drivers own a car.
b) All licensed people drive.
c) Some licensed people who are not drivers own a car.
If I take the bus, then I will arrive late.
If I arrive late, then I will work more hours.
a) If I work more hours, then I will take the bus.
b) I take the bus or I will work more hours.
c) If I take the bus, then I will work more hours.
d) I take the bus and I arrive late.
I will go to the beach or to the movies.
I will not go to the movies.
a) I will not go to the beach.
b) I will go the beach and to the movies.
c) I will go to the movies.
d) I will go to the beach.

i) If you practice, then you will succeed.
ii) You will not succeed.
a) You do not practice.
b) You practice.
c) You practice and you will succeed.
d) You do not practice or you succeed.
25. To qualify for a loan of $100,000, an applicant must have a gross income of $40,000
if single or $50,000 if married and the applicant(s) must have assets of at least
$20,000. Read the requirements and each applicant’s qualifications for obtaining a
$100,000 loan. Select the qualified applicant.
Mr. A and his wife have assets of $25,000. He makes $32,000 and she makes
$23,000.
Mr. B is married with five children and makes $53,000. His wife is unemployed.
Miss C is single and works two jobs. She makes $28,000 on her day job and $13,000
on her night job. She has assets of $17,000.
a) Mr. A
b) Mr. B
c) Miss C
d) No one

Answers to CLAST-LIKE QUESTIONS 1.6
1. b 6. a 11. d 16. a 21. c
2. a 7. b 12. d 17. d 22. c
3. a 8. d 13. b 18. b 23. d
4. c 9. a 14. c 19. b 24. a
5. c 10. c 15. a 20. d 25. a
Explanations
1. B is a subset of A. Therefore, any element of B is an element of A.
2. B and C are disjoint. Therefore, no members of B are members of C.
3. Transferring to symbolic form: “ Not all are p ” implies “ Some are not p ”. The only
similar choice is a.
4. Transferring to symbolic form: “ If p then q ” implies “ If not q, then not p ”.
5. Transferring to symbolic form: “ All are not p ” implies “ None are p ”.
6. Transferring to symbolic form: “ Not ( p or q ) ” implies “ Not p and not q ”.
7. Recall: “ p or q ” negates to “ not p and not q ”. It is not sunny and I am not blind.
8. Recall: “ p and q ” negates to “ not p or not q ”. Jack will not study or he will not pass the
test.
9. Recall: “ if p then q ” negates to “ p and not q ”. Loving you is wrong and I want to be
right.
10. Recall: “ All ” negates to “ Some are not ”. Some teachers are not intelligent.
11. Recall: “ Some are ” negates to “ None/No are ”. No students are taking Algebra I.

12. Recall: “ None are ” negates to “ Some are ”. Some dogs have three legs.
13. Recall: “ Some are not ” negates to “ All are ”. All horses are in the race.
14. Recall: “ If p then q ” negates to “ If not q, then not p ”. If Betty is not short, then Jerry
is not tall.
15. Recall: “ If p then q ” is equivalent to “ Not p or q ”. Florida is not sunny or Chicago is
windy.
16. Recall: “ Not ( p and q ) ” is equivalent to “ Not p or not q ”. I do not like mathematics
or you do not like English.
17. Recall: “ Not ( p or q ) ” is equivalent to “ Not p and not q ”. Bob does not sing and
Jane does not dance.
18. Recall: “ Not ( if p then q ) ” is equivalent to “ p and not q ”. It rains and it does not
pour.
19. From premise 1, we have two circles. Adding premise 2, we get
H
S
T
H
S
Conclusion: No teachers are students.

20. From premise 1, we get Adding premise 2, we get
D
T
D
●
●
T
Bruno can be inside Circle D (dogs) and have a tail, Circle T.
Bruno can be outside Circle D (dogs) and have a tail, Circle T.
Bruno is not necessarily a dog, therefore no proper conclusion can be formed.
21. From premise 1, we get Adding premise 2, we get
D
L T
D
L
Conclusion: Some licensed people who are not drivers own a car.
22. In symbolic form: If p then q
If q then r
Therefore, If p then r. (Form 5)
Conclusion: If I take the bus, then I will work more hours.
23. In symbolic form: p or q
not q
Therefore, p. (Form 4)
Conclusion: I will go to the beach.

24. In symbolic form: If p then q
not q
Therefore, not p. (Form 2)
Conclusion: You do not practice.
25. Mr. A and his wife meet the criteria.
assets of $25,000 > $20,000
income = 32,000 + 23,000 = 55,000
Mr. B has no assets. Therefore, he does not meet the criteria.
Miss C has assets of $17,000 < $20,000. She does not meet the criteria.
Conclusion: The only qualified applicant is Mr. A.

5
1
04
.
0

51
2.1 BASIC OPERATIONS ON RATIONALS Textbook Reference Chapter 3
CLAST OBJECTIVES
" Add, subtract, multiply, and divide rational numbers in fractional form
" Add, subtract, multiply, and divide rational numbers in decimal form
Case 1: Decimals
Operation Rule
Addition/Subtract
Vertically align the decimal points. In subtraction, write
zeroes to the right of the decimal point to make the portion
of the number to the right of the decimal points of equal
length. Add or subtract as usual. The decimal point in the
answer must be vertically aligned with the other decimal
points.
Multiplication
Multiply as usual. In the answer, the number of places after
the decimal point must be the same as the total number of
decimal places in the original problem.
Division
Make the divisor a whole number by moving the decimal to
the right of the last digit. Move the decimal point in the
dividend the same number of places. Divide as usual. The
decimal point in the answer is directly above the decimal
point in the dividend.
Examples Solutions
a) 300.5 + 17.29 + 0.331
331
.
0
29
.
17
5
.
300
318.121
b) 300.5 – 17.29
29
.
17
5
.
300
− 29
.
17
50
.
300
−
283.21
1. Line up decimal points.
2. Add
3. Keep decimal point in
answer aligned with the other
decimal points.
1. Line up decimal points.
2. Place a 0 after the 5 in 300.5
3. Subtract.
3. Keep decimal point in answer
aligned with the other decimal points.

