math made a bit easier workbook: practice exercises, self-tests, and review

MATH MADE

A BIT EASIER

WORKBOOK

MATH MADE

A BIT EASIER

WORKBOOK

Practice Exercises,

Self-Tests, and Review

LARRY ZAFRAN

Self-published by author via CreateSpace

Available for purchase exclusively on Amazon.com

MATH MADE A BIT EASIER WORKBOOK:

Practice Exercises, Self-Tests, and Review

Copyright © 2009 by Larry Zafran

Self published by author via CreateSpace

Available for purchase exclusively on Amazon.com

All rights reserved. No part of this book may be

reproduced or transmitted in any manner whatsoever

without written permission except in the case of brief

quotations embodied in critical articles and reviews.

Book design by Larry Zafran

Printed in the United States of America

First Edition printing December 2009

ISBN-10: 1-4495-9287-2

ISBN-13: 978-1-44-959287-5

Please visit the companion website below for additional

information, to ask questions about the material, to leave

feedback, or to contact the author for any purpose.

www.MathWithLarry.com

CONTENTS

CHAPTER ZERO .................................................... 7

Introduction

CHAPTER ONE .................................................... 15

Is Math Hard, and If So, Why?

Goal-Setting & Assessment Self-Test ................ 23

CHAPTER TWO ................................................... 29

The Foundation of Math:

Basic Skills in Arithmetic

CHAPTER THREE ................................................ 39

Basic Math Topics and Operations

CHAPTER FOUR .................................................. 47

Working with Negative Numbers

CHAPTER FIVE ................................................... 53

Basic Operations with Fractions (+, –, ×, ÷)

CHAPTER SIX ...................................................... 57

More About Fractions

CHAPTER SEVEN ............................................... 65

Other Topics in Fractions

CHAPTER EIGHT ................................................ 71

The Metric System, Unit Conversion,

Proportions, Rates, Ratios, Scale

CHAPTER NINE .................................................. 79

Working with Decimals

CHAPTER NINE AND FIVE-TENTHS .............. 83

More Topics in Decimals

CHAPTER TEN .................................................... 89

Working with Percents

CHAPTER ELEVEN ............................................. 99

Basic Probability and Statistics

CHAPTER TWELVE........................................... 105

How to Study and Learn Math,

and Improve Scores on Exams

End-of-Book Self-Test ....................................... 111

Answers to Exercises and Self-Tests ................ 119

About the Author & Companion Website ....... 143

7

CHAPTER ZERO

INTRODUCTION ABOUT THE MATH MADE A BIT EASIER SERIES

This is the second book in the self-published Math Made a

Bit Easier series which will be comprised of at least nine

books. The goal of the series is to explain math "in plain

English" as noted in the subtitle of the first book.

The series also attempts to explain the truth about why

students struggle with math, and what can be done to

remedy the situation. To write with such candidness is

only really possible as a totally independent author.

Unlike many commercial math books, this series does

not imply that learning math is fast, fun, or easy. It

requires time and effort on the part of the student. It also

requires that the student be able to remain humble as

s/he uncovers and fills in all of his/her math gaps.

THE PURPOSE OF THIS BOOK

The purpose of this book is to provide the reader with

the means to review, practice, and quiz him/herself on

MATH MADE A BIT EASIER W ORKBOOK: PRACTICE EXERCISES, SELF-TESTS, AND REVIEW

8

what s/he has learned in the first book of the series (Math

Made a Bit Easier: Basic Math Explained in Plain English,

ISBN 1449565107, available exclusively on Amazon.com).

The book includes two comprehensive self-tests for the

reader to assess his/her mastery of the math concepts. It

also includes tips for getting oneself into the optimal

mindset to effectively study math and take exams.

HOW THIS BOOK IS ORGANIZED

This book is directly aligned with the first book in the

series. Following this introduction, the book offers some

exercises for practicing and implementing the ideas that

were presented in Chapter One of the first book entitled,

"Is Math Hard, and If So, Why?"

Before the start of the actual math content, the book

presents a self-assessment test which the reader can use

as a means of goal-setting for working through the rest

of the book. This may be a frustrating and humbling

experience, but it is a necessary step on the path toward

math being easier for you.

Chapters Two through Eleven correspond directly to the

chapters of the same names in the first book. Each

chapter includes practice exercises and a review of

concepts that should be memorized. Some of the ques-

INTRODUCTION

9

tions are meant to be answered by way of a definition or

short explanation. For these questions it is not essential

that your answer exactly match the one in the answer

key. It is only important for you to demonstrate that you

fully understand the concept being practiced.

At the end of the math content chapters there is an end-

of-book exam that the reader can take for additional

practice. Aside from the changing of numbers and the

order of the questions, it is almost the same as the pre-

test. The goal, of course, is for problems which proved

difficult on the pre-test to be easier on the post-test.

Chapter Twelve focuses on how to study and learn math,

and improve scores on exams. Some mental exercises are

offered which the reader can use to practice the concepts

from the corresponding chapter in the first book.

HOW TO USE THIS BOOK FOR SELF-STUDY

This book was designed to be used in conjunction with

the first book in the series which can be viewed in its

entirety for free on Google Books if you are unable or

choose not to purchase it. I decided that it would be most

effective to write the first book in a conversational tone

as opposed to turning it into a textbook or commercia-

lized workbook.


10

As mentioned in the first book, the best way to study is

to make up your own examples modeled after the

sample ones provided. Just constantly ask yourself

questions such as, "What if that number had been that

instead?" or "What if that positive had been a negative?"

Go out of your way to "trip yourself up" instead of

waiting for that to happen on an exam.

With that said, many students are just not inclined to

make up their own examples, or are concerned that their

examples are not representative of what they may face

on an exam. This book attempts to address that concern.

As you solve each problem in this book, it is essential

that you constantly think about what you are doing.

Don't take any "stabs in the dark" followed by checking

the answer key to see how your attempt turned out.

Your goal is to learn and master the concepts. Remem-

ber, the book is yours, and no one is grading you or

looking over your shoulder. Progress through the

material as slowly as you need to, and go back and

review to the extent that is necessary.

It is essential that you avoid grading yourself on the

exercises or exams and say, "I got 65, good, I passed."

That mindset is the root of the entire reason why stu-

INTRODUCTION

11

dents struggle with math. Not only should you aim to

answer every question in the book correctly, but your

goal should be to do so with confidence. You should get

to the point where you can clearly explain to someone

else why your correct answers are correct, and why your

wrong answers were wrong. That is what it means to

truly know the material and face exams with confidence.

To get the most benefit from this book, do not work on

the problems in any section which you have recently

reviewed in the main book. Allow at least a day or two to

pass so that you can assess whether you are truly retain-

ing the concepts, and whether you have internalized

them. Don't get into the pattern of mindlessly solving

problems by rote, or by "spitting back," information that

you just saw a moment ago.