52 CHAPTER 2 Arithmetic
Examples Solutions
c) (300.5)(0.27)
27
.
0
5
.
300
×
21035
+ 6010
81135
Answer: 81.135
d) 300.5 ÷ 0.32 5
.
300
32
.
0
Move the decimal in 0.32 to the right of the digit 2.
That’s two places. Move the decimal in 300.5 two
places to the right also. Add zeroes as needed.
Divide as usual and place the decimal point in the
quotient directly over the decimal point in the
dividend.
0625
.
939
0
160
160
64
80
192
200
0
20
288
290
96
125
288
0000
.
30050
32
1. Multiply.
2. The number 300.5 has one
decimal place and the
number 0.27 has two decimal
places. This totals to three
decimal places.
3. Count over three places
from the last digit in the
answer and place a decimal
point there.

SECTION 2.1 Basic Operations on Rationals 53
1. 96.35 + 4.297 2. 7.2 + 498.76 + 22.459
3. 72 – 5.35 4. 856.41 – 25.7
5. (23.9)(4.4) 6. (171.2)(0.35)
7. 9.64 ÷ 0.004 8. 35 ÷ 0.25
Case 2: Fractions
Operation Rule
Addition
Fractions must have the same (common) denominator. Change all
fractions to “fraction form”, i.e. no mixed fractions. Denominators are
never added. The easiest way to find a common denominator is to
multiply the numerator and the denominator of the first fraction by the
denominator of the second fraction. Multiply the numerator and the
denominator of the second fraction by the original denominator of the
first fraction. Finally, add the numerators. Denominators are never
added.
Subtraction
Fractions must have the same (common) denominator. Change all
fractions to “fraction form”, i.e. no mixed fractions. Again, the easiest
way to find a common denominator is to multiply the numerator and
the denominator of the first fraction by the denominator of the second
fraction. Multiply the numerator and the denominator of the second
fraction by the original denominator of the first fraction. Finally,
subtract the numerators. Denominators are never subtracted.
Multiplication
Write all fractions in “fraction form”, i.e. no mixed fractions. Multiply
the numerators and multiply the denominators.
Division
Write all fractions in “fraction form”, i.e. no mixed fractions. Change
division to multiplication. Multiply the first fraction by the reciprocal
of the fraction following the division symbol. Proceed as in
multiplication

Examples Solutions
e) )
3
(
5
2
−
−
−
5
3
2
5
13
5
15
2
5
15
5
2
)
5
(
1
)
5
(
3
5
2
1
3
5
2
)
3
(
5
2
=
=
+
−
=
+
−
=
+
−
=
+
−
=
−
−
−
f) 6
7
1
2 +
−
7
6
3
7
27
7
42
15
7
42
7
15
)
7
(
1
)
7
(
6
7
15
1
6
7
15
6
7
1
2
=
=
+
−
=
+
−
=
+
−
=
+
−
=
+
−
g)
4
3
6
1
1 +
12
11
1
24
22
1
24
46
24
18
28
24
18
24
28
)
6
(
4
)
6
(
3
)
4
(
6
)
4
(
7
4
3
6
7
4
3
6
1
1
=
=
=
+
=
+
=
+
=
+
=
+
h)
4
1
2
7
4
×
7
2
1
7
9
4
7
9
4
4
9
7
4
4
1
2
7
4
=
=
×
×
=
×
=
×
i)
3
1
15
14
−
÷
−
5
4
2
5
14
1
15
3
14
1
3
15
14
3
1
15
14
=
=
×
−
×
−
=
−
×
−
=
−
÷
−
1
1
-1
5

SECTION 2.1 Basic Operations on Rationals 55
9.
9
5
3
9
1
4 +
11.
7
4
3
2
1 +
10.
8
3
3
11−
−
12. ⎟
⎠
⎞
⎜
⎝
⎛
−
−
−
7
3
5
13.
5
3
5
10 − 14.
5
4
2
3 +
−
15.
2
1
3
15
2
×
17. ⎟
⎠
⎞
⎜
⎝
⎛
−
×
−
3
1
8
5
4
16.
7
3
2
35×
−
18.
3
2
2
1
4 ÷
19.
2
1
24
1
÷
− 20.
8
3
32
7
−
÷
−

2.2 INTERCHANGING FRACTIONS, DECIMALS, AND PERCENTS
Textbook Reference Section 3.3
CLAST OBJECTIVE
" Identify equivalent forms of decimals, percents, and fractions
Rule 1: Fraction to Percent
To change a fraction to a percent, multiply the fraction by 100.
Rule 2: Fraction to Decimal
To change a fraction to a decimal, divide the numerator by the denominator.
Examples Solutions
a) Write
8
3
as a percent.
%
5
.
37
%
2
1
37
2
75
1
2
25
3
1
100
8
3
100
8
3
or
=
=
×
×
=
×
=
×
25
2
b) Write
8
3
as a decimal.
0
40
40
56
60
24
000
.
3
8
0.375
Rule 3: Decimal to Percent
To change a percent to decimal, move the decimal point two places to the right.
Rule 4: Decimal to Fraction
To change a decimal to a fraction, remove the decimal point and place the number
over 10 or 100 or 1000, etc. The number of zeroes is equal to the number of digits
to the right of the decimal point in the original number.
Examples Solutions
c) Write 5.43 as a percent. Move decimal two places to the right and add
a percent sign: 5.43 = 543. = 543 %
d) Write 5.43 as a fraction. Note that there are two digits to the right of
the decimal point. We will drop the decimal
point and place the number over 100.
100
43
5
100
543
=

SECTION 2.2 Interchanging Fractions, Decimals, and Percents 57
Rule 5: Percent to Decimal
To change a percent to decimal, drop the percent sign and move the decimal point
two places to the left.
Rule 6: Percent to Fraction
To change a percent to a fraction, drop the percent sign and write the number over
100 and reduce, if possible.
Examples Solutions
e) Write 71 % as a decimal. 71 % = 0.71
f) Write 71 % as a fraction.
100
71
%
71 =
1. Change
4
3
to a percent. 2. Change
5
1
6 to a percent.
3. Write
8
5
as a decimal. 4. Write
5
2
9 as a decimal.
5. Write 0.732 as a percent. 6. Write 1.3 as a percent.
7. Change 0.89 to a fraction. 8. Change 12.4 to a fraction.
9. Change 47 % to decimal. 10. Change 319 % to decimal.
11. Change 52 % to fraction. 12. Change 150 % to fraction.

See If You
Remember
SECTION 2.1
1. 3.25 + 161.9 + 22.831
2. 20.4 – 9.005
3. 0.57 • 6.21
4. 14.4 ÷ 0.024
5. ⎟
⎠
⎞
⎜
⎝
⎛
−
+
−
7
3
4
3
1
1
6. ⎟
⎠
⎞
⎜
⎝
⎛
−
−
9
4
2
5
3
2
7.
4
3
7
2
2 ×
−
8. ⎟
⎠
⎞
⎜
⎝
⎛
−
÷
−
9
2
5
1

SECTION 2.3 Percents 59
2.3 PERCENTS Textbook Reference Section 3.3
CLAST OBJECTIVES
" Solve ‘ a % of b is c ’ where two variables are given
" Calculate percent increase and percent decrease
" Solve real world problems that involve the use of percent
There are many methods which can be used to work percent questions. In this guide, we will
practice
100
%
=
of
is
. Read each problem carefully. Replace “is” with the number before or
after the word “ is ” in the problem. Replace “ of ” with the number after the word “ of ”
in the problem. Replace “ % ” with the percent given in the problem. Proceed to solve by
cross-multiplying and dividing.
Examples Solutions
a) What is 6 % of 50? We are given 6 %. Replace “of” with 50 and
“is” with a variable, x.
3
100
300
100
100
300
100
)
6
(
50
)
100
(
100
6
50
100
%
=
=
=
=
=
=
x
x
x
x
x
of
is
b) 40 is 20 % of what number? “ % ” = 20; “ is ” = 40; “ of ” = x.
x
x
x
x
x
of
is
=
=
=
=
=
=
200
20
20
20
4000
20
4000
)
20
(
)
100
(
40
100
20
40
100
%
c) 15 is what percent of 75? “ % ” = x; “ is ” = 15; “ of ” = 75.
%
20
75
75
75
1500
75
1500
)
(
75
)
100
(
15
100
75
15
100
%
=
=
=
=
=
=
x
x
x
x
x
of
is

1. What is 140 % of 80? 2. 5 is what percent of 35?
3. 35 is what percent of 28? 4. What is 35 % of 70?
5. 15 is 15 % of what number? 6. 24 is 60 % of what number?
Percent Increase:
100
% increase
original
by
increased
=
Percent Decrease:
100
% decrease
original
by
decreased
=
Example Solution
d) If 20 is decreased to 16, what is
the percent decrease?
Decreased by = 20 – 16 = 4
Original = 20
x
x
x
x
x
decrease
original
by
decreased
=
=
=
=
=
=
%
20
20
20
20
400
20
400
)
(
20
)
100
(
4
100
20
4
100
%

SECTION 2.3 Percents 61
Examples Solutions
e) If you increase 30 by 150 % of
itself, what is the result?
Increased by = x
Original = 30; % increase = 150
45
100
4500
100
100
4500
100
)
150
(
30
)
100
(
100
150
30
100
%
=
=
=
=
=
=
x
x
x
x
x
increase
original
by
increased
Result: 30 + 45 = 75
f) A shirt that regularly sells for $20
is on sale at 40% off. How much do
you save, and what is the sale price?
Decreased by = x
Original = 20; % decrease = 40
8
100
800
100
100
800
100
)
40
(
20
)
100
(
100
40
20
100
%
=
=
=
=
=
=
x
x
x
x
x
decrease
original
by
decreased
Saved $8.00
Sale Price: $20 – $8 = $12.00
7. If 50 is increased to 80, what is the percent increase?
8. If 25 is decreased to 5, what is the percent decrease?
9. If you decrease 90 by 30 %, what is the result?

See If You
Remember
10. If you increase 20 by 120 %, what is the result?
11. This month Jack made 60 % as much money as he did last month. If Jack made
$7,200 last month, what was the decrease in income?
12. An item is being sold for 80 % of its regular price of $50. How much would you
save if you bought the item?
13. A game that regularly sells for $40 is on sale at 30 % off. What is the sale price?
14. The new air condition system lets Mary save 90 % on her electricity bill. If Mary’s
previous bill was $85, how much will Mary save on this bill?
SECTIONS 2.1 & 2.2
1. 32.4 – 16.093 2. 36.6 ÷ 0.012
3.
4
3
5
3
1
3 +
− 4. ⎟
⎠
⎞
⎜
⎝
⎛
−
÷
4
3
3
2
1
7
5. Write
20
17
as a decimal. 6. Write
8
7
as a percent.
7. Change 143.5 % to a decimal. 8. Change 0.225 to a fraction.
9. Change 55 % to a fraction. 10. Change 4.39 to a percent.
11. Change %
2
1
14 to a fraction. 12. Write 100 % as a decimal.