Try your best to solve each problem in this book without

resorting to any type of hints, whether by referring to the

main book, or by working backwards after having seen

the answer, or by asking someone for help. It is essential

to understand that even the slightest hint is robbing you

of being able to practice thinking about the material. Try

to simulate typical exam conditions as much as you can.

As you work through this book, remember that you can

e-mail me if you have questions or comments. Take


12

advantage of this opportunity, and don't move past any

concept that is not fully clear to you. A huge component

of why students struggle with math is the mindset of,

"I'll just move on and get back to this later." Math simply

doesn't work that way. It must be learned step by step.

A NOTE ABOUT ERRORS / TYPOS IN MATH BOOKS

Virtually all books go to print with some undiscovered

errors or typos. Math books are especially prone to this.

Unfortunately, many commercial publishers rush to get

their books on the shelf before their competitors, result-

ing in an even greater number of errors.

When studying from any math book, don't assume that

the book is flawless. This is especially important when

checking your answers in an answer key. While it's

possible that you made a careless error, or misunders-

tood a question, or got tricked by a "trick" question, it's

also possible that the editor typed "C" instead of "B," or

misinterpreted a handwritten negative sign or decimal

point. If you're finding yourself in doubt about a ques-

tion, try to speak with someone before giving up in

frustration. You can also contact me for this purpose.

Many publishers are confident enough to include an

errata section on their website where they list errors in

INTRODUCTION

13

their books that weren't caught before publication. If a

publisher doesn't do that, there is no harm in e-mailing

the company with your question or concern. In the case

of this book and all other books in the series, any errors

discovered after publication will be noted and explained

on my website. I also offer free copies of my books to

anyone who catches and informs me of a major error.

THE BOOK’S POSITION ON CALCULATOR USE

As described in the first book, this book takes a realistic

and modern position on calculator use. Unless an exer-

cise in this book specifically states to not use a calculator,

you should feel free to use one unless you are studying

for an exam which does not allow their use.

All of your effort should be to master the concepts being

taught. If you do not fully understand a concept, not

only will the use of a calculator not help you, it will

almost certainly hinder you.

THE BOOK’S POSITION ON WORD PROBLEMS

As described in the first book, it is inefficient to prepare

for word problems on an exam by repeatedly reading

and solving a handful of sample word problems in a

book. The word problems you face on an exam will


14

almost certainly be different, and changing even one

word can drastically alter an entire problem.

The only way to prepare for word problems is to make

sure that you have fully mastered all of the topics that

you will be tested on. Of course it is essential that you

become a skilled and careful reader, but that is some-

thing that cannot be learned from a book. That is some-

thing which simply has to develop over time. It is also

important to develop good test-taking skills in general

which is discussed in Chapter Twelve.

HOW TO GET MORE HELP ON A TOPIC

I maintain a free math website with extensive content

including the means for students to e-mail their math

questions. That will continue with the publication of this

series, although I'm working to redesign the website to

better align it with the series. The old content will still be

available, and new content will be added as students ask

questions or make comments about the books.

My goal is for the website to serve as an interactive

companion to the series so that students’ questions can

be addressed. The website and my question-answering

service will continue to be free for all. The address is

www.MathWithLarry.com.

15

CHAPTER ONE

Is Math Hard,

and If So, Why?

TAKING INVENTORY OF YOUR MATH HISTORY

If you are reading this book, you are probably of the

opinion that "math is hard." If you want math to start

being a bit easier, you will need to go through the

therapeutic exercise of taking inventory of your math

history. This has the potential to be a very painful

experience, and may bring up sensitive matters that you

have either repressed, or have never given much thought

to. This is a very personal exercise, and is no one's

business but your own unless you'd like to e-mail me

and share anything that is on your mind.

Try to recall your very earliest memories, and think

about how you felt about math at that time. Did you

watch children's learning programs on TV and have fun

counting along with the furry characters on the screen?

Do you remember arranging blocks or counters in


16

various patterns? Did you have any strong feelings

about numbers or math at that time? Everyone will have

a different answer to these questions. Just think about

them and see what comes up for you.

Continue sifting through your memories to kindergarten

and the lower grades. What was math like at that time?

It is rare for a kindergartener to proclaim that "math is

hard." If anything, most say that "numbers are fun," but

that may not have been your experience.

See if you can get a clear picture of when exactly the

trouble started. When did you start scratching your head

in confusion, or first use the phrase, "I hate math!" or "I

don't get it?" When did you start getting lower grades

than you were hoping for? For many students the trouble

sets in at roughly the third grade, but everyone is differ-

ent. Just see what you come up with.

Some readers will know right away what the root of the

problem is. Perhaps you had an awful math teacher in

elementary school who either didn't know the material

him/herself, or didn't know how to convey it, or was

condescending to you. Perhaps you were the victim of

gender-based discrimination. Perhaps your family

presented math in a bad light because of their own

IS MATH HARD, AND IF SO, W HY?

17

experiences with the subject. Perhaps none of those

things applied at all, or perhaps you're just not certain,

and all of your memories on the matter or a blur or

locked away in a place that you aren't willing to go to

just now. Just keep thinking about it as best as you can.

Doing this exercise will not magically make math any

easier for you. What it will do, though, is give you a

starting point from which to move forward. It is said that

you can't know where you are going if you don't know

where you have been. Once you have a better idea of

your personal history with math, it will be easier to make

goals for yourself since you will understand precisely

what struggles you are up against.

ASSESSING AND ACCEPTING YOUR CURRENT

MATH ABILITY LEVEL

Another aspect of overcoming the struggle with math is

the ability to assess and accept your current math ability

level. For most students this will come as a huge blow to

their ego. It is possible that you will take the assessment

pre-test in this book, and realize that you can barely

answer one question, let alone with complete confidence.

This will especially not sit well if you earned passing or

even high math grades throughout elementary and


18

middle school, and cannot make sense of why you're

having so much trouble now.

What is important is to bring yourself to the point where

you can honestly assess and accept your current know-

ledge and ability level. There is no need to assign a

grade or a grade level to the matter. What is important is

to determine the appropriate starting point.

For most students, the appropriate starting point is at the

beginning. That is not at all to say that you must return

to the first grade and sit through twelve years of school-

ing. It just means that by starting your review of math at

the beginning, you can fill in all of the gaps in material

which were never addressed. As you do this, math will

slowly become easier and easier.

IS THE TIMETABLE FOR YOUR GOALS REALISTIC?

The most common e-mail that I get from prospective

tutoring clients is to the effect of, "I'm scheduled to take

my GED exam in two weeks, and I absolutely have to

pass it but I really suck at math. Can you fit me in for one

or two sessions before my exam?" As you may guess, I

do not take on such clients regardless of my availability

or how much money they are willing to pay me.