SECTION 2.4 Place Value, Rounding, and Estimations 63
2.4 PLACE VALUE, ROUNDING, AND ESTIMATIONS
Textbook Reference Chapter 3
CLAST OBJECTIVES
" Recognize the role of the base number in the base – ten numeration system
" Identify a reasonable estimate of sum, average, or product
Place Value
The place value of any digit can be found by following the format in the chart above.
Examples Solutions
a) What is the place value of 8 in 5.3897? The 8 is two places right of the decimal
point. Thus, the place value of 8 is 2
10
1
.
b) What is the place value of 4 in 7,493.32? The 4 is three places left of the decimal
point. Thus, the place value of 4 is 10 2
.
c) Write the expanded form of 407.02. Consider the place values of the 4, 7, and 2.
( ) ( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
+
×
+
× 2
0
2
10
1
2
10
7
10
4
d) Write the numeral for
( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
×
10
1
5
10
3
10
2 3
2 is four places left of the decimal point.
3 is two places left of the decimal point.
5 is one place right of the decimal point.
2__3__.5
Place zeroes in the hundreds place and the
ones place.
Result: 2030.5

For Questions 1 – 4, give the place value of the underlined digit.
1. 8,394.5126 2. 8,394.5126
3. 8,394.5126 4. 8,394.5126
For Questions 5 – 7, write the numeral in expanded form.
5. 24.38
6. 7,040.09
7. 508.4
For Questions 8 – 10, write the numeral.
8. ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
×
10
1
7
10
5
10
1 0
2
9. ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
× 2
2
4
10
1
1
10
2
10
9
10. ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
⎟
⎠
⎞
⎜
⎝
⎛
×
+
× 3
2
0
10
1
7
10
1
9
10
4
Rounding
Note the digit x that is specified.
• If the digit after x is 5 or greater than 5, increase x by one and replace all the digits
after x with zeroes.
• If the digit after x is less than 5, leave x as it is and replace all the digits after x with
zeroes.
• Note that some zeroes will be insignificant.
Examples Solutions
e) Round 73.159 to the nearest tenth. The 1 is in the tenth place. The digit after 1
is 5, so increase 1 to 2.
Result: 73.200 or 73.2
f) Round 73.159 to the nearest one. The 3 is in the ones place. The digit after 3
is 1 which is less than 5, so do not increase 3.
Result: 73.000 or 73

11. Round 8,490.372 to the nearest hundredth.
12. Round 8,492.372 to the nearest one.
13. Round 8,490.372 to the nearest hundred.
14. Round 8,490.372 to the nearest tenth.
Estimation: A rough calculation using rounded numbers.
Examples Solutions
g) Gas is sold at $1.89 a gallon and Tom’s
car takes 19 gallons. What is a reasonable
estimate of Tom’s gas bill?
Round $1.89 to $2.00
Round 19 gallons to 20 gallons
Estimate: ($2)(20) = $40
h) Jenny bought perfume for $12.99, nail
polish for $3.15, lipstick for $7.95, and
earrings for $7.19. About how much money
did Jenny spend?
Round $12.99 to $13.00
Round $3.15 to $3.00
Round $7.95 to $8.00
Round $7.19 to $7.00
Estimate: $13 + $3 + $8 + $7 = $31
i) Three hundred students took an English
examination. All of the students scored less
than 93, but more than 61. What is a
reasonable estimate of the average score of
the students?
Estimate: 77
2
154
2
61
93
=
=
+
j) A music store has 30 employees. The
largest gross pay is $300 per week, and the
lowest is $200 per week. Estimate the
weekly payroll at the music store.
Average pay: 250
2
500
2
200
300
=
=
+
Estimate: 30($250) = $7,500
15. Bill owns 432.87 shares of a mutual fund valued at $40.17 per share. Find a
reasonable estimate of the value of Bill’s stock to the nearest hundred dollars.
16. A unit of water costs $2.09 and 60.34 units are used. Find a reasonable estimate of
the bill to the nearest hundred dollars.

17. Seventy students took a test. If the highest score was 96 and the lowest score was 72,
what was the class average.
18. On a certain day, the college cafeteria sold 70 lunch specials. The lowest special was
$3 and the highest was $7. What is the estimate of income from lunch specials that day?
19. A bag of fertilizer covers 1.75 acres. What is a reasonable estimate of the number of
acres that could be covered with 159.5 bags of fertilizer?
20. Below are the prices of the four most active stocks on a certain Exchange and their
closing prices per share in dollar.
Tallar Corp
8
5
11
Paxin Inc
4
3
3
Fun NY
2
1
4
3
1
14
Aninus
What would be a reasonable estimate of the average closing price per share for these
stocks?
See If You
Remember
1. 349.248 + 2.9 + 51.73 2. (22.8)(1.4)
3. ⎟
⎠
⎞
⎜
⎝
⎛
−
−
7
1
5
2
4 4. ⎟
⎠
⎞
⎜
⎝
⎛
−
÷
−
5
3
4
1
2

5. Change 0.25 % to decimal.
6. Change
8
7
to decimal.
7. Write 0.655 as a fraction.
8. 18 is 30 % of what number?
10. If 32 is decreased to 24, what is the percent decrease?
11. If you increase 60 by 150 %, what is the result?
12. Peter weighed 160 pounds. After the vacation, his weight increased by 20 %. What is
his new weight?
13. A car that originally cost $25,000.00 has now depreciated by 10%. What is the
present value of the car?

2.5 EXPONENTS, COMPARISONS AND SEQUENCES
Textbook Reference Chapter 3
CLAST OBJECTIVES
" Recognize the meaning of exponents
" Determine the order relation between real numbers
" Infer relations between numbers in general by examining pairs
Exponents
Consider a n
, where a is the base and n is the exponent. The exponent n indicates how many
times the base a must be used as a factor.
a
a
a
a
n
a ×
×
×
×
= ...
n factors of a
• a 0
= 1
• a 1
= a
• ( - a ) n
= - a ×- a ×…×- a
• - a n
= - (a × a ×…× a)
Examples Solutions
a) 2 4
2 4
= 2 × 2 × 2 ×2 = 16
b) ( - 2 ) 4
( - 2 ) 4
= - 2 × - 2 × - 2 × - 2 = 16
c) - 2 4
- 2 4
= - ( 2 × 2 × 2 × 2 ) = - 16
Write in expanded form.
d) ( 5 3
) 2
( 5 3
) 2
= ( 5 3
)( 5 3
) = ( 5 × 5 × 5 ) ( 5 × 5 × 5 )
e) ( 7 2
) ( 3 4
) ( 7 2
) ( 3 4
) = ( 7 × 7 ) ( 3 × 3 × 3 × 3 )
f) ( 7 2
) + ( 3 4
) ( 7 2
) + ( 3 4
) = ( 7 × 7 ) + ( 3 × 3 × 3 × 3 )
Write in expanded form.
1. ( 6 2
) ( 9 5
) 2. ( 7 3
) 2
3. ( 4 2
) + ( - 3 ) 4
4. ( 3 + x ) 2
5. ( 11 3
) ( 8 3
) 6. - 2 2

SECTION 2.5 Exponents, Comparisons, and Sequences 69
Comparison
Numbers are compared by placing the appropriate sign between them: an equal sign ( = ), a
greater than sign ( > ), or a less than sign ( < ).
Case 1: Comparing Fractions
Rewrite the fractions so that they have the same denominator. Then compare the numerators.
Case 2: Comparing Decimals
Rewrite the decimals to the same number of places after the decimal point. If there are
repeating decimals, expand the pattern. Then compare the numbers.
Case 3: Comparing Radicals
Compare the radicands, the numbers under the radical sign.
Note that numbers must be of the same type before comparisons can be made. Both numbers
must be fractions or decimals or radicals.
Examples Solutions
Compare the following numbers by placing
= , < , or > between them.
g)
11
6
?
5
2 First, get a common denominator.
11
6
5
2
55
30
55
22
55
30
?
55
22
)
5
(
11
)
5
(
6
?
)
11
(
5
)
11
(
2
11
6
?
5
2
<
⇒
<
h) 3
1
.
7
?
13
.
7 Expand the decimals.
3
1
.
7
13
.
7
1333
.
7
1313
.
7
1333
.
7
?
1313
.
7
3
1
.
7
?
13
.
7
<
⇒
<
i) 5
?
29 Make the numbers the same type by squaring
both of them.
( ) ( )
5
29
25
29
25
?
29
5
?
29
5
?
29
2
2
>
⇒
>
Compare numerators.

Examples Solutions
j) 85
?
9
.
8 Estimate 8.9 to the nearest whole number,
which is 9.
( ) ( )
85
9
.
8
85
81
85
?
81
85
?
9
85
?
9
2
2
<
⇒
<
k) 0.5 ? - 1.2 A positive number is always greater than a
ative number.
neg
0.5 > - 1.2
Compare each pair of numbers by placing = , < , or > between them.
7.
5
1
?
100
17
− 8.
33
25
?
4
3
9. 8.62 ? 8.672 10. 97
.
9
?
7
9
.
9
11. 53
?
50 12. 120
?
121
13. 7
.
9
?
80 14. 37
?
4
.
5
Sequences
A sequence is a series of numbers that have been arranged by a pattern. The pattern often
varies. It could be as simple as adding or multiplying by a constant. It could be a change that
takes place in the numerator or denominator or both. In working with sequences, remember
the following:
1. Examine the numbers.
2. Look for a pattern.
3. Test the pattern on all numbers that are given. If all the numbers test correctly, use
the pattern to predict the next numbers in the sequence.
4. If the test fails, start over with Step 1 and look for a different pattern.

Examples Solutions
Find the missing term in the
sequence.
l) ?
,
81
1
,
27
1
,
9
1
,
3
1
,
1 −
−
Notice that all the numbers are fractions except the
number 1. Make the number 1 look like a fraction and re-
examine the sequence.
?
,
81
1
,
27
1
,
9
1
,
3
1
,
1
1
−
−
In examining the sequence, notice the following:
1) The signs alternate. This indicates multiplication
by –1.
2) The numerators are the same. They are all 1.
3) The denominators look like the multiplication
table for the number 3. This means we can get the
next denominator by multiplying the previous
denominator by 3.
Now put it all together:
( )
r
denominato
get
to
numerator
in
always
signs
g
alternatin
←
←
⎟
⎠
⎞
⎜
⎝
⎛
−
→
3
1
1
Test the pattern of multiplying by
3
1
− .
1st
term: 1
2nd
term: 1 •
3
1
− =
3
1
−
3rd
term:
3
1
− •
3
1
− =
9
1
3
3
1
1
=
×
−
×
−
4th
term:
27
1
27
1
3
9
1
1
3
1
9
1
−
=
−
=
×
−
×
=
−
•
5th
term:
81
1
3
27
1
1
3
1
27
1
=
×
−
×
−
=
−
•
−
The pattern tests correctly. The next term is
243
1
3
81
1
1
3
1
81
1 −
=
×
−
×
=
−
•
m) Find the missing term in the
sequence: 7, 11, 15, 19, ?
In examining the sequence, it looks like the pattern is to
add 4. Now test the pattern.
1st
term: 7 2nd
term: 7 + 4 = 11
3rd
term: 11 + 4 = 15 4th
term: 15 + 4 = 19
The pattern tests correctly. The next term is 19 + 4 = 23.

1st
term:
2nd
term:
3rd
term:
4th
term:
Complete 5th
term:
Examples
Examples Solutions
Solutions
n) Find the missing term in the
sequence:
?
,
26
1
,
20
1
,
14
1
,
8
1
,
2
1
After examining the sequence, it looks like the numerator
stays the same, which is 1. The denominators appear to
change by adding 6. Now, test the pattern.
1st
term:
2
1
2nd
term:
8
1
6
2
1
=
+
3rd
term:
14
1
6
8
1
=
+
4th
term:
20
1
6
14
1
=
+
5th
term:
26
1
6
20
1
=
+
The pattern tests correctly. The next term is
32
1
6
26
1
=
+
o) Look for a linear relationship
between each pair and fill in the
blank:
( ) )
(
⎜
⎜
⎝
⎛
−
−
⎟
⎠
⎞
,
3
,
15
,
2
1
,
2
5
,
2
,
10
( ) ⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
?
,
2
15
,
1
.
0
,
5
.
0
These are ordered pairs ( x , y ). How does x change to y ?
After examining the pairs, it looks like the pattern to get y
is to divide x by 5. Test the pattern.
)
(
)
(
)
(
⎜
⎜
⎝
⎛
⎟
⎠
⎞
=
×
=
÷
=
÷
−
−
−
=
÷
−
⎟
⎠
⎞
⎜
⎝
⎛
=
×
=
÷
=
÷
2
3
,
2
15
2
3
5
1
2
15
5
2
15
1
.
0
,
5
.
0
1
.
0
5
5
.
0
3
,
15
3
5
15
2
1
,
2
5
2
1
5
1
2
5
5
2
5
2
,
10
2
5
10
The pattern tests correctly.
Find the missing term in each sequence.
15. – 1 , 2 , 5 , 8 , 11 , ? 16. ?
,
32
1
,
16
1
,
8
1
,
4
1
,
2
1
17. – 3 , 9 , -27 , 81 , ? 18. ?
,
15
1
,
22
1
,
29
1
,
36
1
,
43
1