19

In order to end the pattern of struggling with math, it is

important to have a very realistic timetable for your

goals. Since every student is different, it is impossible to

offer any specific or personalized timetables on the

matter. However, a few points are worth mentioning.

First of all, if you are willing to study effectively and

diligently, it won't take you "years and years" to achieve

your math goals no matter how far behind you are or

think you are. There is very little material covered in a

typical school math lesson, and much of the material is

repeated and reviewed year after year. For most students

there is not an insurmountable material to learn.

With that said, if you are struggling with math, it is

preposterous to think that taking one or two last-minute

sessions with a tutor will somehow make any difference.

If anything, those sessions will only serve to confuse and

fluster you, because you will quickly realize the extent to

which you are unprepared for your exam.

Based on my experience, a typical secondary school or

adult student requires about six to twelve months of

dedicated study to "catch up" in math to the point where

they feel prepared for whatever exams or coursework

they are facing. If such a student wishes to work with a


20

private tutor, the student must plan on taking at least

two sessions per week, with those sessions being used

for systemically progressing through a well-planned

roadmap of the material.

As mentioned, everyone is different. The point is that if

you do not have a realistic time table for your goals, you

will just end up wasting your time and money on

whatever help you may seek out, and you will become

very frustrated in the process.

Most people want to achieve their goals in a manner that

is fast, fun, and easy, but math just doesn't work like

that. At some level you probably accept that, otherwise

you would not still be reading this book. Try to come up

with a realistic timetable for your goals either on your

own or with the guidance of a tutor. It is unlikely that

your life is going to change that much if you postpone an

exam for a relatively short amount of time to ensure that

you get the help that you need to succeed on it.

DO YOU HAVE A REALISTIC STUDY PLAN?

To end the pattern of struggling with math, you will

need to have a realistic study plan, and you will need to

stick to it. One of the main reasons why I wrote this


21

series of books is so that students will have an organized

and well-planned roadmap that they can follow on the

path to success with math. By systematically working

through the books, you will avoid the trap of studying in

a haphazard manner without filling in your gaps.

The concern, then, is making sure that you have the time,

energy, and environment necessary for effective study. If

you come home after a long, hard day of work to a noisy

and chaotic home with chores waiting for you, don't fool

yourself into thinking that you will be able to do exten-

sive work on your math. It just will not happen. Also

don't fool yourself into thinking that it will make a huge

difference if you sit for two or three minutes each day

flipping through the pages of this or any other book.

Everyone's life circumstances are different, but the last

thing you need is to allow math to frustrate you more

than it probably already does. Don't even attempt to

study unless you have a quiet environment, and some

minimal quality time and energy to devote to it.

23

Goal-Setting &

Assessment Self-Test

Take this self-test after you have read the first book, but

not immediately after. Use it to determine how much of

the material you are retaining, and what concepts you

haven't yet fully internalized. Don't be concerned about

how many questions you get right or wrong. Just make

sure that you understand why the right answers are right

and why the wrong answers are wrong.

After completing this test and checking your answers,

proceed with the exercises in this book with an emphasis

on the topics that you either forgot or had trouble with.

Refer back to the first book for review as needed.

1) Which basic operations are not commutative?

2) Find the product of 4 and 6

3) Find the sum of 7 and 8

4) List the first 10 multiples of 7

5) Compute 71 ÷ 9 in mixed number format

6) Is the number 791,350 even or odd?

7) What is the result of multiplying an even times an

odd number? (Even or Odd)


24

8) Insert "<" or ">": 73,001 72,999

9) Evaluate 6 × [10 ÷ (4 + 1)]

10) Is 2 a composite number? Why?

11) Write "Nine billion, five hundred three million, forty

thousand seventeen" as a number

12) Round 27,815 to the nearest hundred

13) Round 139,501 to the nearest thousand

14) Evaluate: 121. Include both roots.

15) Write "Two hundred three and fifty-nine hun-

dredths" as a number

16) Insert "<" or ">": 5.99999 6.18

17) Insert "<" or ">": 0.29 0.2876

18) Convert to a fraction: 0.021

19) Convert 2/7 to a decimal (round to the nearest

hundredth)

20) Convert to a decimal by hand: 3/25

21) Convert to a decimal: 1/3

22) True/False: 0.7 = .7 = 0.70

23) True/False: 0.9 is a repeating decimal.

24) Insert "<" or ">" (no calculator): 6/25 251/500

25) Insert "<" or ">" (use calculator): 425/639 541/803

26) Multiply (no calculator): 7.23 × 1000

27) Divide (no calculator): 12.3 ÷ 10,000

28) Round 12.3456 to the nearest hundredth

29) Round 79.9912 to the nearest tenth

30) Express 1,234,000,000 in scientific notation

GOAL-SETTING & ASSESSMENT SELF -TEST

25

31) Express 7.89 × 10−5 in standard notation

32) Convert 14% to a reduced fraction

33) Convert 107% to a decimal

34) Convert 0.567 to a percent

35) Convert 11/17 to a percent (round to nearest tenth)

36) Convert 3/50 to a percent (don't use calculator)

37) Convert 0.07% to a decimal

38) Compute the percent of change from 23 to 37 (round

to the nearest tenth of a percent)

39) How much money will you save on a $29.95 item

during a "30% Off" sale?

40) What will a person's monthly rent be after a 2.7%

increase if it is currently $817?

41) What is the cost including tax on a $195 item if the

tax rate is 8.25%?

42) What percent of 87 is 35? Round to the nearest tenth.

43) What is the reciprocal of 7/11?

44) Convert 7 ¼ to an improper fraction.

45) How is a millimeter related to a meter?

46) Apples are being sold at the rate of 34 apples for $19.

How much does one apple cost at that rate?

47) Convert 102 inches to feet

48) Solve for the unknown value: 3

7=

12

?

49) Simplify to a single fraction: ( 3

7 ) / 8

50) Find the mean of this list (rounded to the nearest

tenth): 27, 94, 85, 0, 62


26

51) Find the median of this list: 36, 7, 7, 12, 12

52) Find the median of this list: 108, 92, 86, 84, 72, 61

53) Find the mode of this list: 17, 12, 38, 45, 12, 91, 38

54) Find the mode of this list: 1, 2, 7, 4, 5

55) Find the range of this list: 78, 50, 32, 19, 42

56) Find the probability of rolling a 5 or 6 on a single

roll of one standard die

57) Find the probability of rolling a 7 on a single roll of

one standard die

58) What are the chances that a flipped fair coin will

land on heads or tails?

59) If there is a 3/20 chance that it will rain tomorrow,

what is the chance that it will not rain?