Look for a common relationship between each pair and find the missing value?
19. ( 20 , 4 ) , ⎟
⎠
⎞
⎜
⎝
⎛
50
1
,
10
1
, ( -55 , -11 ) , ( 6.0 , 1.2 ) , ⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
?
,
2
1
20. ( 28 , 7 ) , ⎟
⎠
⎞
⎜
⎝
⎛
16
1
,
4
1
, ( 36 , 9 ) , ( - 0.8 , - 0.2 ) , ⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
?
,
3
1
See If You
Remember
1. – 72.55 ÷ 2.5 2. ⎟
⎠
⎞
⎜
⎝
⎛
−
⎟
⎠
⎞
⎜
⎝
⎛
−
4
3
5
1
3
3. What is 140 % of 90 ? 4. 25 is what percent of 40 ?
5. Change 305 % to a fraction. 6. Change
50
13
to a decimal.
7. Write 410.03 in its expanded form.
8. An item that usually sells for $350 is on sale for $300. What is the percent decrease?
9. What is the numeral for ( )
3
10
7 × + ( )
1
10
3 × + ⎟
⎠
⎞
⎜
⎝
⎛
×
10
1
5 .
10. Harry bought shaving cream for $4.89, after-shave lotion for $12.20, shampoo for
$3.95, and cologne for $48.15. What is a reasonable estimate of Harry’s bill to the
nearest dollar?

2.6 WORD PROBLEMS Textbook Reference Chapter 3
CLAST OBJECTIVES
" Solve real world problems that do not involve the use of percent
" Solve problems that involve the structure and logic of arithmetic
Case 1: Number Theory
FACTS
1.
A factor is one of two or more numbers that multiply to give another number. For
example, 5 and 3 are factors of 15 because (5)(3) = 15.
2.
A prime number only has factors 1 and itself. For example, the only factors of 3 are 1
and 3. Here is a partial list of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, etc.
3.
A multiple is a number that contains another number an integral number of times
without a remainder. For example, 15 is a multiple of 3 because 15 ÷ 3 = 5.
Examples Solutions
a) Find the smallest
positive multiple of 6
which leaves a
remainder of 4 if
divided by 10 and a
remainder of 3 if
divided by 7.
Multiples of 6 are listed below:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, etc.
Determine which multiples leave a remainder of 4 when divided by
10. They are 24 and 54.
Which of the two, 24 or 54, leaves a remainder of 3 when divided
by 7?
The answer is 24.
b) Find the largest
positive integer which
is both a factor of 24
and 40 and is also a
factor of the difference
of 24 and 40.
Difference: 40 – 24 = 16.
List the factors of 24, 40, and 16.
24: 1, 2, 3, 4, 6, 8, 12, 24
40: 1, 2, 4, 5, 8, 10, 20, 40
16: 1, 2, 4, 8, 16
The largest factor common to 24, 40, and 16 is 8.
c) How many whole
numbers leave a
remainder of 3 when
divided into 19 and a
remainder of 1 when
divided into 25?
Determine which whole numbers leave a remainder of 3 when
divided into 19.
19 ÷ 2 = 9 R 1 19 ÷ 3 = 6 R 1 19 ÷ 4 = 4 R 3
19 ÷ 5 = 3 R 4 19 ÷ 6 = 3 R 1 19 ÷ 7 = 2 R 5
19 ÷ 8 = 2 R 3 19 ÷ 9 = 2 R 1 19 ÷ 10 = 1 R 9
and so on. The successful trials were 4, 8, and 16. Divide these
into 25 to determine if the remainder is 1.
25 ÷ 4 = 6 R 1 25 ÷ 8 = 3 R 1 25 ÷ 16 = 1 R 9
The successful trials were 4, and 8. Thus, there are two whole
numbers that meet the criterion.

SECTION 2.6 Word Problems 75
1. How many positive factors of 42 are even and less than 42 and also divisible by 3?
2. Find the smallest positive multiple of 7 that yields a remainder of 1 when divided by 3.
3. Which whole numbers are divisible by 5 and also a factor of 20?
4. Find the smallest positive multiple of 8 that yields a remainder of 2 when divided by 5.
5. How many factors of 126 are also factors of 15?
6. How many whole numbers leave a remainder of 3 when divided into 33 and a remainder
of 1 when divided into 46?
7. How many whole numbers leave a remainder of 2 when divided into 34 and a remainder
of 3 when divided into 19?
8. Find the smallest positive multiple of 5 that leaves a remainder of 3 when divided by 7
and a remainder of 5 when divided by 8.