60) In an experiment comprised of a coin toss followed

by a roll of a single die, what is the probability of

flipping heads and rolling a 3?

61) An urn has 3 red marbles and 5 blue marbles. Find

the probability of drawing a red marble followed by

a blue marble, with replacement.


the probability of drawing a blue marble followed

by a blue marble, without replacement.

63) A man has three shirts, four pairs of pants, and five

ties. How many different outfits comprised of a

shirt, a pair of pants, and a tie can he create?

64) Convert 17 to an equivalent fraction.

GOAL-SETTING & ASSESSMENT SELF -TEST

27

65) What is the GCF of 4 and 6?

66) What is the LCM of 20 and 30?

67) Reduce 3/17 to lowest terms

68) Multiply: 4 ×2

9

69) Add: 1

2+

1

3

70) True/False: 14

21=

30

45

71) True/False: 11

12=

13

14

72) True/False: −2

−7=

2

7

73) True/False: −3

5=

3

−5

74) Add: 5

13+

7

13

75) Multiply: 2

7×

3

5

76) Divide: 4

5÷

5

4

77) Multiply: 524

839×

839

524

78) Compute 37 + 89

79) Compute 81 - 25

80) Define: Integer.

81) Evaluate 91

82) Evaluate 64

83) Evaluate 3 + 4 × 2

84) Evaluate 8 – 5 + 1

85) Compute: 4 + (-7)

86) Compute: (-8) + (-6)

87) Compute: 3 – 10

88) Compute: (-4) – 2

89) Compute: (-5) – (-1)

90) Compute: (-9) × 6

91) Compute (-5) × (-8)

92) Compute: 20 ÷ (-5)

93) Compute: (-8) ÷ (-2)

94) Evaluate: (−4)2

95) Evaluate: −4

96) Evaluate: 34

97) Evaluate: |8|

98) Evaluate: |(-3) – 4|

99) List the factors of 46


101) Is 27 prime? Why?

102) Compute: 0 ÷ 23

103) Compute: 2 ÷ 0

104) What is 3 squared?

105) What is 71% of 539?

(round to the near-

est whole number)

29

CHAPTER TWO

The Foundation of Math:

Basic Skills in Arithmetic

ADDING SINGLE-DIGIT NUMBERS

1) When we add numbers, what do we call the result?

2) Is addition commutative? Support with an example.

3) What happens when we add 0 to a number?

Practice adding single-digit numbers together until you

can do so easily. The best way to do this is by using

either store-bought or homemade flashcards. Here are

some exercises to quiz yourself.

4) 8 + 7 7) 5 + 0 10) 7 + 4

5) 6 + 6 8) 4 + 5 11) 1 + 9

6) 3 + 8 9) 2 + 8 12) 8 + 8

A "TRICK" FOR ADDING 9 TO A NUMBER

In the main book we learned that we can rearrange

addition problems by regrouping the items to be added.

For example, instead of computing 9 + 7, we can move


30

one of the items from the second group into the first

group, thereby making the problem into 10 + 6 which is

equivalent and easier. Try using that technique for the

following addition problems.

13) 9 + 7 16) 9 + 0 19) 1 + 9

14) 6 + 9 17) 4 + 9 20) 9 + 9

15) 3 + 9 18) 2 + 9 21) 8 + 9

SUBTRACTING SINGLE-DIGIT NUMBERS

22) What is the answer called in a subtraction problem?

23) Is subtraction commutative? Support with an example.

24) What happens when we subtract 0 from a number?

25) Does the above trick about adding 9 to a number also

apply to subtraction? Why or why not?

As with addition, use homemade or store-bought flash-

cards to practice basic subtraction facts. For now we'll

stick to problems in which we don't subtract a larger

number from a smaller one. Here are some exercises to

practice and quiz yourself.

26) 9 – 7 29) 0 – 0 32) 8 – 7

27) 8 – 8 30) 6 – 5 33) 7 – 4

28) 7 – 5 31) 8 – 4 34) 9 – 0

THE FOUNDATION OF MATH: BASIC SKILLS IN ARITHMETIC

31

ADDITION AND SUBTRACTION ARE OPPOSITES

Remember: Addition and subtraction are inverse opera-

tions. This means is that they “undo” each other. When

computing a subtraction problem, it is sometimes best to

look at the problem in reverse, and ask yourself what

number you must add to the second number in order to

get back to the first number. For example, instead of

computing 8 – 5, you could ask yourself what number

you must add to 5 in order to get back to 8.

For practice, redo the previous exercises but compute the

related addition problem for each one as described.

MULTIPLYING SINGLE-DIGIT NUMBERS

35) What is the answer called in a multiplication problem?

36) Is multiplication commutative? Support with an example.

37) What is multiplication a shortcut for?

38) What do we get when we multiply a number times 0?

39) What do we get when we multiply a number times 1?

40) How do we compute the positive multiples of a number?

For practice, list the first 12 multiples of each number in

the following exercises. If you need help with the larger

numbers and larger multiples, see the section on two-

digit addition later in this chapter.


32

Multiples of 1: ____________________________________

Multiples of 2: ____________________________________

Multiples of 3: ____________________________________

Multiples of 4: ____________________________________

Multiples of 5: ____________________________________

Multiples of 6: ____________________________________

Multiples of 7: ____________________________________

Multiples of 8: ____________________________________

Multiples of 9: ____________________________________

Multiples of 10: ____________________________________

Multiples of 11: ____________________________________

Multiples of 12: ____________________________________

Recall from the main book that the rows and columns of

the multiplication table contain the multiples of the

numbers that are in the row and column headers. For

practice, fill in the blank multiplication table on the next

page which should be quick and easy since you just

finished listing all of the required multiples.

After you have done that, try quizzing yourself by

finding the following products from memory. Of course

be sure to memorize the whole multiplication table.

41) 8 × 7 44) 5 × 11 47) 9 × 6

42) 9 × 4 45) 3 × 12 48) 6 × 7

43) 8 × 8 46) 7 × 9 49) 3 × 0


33

DIVISION IS THE INVERSE OF MULTIPLICATION

50) What is the answer called in a division problem?

51) Is division commutative? Support with an example.

Remember: Division is the inverse of multiplication, just

like subtraction is the inverse of addition. For example, if

we want to compute 24 ÷ 3, we can determine what

number must be multiplied by 3 to get back to 24.

For practice, perform these division exercises using the

method described above.

52) 49 ÷ 7 55) 99 ÷ 11 58) 40 ÷ 5

53) 24 ÷ 4 56) 56 ÷ 7 59) 12 ÷ 1

54) 8 ÷ 8 57) 72 ÷ 9 60) 21 ÷ 3


34

DIVISION WITH A REMAINDER

Remember: Quantities don’t always divide evenly. For

example, think about the problem 27 ÷ 4. Using the

technique of looking at the problem in reverse, ask

yourself, “What number can I multiply 4 by so that I can

get as close to 27 as possible, but without actually going

over it?” The answer is 6. That gets us back to 24, but we

have 3 left over. For now, we can say that the answer to

the problem is “6 remainder 3,” (6 R 3). In Chapter Five

we’ll practice converting that remainder into a fraction.

Use the above method to find the quotients with re-

mainders in the practice exercises below.

61) 57 ÷ 8 64) 94 ÷ 11 67) 71 ÷ 5

62) 29 ÷ 6 65) 58 ÷ 7 68) 26 ÷ 4

63) 11 ÷ 7 66) 88 ÷ 10 69) 38 ÷ 3

TWO-DIGIT ADDITION WITH CARRYING

70) What are the first four place values called as we look

at a whole (non-decimal) number from right to left?

71) How must we line up numbers if we're adding

numbers that don't have the same quantity of digits?

72) When adding, do we make our way from leftmost

column to rightmost, or vice-versa?


35

Remember: If the sum of the ones place is more than 10,

the result must be broken up into ones and tens. The tens

must be "carried" into the tens column to be added in

when we get to that column. The procedure is the same

when adding other columns such as the tens place. We

always carry into the column on the left of our current

column, and we add our columns from right to left.

For practice, compute these addition problems:

73) 42 + 79 76) 86 + 45 79) 62 + 88

74) 99 + 99 77) 27 + 85 80) 94 + 59

75) 32 + 85 78) 15 + 55 81) 83 + 17

TWO-DIGIT SUBTRACTION WITH BORROWING

Remember: When subtracting, line up the numbers by

place value just like when adding. Subtract the columns

working from right to left, just like with addition.

Remember: If a column requires you to

subtract a larger number from a smaller

number such as 3 – 5, it is totally wrong to

just reverse the digits into 5 – 3. Instead,

we must "borrow" from the column on the

left. In the example at left, we borrowed 10 from the top

number, turning the 7 into a 6. We then "returned" the

10 in the form of 10 ones, making the 2 into 12.


36

For practice, compute these subtraction problems.

82) 87 – 19 85) 71 – 28 88) 83 – 26

83) 56 – 47 86) 64 – 38 89) 35 – 15

84) 99 – 59 87) 52 – 7 90) 40 – 19

TWO-DIGIT BY ONE-DIGIT MULTIPLICATION

Review the main book for details about multiplying a

two-digit number by a one-digit number.

Remember: We perform our multiplication

in stages, essentially "distributing" the

second number over the columns of the first.

In the example at left, the 8 multiplies the 9

giving us 72 which is 2 ones and 7 tens. The

2 goes in the ones place of the answer, and

the 7 gets carried into the tens place to remind us that we

must later add 7 tens (or 70) to our answer.

Remember: Even though the next step in the example is

for the 8 to multiply the 3 in the first number, the 3 is

really worth 30 because of its place. We get 240, which

equals 24 tens. We must add in the 7 tens that we carried,

giving us 31 tens which is really 1 ten and 3 hundreds.

We can write 1 in the tens place of our answer, and we

can just write the 3 hundred next to it in the answer.


37

For practice, compute these multiplication problems.

91) 99 × 9 94) 53 × 8 97) 50 × 8

92) 85 × 6 95) 87 × 6 98) 46 × 9

93) 49 × 7 96) 76 × 2 99) 96 × 0

SO NOW WHAT?

Please refer to the main book for an explanation of why

computations like multi-digit multiplication and long

division have been omitted from the series. In the

unlikely case that you are taking an exam which requires

you to do such computations by hand, please contact me

and I'll guide you to sources of free help on the matter.

Before progressing to the next chapter, it is essential that

you fully understand all of the concepts in this one. Take

time to review the material. See the last page of the book

for the companion website that you can use to contact me

for additional information or help.

39

CHAPTER THREE

Basic Math Topics

and Operations

WHAT IS AN INTEGER?

1) What is an integer?

2) List some integers which support the definition.

3) List two examples of values that are not integers.

EVEN AND ODD NUMBERS

4) How do we know if a number is even?

5) How do we know if a number is odd?

Use small, simple numbers such as 1 or 2 to determine

whether the following computations will result in an

even or odd answer.

6) Even + Even 9) Even × Even

7) Even + Odd 10) Even × Odd

8) Odd + Odd 11) Odd × Odd


40

GREATER THAN AND LESS THAN

Insert the appropriate symbol (<) or (>) in each of these

comparisons:

12) 635 97 13) 205 1,999 14) 799 913

INTRODUCING EXPONENTS (POWERS)

15) What is the significance of an exponent (or power)?

For practice, evaluate these bases which have been raised

to various powers. Use a calculator if you need to since

the most important thing is to understand the concept,

but don't use the exponent key if your calculator has one.

16) 34 18) 16 20) 28

17) 45 19) 010 21) 97

SQUARE, CUBE, AND OTHER SPECIAL POWERS

22) How do we usually read an exponent of 2?

23) What does an exponent of 2 mean?

24) How do we usually read an exponent of 3?



27) What is any number (other than 0) to the power of 0?

BASIC MATH TOPICS AND OPERATIONS

41

For practice, evaluate these bases which have been raised

to powers of either 0, 1, 2, or 3.

28) 72 31) 152 34) 10

29) 130 32) 33 35) 23

30) 91 33) 53 36) 122

WHAT IS A PERFECT SQUARE?

37) Explain the concept of "perfect square."

It’s very important to memorize the perfect squares

between 1 and at least 144 since they come up so often in

math. When you see a number in that range, you should

be able to instantly recognize if it is a perfect square, and

if so, what number it is the square of.

For practice, fill in the perfect squares in the chart below,

ideally without using a calculator, and without referring

to the chart in the main book.

12 = 52 = 92 = 132 = 252 =

22 = 62 = 102 = 142 = 302 =

32 = 72 = 112 = 152 = 402 =

42 = 82 = 122 = 202 = 502 =


42

THE SQUARE ROOT OF A NUMBER

38) Explain the concept of "square root."

39) What symbol is used for the square root operation?

40) What is the relationship between squaring and

"square rooting?"

41) When we evaluate a typical square root, do we

sometimes use the squaring or square root notation

in the answer?

For practice, fill in the square roots in the chart below

without using a calculator or referring to any charts:

1 = 36 = 121 = 400 =

4 = 49 = 144 = 625 =

9 = 64 = 169 = 900 =

16 = 81 = 196 = 1600 =

25 = 100 = 225 = 2500 =

ORDER OF OPERATIONS (PEMDAS)

42) What do the letters of PEMDAS stand for?

43) How do we handle the case of nested parentheses or

brackets in a PEMDAS problem?