Case 2: Real World Problems
Read the problem carefully. Re-read if necessary. Determine what information you are
given and what you are asked to find. Find a mathematical relationship between the
information given and solve.
Examples Solutions
d) An AC repairman charges
$75 for a house call and $20
an hour for labor. Find the
cost when a repairman visits a
home for three hours?
Given: $75 fee for a house call
$20 per hour for labor
Find: Cost of a house call that takes 3 hours
$75 + $20(hours) = Cost of a house call
$75 + $20(3) = $75 + $60 = $135
The cost of a three-hour house call is $135.
e) Jack and Jill fold 500
flyers each per day. On
Tuesday, Jack did
5
4
of the
his regular production and Jill
did
10
7
of her regular
production. How many flyers
went unfolded?
Given: Jack usually folds 500 flyers
Jack folded
5
4
of his 500 →
5
1
unfolded
Jill usually folds 500 flyers
Jill folded
10
7
of her 500 →
10
3
unfolded
Find: Total number of flyers that were not folded
No. Jack didn’t fold + No. Jill didn’t fold = Total number of
flyers not folded.
( ) ( ) 250
150
100
500
10
3
500
5
1
=
+
=
⎟
⎠
⎞
⎜
⎝
⎛
×
+
⎟
⎠
⎞
⎜
⎝
⎛
×
There were 250 flyers that were not folded.
f) A 2-pound bag of sugar
costs $0.90 and a 5-pound bag
costs $2.00. How much
money can be saved by
buying 50 pounds of the more
economical size?
Given: 2-lb bag costs $0.90
5-lb bag costs $2.00
Will purchase 50 pounds of sugar
Find: Money saved on the “best” buy
(Money saved per pound)(50 pounds) = Total savings.
Cost per pound: 2-lb bag: 0.90 ÷ 2 = 0.45
5-lb bag: 2.00 ÷ 5 = 0.40 “best” buy
Money saved per pound: 0.45 – 0.40 = 0.05
(Money saved per pound)(50 pounds) = Total savings
(0.05)(50) = $2.50.
We save $2.50 by purchasing 50 pounds of the 5-lb bags of
sugar.

SECTION 2.6 Word Problems 77
Example Solution
g) Bob’s phone bill is
calculated as $30 flat rate and
$0.25 for each call made.
Bob’s bill is $50.00. How
many calls did Bob make?
Given: $30 flat rate
$0.25 per call
Phone bill is $50
Find: Number of calls Bob made
Flat Rate + 0.25(no. of calls) = Total Phone Bill
$30 + 0.25(x) = $50
0.25x = 50 – 30
0.25x = 20
x = 80
25
.
0
20
=
Bill made 80 calls.
9. A truck rents for $320 per week plus $0.25 per mile. Find the cost of renting the truck
for two weeks to travel 1,200 miles.
10. An 8 – ounce can of soup costs $1.20, while a 12 – ounce can costs $2.00. How much
would be saved by buying 60 ounces of the more economical size?
11. A phone company charges a $29 flat rate and $0.15 per call. How many calls were
made for a phone bill of $38.00?
12. A bookstore ordered 30 books at the cost of $40 each. The store sold 24 of these
books at $70 each and returned the other 6 to the publisher at a service charge of $2
each. How much profit did the bookstore make?
13.
Two containers can each hold 30 gallons of water. On a certain day,
6
1
of one
container is used and
10
9
of the other container is used. How much water is not used
(in gallons)?

See If You
Remember
14. A caterer charges $250 per banquet plus $7.00 per plate. Find the cost of two
banquets in which the attendance was 60 and 75 persons respectively.
1. 94.2 – 17.483 2. ⎟
⎠
⎞
⎜
⎝
⎛
−
÷
2
1
4
2
1
13
3. Write
5
2
7 as a percent. 4. Write
4
3
9 % as a decimal.
5. If you decrease 80 by 30 %, what is the result?
6. 90 is what percent of 60 ?
7. Write the numeral for ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
⎟
⎠
⎞
⎜
⎝
⎛
×
+
× 3
0
10
1
9
10
1
3
10
5
8. In a class of 30 students who took a History test, some scored 89 and some scored 63.
What was the class average on that History test?
9. Write the expanded form: 7 4
– 3 4
10. Compare the numbers by placing = , < , or > between them: 1
.
5
?
20
11. Find the missing term in the sequence:
?
,
21
7
,
16
7
,
11
7
,
6
7

SECTION 2.7 CLAST-Like Questions 79
2.7 CLAST-LIKE QUESTIONS No Calculator Allowed
1. 82.965 + 3.47 + 108.2
a) 192.035 b) 194.635 c) 843.94 d) 84.394
2. 2.28 ÷ 0.012
a) 190 b) 19 c) 1.9 d) 0.19
3. ⎟
⎠
⎞
⎜
⎝
⎛
−
−
5
1
2
4
3
7
a)
20
19
5 b)
20
11
9 c)
20
11
5 d)
20
19
9
4. ⎟
⎠
⎞
⎜
⎝
⎛
−
÷
2
1
2
4
3
8
a)
2
1
3
− b)
2
1
3 c)
8
7
21
− d)
8
7
21
5. 407 % is equivalent to
a) 407 b) 40.7 c) 4.07 d) 0.407
6.
25
11
is equivalent to
a) 4.4 % b) 4.4 c) 0.44 d) 0.44 %
7. 0.28 is equivalent to
a)
7
25
b) 2.8 % c)
25
7
% d)
25
7

8.
20
17
is equivalent to
a) 8.5 b) 0.85 c) 0.085 d) 8.5 %
9. What is 130 % of 90?
a) 117 b) 27 c) 11.7 d) 270
a) 150 % b) 50 % c) 6
.
66 % d)
2
3
%
11. An item that regularly sells for $80 is on sale for $60. What is the percent decrease?
a) 250 b) 25 c) 2.5 d) 0.25
12. Jim paid $70,000 for his house 15 years ago. The house has increased value by 15%.
What is the new value of the house?
a) $10,500 b) $71,500 c) $80,500 d) $79,500
13. Round to the nearest hundredth: 547.3951
a) 547 b) 547.395 c) 500 d) 547.4
14. Select the expanded form for 27.09.
a) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
⎟
⎠
⎞
⎜
⎝
⎛
×
+
× 2
3
10
1
9
10
1
7
10
2 b) ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
× 2
2
10
1
9
10
7
10
2
c) ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
× 2
0
10
1
9
10
7
10
2 d) ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
×
10
1
9
10
7
10
2 0
15. Select the numeral for ( ) ( ) ⎟
⎠
⎞
⎜
⎝
⎛
×
+
×
+
×
10
1
2
10
2
10
2 0
2
a) 2.22 b) 2.022 c) 20.22 d) 202.2