44) What do we do if a problem has more than one pair

of parentheses which aren't nested?


43

45) Do we always handle multiplication before division

since M comes before D? Why?

46) Do we always handle addition before subtraction

because A comes before S? Why?

47) Do we always handle multiplication before addition

even if addition appears first in an expression? Why?

For practice, use PEMDAS to evaluate the expressions

below. Use a separate sheet of paper to carefully simplify

each expression one step at a time.

48) 10 – 2 + 1 51) 1 + 2 × 1000 54) 100 + 0 × 7

49) 6 × (4 + 3) 52) 50 ÷ 5 × 2 55) 12 + 3 × 4 ÷ 6

50) 3 + 52 × 2 53) 10 + [7×(3+1)] 56) 14 – 49 + 7

WHAT IS A FACTOR?

57) How do we find the factors of a number?

58) What number has only one factor?

59) What numbers are guaranteed factors of any number?

For practice, list the factors of these numbers. Make sure

that this is a task you can do quickly and easily.

60) 36 63) 48

61) 100 64) 41

62) 2 65) 27


44

PRIME AND COMPOSITE NUMBERS

66) What does "prime" mean?

67) What does "composite" mean?

68) Are there any even prime numbers? Elaborate.

69) Is 1 a prime or a composite number? Why?

For practice, state if these numbers are prime or composite:

70) 23 73) 2 76) 13

71) 41 74) 1 77) 27

72) One Million 75) 791,354 78) Ninety-Nine

THE PLACE VALUE CHART UP TO BILLIONS

79) What are the first ten place values called as we look

at a whole (non-decimal) number from right to left?

80) Where is the "zillions" place?

81) How are commas used when writing large numbers?

READING LARGE NUMBERS WRITTEN IN WORDS

Review the main book for instruction on reading and

writing large numbers with words. For practice, convert

these written numbers to their numeric form:

82) Three hundred four thousand:

83) One hundred one:

84) Twenty-seven million, thirty:

85) Two billion, forty-eight thousand :


45

ROUNDING NUMBERS TO VARIOUS PLACES

86) List some times when we round a number.

87) If we are asked to round a number to a given place,

what place will we actually examine?

88) What digits in that place tell us to round up?

89) What digits in that place tell us to round down?

For practice, round these numbers to the specified places.

Refer to the main book for a detailed instructions.

90) 23,552 to the nearest hundred:

91) 4,567,890 to the nearest ten thousand:

92) 6,357,498,765 to the nearest million:

93) 9,700,000,000 to the nearest billion:

SO NOW WHAT?


you fully understand all of the concepts in this one. Take

time to review the material. See the last page of the book

for the companion website that you can use to contact me

for additional information or help.

47

CHAPTER FOUR

Working with

Negative Numbers

WHAT IS A NEGATIVE NUMBER?

1) Using money as an analogy, how should we think

about positive and negative numbers (according to

the main book)?

ADDING SIGNED NUMBERS

2) When we add a positive plus a positive, what sign is

the answer? How should we think about this using

money as an analogy (according to the main book)?

3) When we add a positive plus a negative (or vice-

versa), how can we determine the sign of the answer?

How do we determine the numeric portion of the an-

swer? How should we think about this using money

as an analogy (according to the main book)?


48

4) When we add a negative plus a negative, what sign

is the answer? How do we determine the numeric

portion of the answer? How should we think about

this using money as an analogy (according to the

main book)?

For practice, compute these addition problems involving

adding signed numbers:

5) 7 + (-4) 8) (-5) + 6 11) (-7) + (-8)

6) (-9) + (-5) 9) (-17) + (-2) 12) (-3) + (-3)

7) (-8) + 3 10) 10 + (-10) 13) (-6) + 6

SUBTRACTING SIGNED NUMBERS (AN OVERVIEW)

14) (True/False): The problem 2 – 5 cannot be done.

15) (True/False): The problem 2 – 5 should be converted

to 5 – 2 to get an answer of 3.

16) Use a number line and your knowledge of how

subtraction works to compute 2 – 5.

Remember: The main book asserted that it is best to

convert signed number subtraction problems into

equivalent addition problems. Once that is done, we can

follow the addition procedures that you just practiced.

W ORKING W ITH NEGATIVE NUMBERS

49

HOW TO SUBTRACT TWO SIGNED NUMBERS

Remember: The main book presented a four-step proce-

dure for signed number subtraction problems. It's best to

use this procedure even for "double-negative" problems

such as (4) – (-3) which are associated with a "shortcut."

Step 1 of 4: Leave the first number alone. Don’t touch it.

Step 2 of 4: Change the subtraction (minus) sign to an

addition (plus) sign. This does not involve changing the

sign of either number. It involves changing the actual

operation of the problem from subtraction to addition.

Step 3 of 4: Change the sign of the second number to its

opposite. If it was negative, make it positive. If it was

positive, make it negative.

Step 4 of 4: You have converted the subtraction problem

into an equivalent addition problem which you can solve

as previously described and practiced.

For practice, compute these signed number subtraction

problems by following the four-step procedure above:

17) (-4) – (-5) 20) (-4) – 3 23) (-10) – (-6)

18) (-6) – 6 21) (-5) – 7 24) 7 – (-9)

19) 5 – 7 22) 10 – (-10) 25) 0 – 3


50

MULTIPLYING SIGNED NUMBERS

26) What is the sign of a positive times a positive?

27) What is the sign of a positive times a negative?

28) What is the sign of a negative times a positive?

29) What is the sign of a negative times a negative?

30) Does anything else affect the sign of the answer?

31) According the main book, what analogy can you use

to help remember these rules?

For practice, compute these signed number multiplica-

tion problems by following the above rules:

32) 7 × -8 35) -2 × -15 38) 1 × -1

33) -6 × -7 36) -7 × 0 39) 9 × -8

34) -9 × -10 37) -3 × -1 40) -2 × -2

DIVIDING SIGNED NUMBERS

41) What is the sign of a positive divided by a positive?

42) What is the sign of a positive divided by a negative?

43) What is the sign of a negative divided by a positive?

44) What is the sign of a negative divided by a negative?

45) Does anything else affect the sign of the answer?

46) What operation follows these exact same rules?

For practice, compute these signed number division

problems by following the above rules:

W ORKING W ITH NEGATIVE NUMBERS

51

47) -24 ÷ 6 49) -10 ÷ -2 51) 18 ÷ -9

48) 21 ÷ -7 50) 50 ÷ -5 52) -32 ÷ -32

THE SQUARE OF A NEGATIVE NUMBER

Remember: When we square a negative number in

parentheses, we get a positive answer since we really

have a negative times a negative which is positive. When

we square a number that is not in parentheses but has a

negative sign to the left of it, we must apply the squaring

operation first, and then make the answer negative.

(−8)2 = −8 × (−8) = 64

−82 = −(82) = −64

THE SQUARE ROOT OF A NEGATIVE NUMBER

53) What is the square root of a negative number? Why?

POSITIVE NUMBERS HAVE TWO SQUARE ROOTS

54) What are two valid ways to evaluate 16? Why?

55) How do we typically write and say such an answer?

56) How do we answer a square root problem "by

default," and what do we call such an answer?

ABSOLUTE VALUE

57) What is the significance of absolute value? What is

always the sign of the answer in such problems?


52

58) What symbol(s) do we use to represent the absolute

value operation?

For practice, evaluate these expressions and single terms

which involve the absolute value operation.

59) |18| = 61) |(-3) – 2| = 63) |(-9) – (-9)| =

60) |-27| = 62) |-56 ÷ -7| = 64) |2 – 5| =

SO NOW WHAT?

Before progressing to the next chapter, it is absolutely

essential that you fully understand all of the concepts in

this one. If you do not fully master how to perform the

four basic arithmetic operations with signed numbers,

you will run into endless difficulty with all of the math

that you will study from this point forward. None of this

is “busy” or “baby” work. It is the foundation of math.

Take as much time as you need to review the material in

this chapter, and return to it as often as necessary until

all of it becomes second nature to you, and you are no

longer confused or intimidated by the sight of a negative

number. See the last page of the book for the companion

website that you can use to contact me for additional

information or help.

53

CHAPTER FIVE

Basic Operations

with Fractions (+, –, ×, ÷)

WHAT IS A FRACTION?

1) Define or explain what a fraction is.

2) (True/False): Sometimes a basic fraction problem can

become a matter of, “What if my pizza pie was bigger

than yours in the first place.”

3) (True/False): Sometimes a basic fraction problem can

become a matter of “What if the pie wasn’t cut even-

ly, and my slice was bigger than yours.”

4) What do we call the top part of a fraction?

5) Define or explain its significance.

6) What do we call the bottom part of a fraction?

7) Define or explain its significance.

8) (True/False): a/b is another way of representing 𝑏𝑎.


54

THE EFFECT OF INCREASING / DECREASING THE

NUMERATOR / DENOMINATOR OF A FRACTION

9) As the numerator of a fraction increases, what

happens to the the value of the fraction?

10) As the numerator of a fraction decreases, what

happens to the value of the fraction?

11) As the denominator of a fraction increases, what


12) As the denominator of a fraction decreases, what


Review the main book for a detailed explanation of why

this is the case, and be sure that it makes sense to you.

ADDING AND SUBTRACTING FRACTIONS WITH

LIKE (MATCHING) DENOMINATORS

13) Describe how we add and subtract fractions with like

(matching) denominators.

For practice, add/subtract these fractions with like

denominators. For now, don't worry about reducing the

answers to lowest terms which we'll practice later.

14) 2

15+

7

15 16)

17

21−

9

21 18)

1

3+

1

3+

1

3

15) 7

51−

2

51 17)

8

9−

8

9 19)

10

39+

28

39

BASIC OPERATIONS W ITH FRACTIONS (+ , – , × , ÷ )

55

MULTIPLYING A FRACTION TIMES A FRACTION

20) Describe how we multiply fractions. Does it matter

whether or not the denominators match?

For practice, multiply these fractions. For now, don't

worry about any "cross cancelling" or reducing the


21) 1

2×

1

3 23)

3

8×

3

8 25)

4

5×

5

4

22) 5

7×

2

9 24)

6

15×

1

2 26)

10

10×

6

7

WHAT IS A RECIPROCAL?

27) Define or explain what a reciprocal is.

28) When will a fraction and its reciprocal be equal?

For practice, write the reciprocal of each fraction:

29) 7/2 30) -5/6 31) 17/17

DIVIDING A FRACTION BY A FRACTION

Remember: We have a four-step procedure to follow for

dividing a fraction by a fraction.


56

Step 1 of 4: Leave the first fraction alone

Step 2 of 4: Change the division to multiplication

Step 3 of 4: “Flip” the second fraction to its reciprocal

Step 4 of 4: Multiply as previously described

For practice, compute these fraction division problems.

For now, don't worry about reducing the answers to

lowest terms which we'll practice later.

32) 1

2÷

1

3 34)

5

6÷

5

6 36)

1

3÷

1

2

33) 7

8÷

8

7 35)

3

17÷

2

7 37)

2

5÷

3

4

SO NOW WHAT?


you fully understand all of the concepts in this one. The

next chapter introduces more advanced fraction topics. If

you don’t fully understand this chapter, the next one will

probably be confusing and difficult.

Be sure to also study the multiplication table from

Chapter Two since that will play a large role in the

upcoming material. See the last page of the book for the

companion website that you can use to contact me for

additional information or help.

57

CHAPTER SIX

More About Fractions

A FRACTION IS ACTUALLY A DIVISION PROBLEM

1) Describe how a fraction is really a division problem,

and the significance of the horizontal line.

CONVERTING AN INTEGER TO A FRACTION

2) How do we convert an integer into a fraction?

3) Why is there no harm in doing such a thing?

4) Under what circumstances might we do that?

MULTIPLYING AN INTEGER TIMES A FRACTION

5) When multiplying an integer times a fraction such as

2 × 37 what is the mistake that is commonly made?

6) Why is it a mistake?

7) What is the actual procedure for multiplying an

integer times a fraction?

For practice, compute these problems involving an

integer times a fraction. Don't worry about reducing the



58

8) 3 ×2

5 = 10) 2 ×

3

10 = 12) 1 ×

2

3 =

9) 7 ×4

7 = 11) 4 ×

4

5 = 13) 2 ×

9

18 =

FRACTIONS WITHIN FRACTIONS

Remember: A fraction is just a value divided by a value.

Sometimes the values in question may be fractions

themselves. Since a fraction can be thought of and

rewritten as “top divided by bottom,” we can rewrite the

"fraction of fractions" in the more conventional form as

shown below, and then evaluate it as you learned.

34

89

=3

4÷

8

9=

3

4×

9

8=

3 × 9

4 × 8=

27

32

For practice, convert these "fractions of fractions" into

fraction division problems like the ones we worked with.

Then follow the four-step procedure for converting a

fraction division problem into a multiplication problem,

and then multiply the fractions as you've learned. Don't

worry about reducing the fractions to lowest terms

which we'll practice later.

14) 89

1011

15)

25

79

16)

6

13

12

MORE ABOUT FRACTIONS

59

DIVIDING AN INTEGER BY A FRACTION

Remember: Sometimes a fraction is comprised of an

integer over a fraction, or vice-versa. To simplify such a

fraction, rewrite it as a "top divided by bottom" division

problem, remembering to put the integer over 1 so that it

will be in fraction format. Then proceed as you did with

the previous practice problems. Study the sample

problems below, and then simplify the ones presented.