16. An investor buys 203 shares of stock. Each share costs $52.50. What is a reasonable
estimate of the purchase?
a) $10,000 b) $12,000 c) $8,000 d) $14,000
17. A company employs 10 people. The lowest pay is $200 per week and the highest pay is
$500 per week. Which is a reasonable estimate of the company’s weekly payroll for
those 10 people?
a) $5,000 b) $4,000 c) $2,000 d) $3,500
18. 3 3
+ 5 3
is equivalent to
a) ( 3 + 5 ) 3
b) ( 3 • 3 ) + ( 5 • 3 ) c) ( 3 + 5 ) 6
d) ( 3 • 3 • 3 ) + ( 5 •5 • 5 )
.
9
?
7
1
.
9
a) < b) > c) =
.
5
?
30
a) < b) > c) =
21. Find the next term in the sequence:
?
,
256
7
,
64
7
,
16
7
,
4
7
,
7 −
−
−
a)
1024
7
b)
1024
7
− c)
4
1
d)
4
1
−
22. Look for a common linear relationship between the numbers in each pair. Identify the
missing term. ( 34 , 17 ) , ( 0.6 , 0.3 ) , ( – 18 , – 9 ) , ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛ ?
,
3
1
,
5
1
,
5
2
a)
2
3
b)
6
1
c)
3
2
d) 6

23. A passenger van rents for $350 per week plus $0.20 per mile. Find the cost of renting
this van for a three-week trip covering 700 miles.
a) $2,450 b) $1,400 c) $1,050 d) $1,190
24. A checking account charges $6 per month plus $0.20 per check. How many checks were
written in a month in which the total charges amounted to $22.00?
a) 6 b) 140 c) 80 d) 110
25. Find the largest positive integer which is a factor of 24 and 40 and is also a factor of the
sum of 24 and 40.
a) 4 b) 8 c) 16 d) 32
Answers to CLAST-LIKE Questions 2.7
1. b 6. c 11. b 16. a 21. a
2. a 7. d 12. c 17. d 22. b
3. d 8. b 13. d 18. d 23. d
4. a 9. a 14. c 19. b 24. c
5. c 10. a 15. d 20. a 25. b
Explanations
1.
635
.
194
2
.
108
47
.
3
965
.
82
+
2.
190
0
108
108
12
.
2880
12
28
.
2
012
.
0

3.
20
19
9
20
199
20
44
20
155
4
5
4
11
5
4
5
31
5
11
4
31
5
1
2
4
3
7 =
=
+
=
•
•
+
•
•
=
+
=
⎟
⎠
⎞
⎜
⎝
⎛
−
−
4.
2
1
3
2
7
5
2
4
35
2
5
4
35
2
1
2
4
3
8 −
=
−
=
−
×
=
−
÷
=
−
÷
5. Move decimal two places to the left: 407 % = 4.07
6. 44
.
0
0
100
100
100
00
.
11
25
7.
25
7
100
28
%
28
28
.
0 =
=
=
8. 85
.
0
0
100
100
160
00
.
17
20
9.
117
100
11700
11700
100
130
90
100
100
130
90
100
%
=
=
=
•
=
•
=
=
x
x
x
x
of
is
10.
%
150
40
6000
40
6000
40
100
60
100
40
60
100
%
=
=
=
•
=
•
=
=
x
x
x
x
of
is

11. Original = $80
Decreased by: $80 – $60 = $20
%
25
80
2000
80
2000
80
100
20
100
80
20
100
%
=
=
=
•
=
•
=
=
x
x
x
x
original
by
decreased
12.
500
,
10
100
1050000
1050000
100
15
000
,
70
100
100
15
000
,
70
100
%
=
=
=
•
=
•
=
=
x
x
x
x
original
by
increased
New Value: $70,000 + $10,500 = $80,500
13. The number 9 is in the hundredth place. The number after it is 5, so increase nine by
one.
547.40 = 547.4
40
.
547
01
.
0
39
.
547
+
14. 2 is in the tens place. 7 is in the ones place. 9 is in the hundredths place.
( ) ( ) ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×
+
×
+
×
2
0
10
1
9
10
7
10
2
15. 2 hundreds, 2 ones, 2 tenths = 202.2
16. Round 203 to 200. Round $52.50 to $50.
200 • $50 = $10,000

17.
The average pay: 350
$
2
500
200
=
+
Estimation of weekly payroll: $350 • 10 = $3,500
18. 3 3
= 3 • 3 • 3 5 3
= 5 • 5 • 5
3 3
+ 5 3
= 3 • 3 • 3 + 5 • 5 • 5
19. Expanding: 9.1777 ? 9.1717
9.1777 > 9.1717
20.
( ) ( )
25
.
30
30
5
.
5
?
30
5
.
5
?
30
2
2
<
21.
Multiplying by ⎟
⎠
⎞
⎜
⎝
⎛
−
4
1
is the pattern.
The term after
256
7
− is
1024
7
4
1
256
7
=
−
•
−
22. ( x , y ): To get y, divide x by 2.
⎜
⎜
⎝
⎛
⎟
⎠
⎞
=
•
=
÷
⋅
⋅
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
6
1
,
3
1
:
6
1
2
1
3
1
2
3
1
?
,
3
1
Answer
23. Given: $350 rental fee
$0.20 per mile
3 – week trip
Traveling 700 miles
Find: cost of rental van
( $350 ) ( number of weeks ) + ( $0.20 ) ( number of miles ) = Cost of rental van
$350 • 3 + $0.20 • 700 = $1050 + $140 = $1190

24. Given: $6 monthly charge
$0.20 per check
total charges for a month $22
Find: number of checks written
( $6 ) + ( $0.20 ) ( number of checks ) = Total Charges
$6 + $0.20 • x = $22
0.20 x = 22 – 6
0.20 x = 16
x = 16 ÷ 0.20 = 80 checks
25. Sum: 24 + 40 = 64
Factors of 24: 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24
Factors of 40: 1 , 2 , 4 , 5 , 8 , 10 , 20 , 40
Factors of 64: 1 , 2 , 4 , 8 , 16 , 32 , 64
The largest common factor is 8.

18

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