Don't worry about reducing the fractions to lowest terms

which we'll practice later.

2

34

= 2 ÷

3

4=

2

1÷

3

4=

2

1×

4

3=

2 × 4

1 × 3=

8

3

23

5 =

2

3÷ 5 =

2

3÷

5

1=

2

3×

1

5=

2 × 1

3 × 5=

2

15

17) 3

45 19)

811

5

18) 1

7

6 20)

1

34

FRACTIONS THAT ARE EQUAL TO 1

21) Under what circumstances is a fraction equal to 1?


60

FINDING THE GREATEST COMMON FACTOR (GCF)

22) Define or explain "greatest common factor (GCF)".

23) When do we often use the GCF?

Remember: The main book listed four steps for finding

the greatest common factor (GCF) of two numbers:

Step 1: List all of the factors of the first number

Step 2: List all of the factors of the second number

Step 3: Take note of the factors that appear on both lists

Step 4: Choose the largest of those common factors

For practice, follow the given four steps to compute the

GCF for each pair of numbers below:

24) 2, 4 27) 1, 12 30) 36, 70

25) 24, 36 28) 14, 21 31) 40, 80

26) 1, 100 29) 20, 100 32) 17, 23

REDUCING (SIMPLIFYING) FRACTIONS

Remember: To reduce (simplify) a fraction to lowest

terms, divide both numerator and denominator by their

GCF. When we reduce (simplify) a fraction, we do not

actually change its value. We just convert the fraction

into an equivalent one which does not have any common


61

factors that can be "pulled out." A common and simple

example is reducing 4/8 to the equivalent 1/2.

33) How do we know when a fraction is fully reduced?

For practice, reduce these fractions to lowest terms by

dividing numerator and denominator by the GCF:

34) 14/21 37) 3/17 40) 5/24

35) 1/682 38) 24/24 41) 12/18

36) 32/64 39) 33/88 42) 80/100

MULTIPLES VERSUS FACTORS

43) How do the positive multiples of a number compare in

size to the number (less than, greater than, equal to, etc.)?

44) How do the factors of a number compare in size to

the number (less than, greater than, equal to, etc.)?

COMPUTING THE LEAST COMMON MULTIPLE /

LOWEST COMMON DENOMINATOR (LCM / LCD)

45) Define or explain "least common multiple (LCM)".

46) When do we often use the LCM?

47) What is the relationship between LCM and LCD?

Remember: To add or subtract two fractions with unlike

(non-matching denominators), we must compute the

least common multiple (LCM) of the two unlike denomi-


62

nators. We then “convert” each fraction so that each one

has the LCM as its common denominator, thereby

allowing us to add or subtract them as we’ve learned

how to do. This is practiced in the next section.

ADDING AND SUBTRACTING FRACTIONS WITH

UNLIKE (NON-MATCHING) DENOMINATORS

Remember: We cannot directly add/subtract fractions

with unlike denominators such as 2/3 + 1/4. We must

"convert" the fractions so they have a common denomi-

nator. We compute the least common multiple (LCM) of

the two denominators, and use that as our lowest com-

mon denominator (LCD). The main book outlined a four-

step procedure for computing the LCM of two numbers:

Step 1: List the first few multiples of the first number

Step 2: List the first few multiples of the second number

Step 3: Note the multiplies that appear on both lists

Step 4: Choose the smallest of the common multiples

If necessary, extend the lists in Steps 1 and 2 until you find the

first common multiple.

For practice, find the LCM of these pairs of numbers:

48) 4, 6 51) 5, 10 54) 3, 21

49) 1, 7 52) 2, 23 55) 12, 18

50) 7, 11 53) 6, 8 56) 2, 4


63

Remember: We use the LCM as the “target” number for

each denominator. We must “convert” each fraction to

an equivalent fraction with the LCM as its denominator.

We do that using this four-step procedure:

Step 1 of 4: Multiply the denominator of the first fraction

by whatever number is necessary so it be-

comes the “target” denominator (the LCD).

Step 2 of 4: Multiply the numerator of the first fraction by

same number that you used to multiply the

denominator. Remember that whatever we do

to the bottom, we must also do to the top.

Step 3 of 4: Repeat steps 1 and 2 for the second fraction.

You will multiply top and bottom by a num-

ber other than the one used for the first frac-

tion, but you must still end up with the same

“target” denominator.

Step 4 of 4: You now have two fractions with like (match-

ing) denominators. Add (or subtract) them as

you practiced in the previous chapter.

2

5+

3

7=

2 × 𝟕

5 × 𝟕+

3 × 𝟓

7 × 𝟓=

14

35+

15

35=

29

35

For practice, add/subtract these fractions with unlike

denominators, and reduce your answers.


64

57) 1

2+

1

3 = 59)

2

3−

2

21 = 61)

2

5+

3

8 =

58) 5

6−

1

4 = 60)

3

4−

1

8 = 62)

7

8+

5

6 =

63) What happens if you choose a common denominator

that is larger than the LCM?

SO NOW WHAT?



next chapter introduces some more advanced fraction

topics that are very important. If you don’t fully under-

stand this chapter, the next chapter will probably be very

confusing and difficult for you. Take time to review the

material. See the last page of the book for the companion

website that you can use to contact me for additional

information or help.

65

CHAPTER SEVEN

Other Topics in Fractions

MIXED NUMBERS AND IMPROPER FRACTIONS

1) Define or explain what an improper fraction is.

2) Define or explain what a mixed number is.

3) In a mixed number such as 5 ½, what operation is

implied between the two values?

CONVERTING MIXED NUMBERS TO IMPROPER

FRACTIONS

Remember: When we have to do mathematical opera-

tions involving mixed numbers, it is usually best to

convert them into improper fractions. The main book

outlined a five-step procedure for doing this which in

practice is much simpler than it sounds.

Step 1 of 5: Put a plus sign between the integer and

fractional components of the mixed number.

Step 2 of 5: Convert the integer to a fraction by putting it

over a denominator of 1.


66

Step 3 of 5: Note that the LCM of the two denominators

involved will be the denominator of the

second fraction, since the first fraction has a

denominator of 1.

Step 4 of 5: Multiply top and bottom of the first fraction

by the LCM (i.e., the denominator of the

second fraction) so that the fractions will

have a common denominator.

Step 5 of 5: Add the fractions as previously described

3 +2

5=

3

1+

2

5=

3 × 𝟓

1 × 𝟓+

2

5=

15

5+

2

5=

17

5

For practice, convert these mixed numbers to improper

fractions:

4) 41

7 = 6) 9

3

8 = 8) 1

1

8 =

5) 32

5 = 7) 11

1

2 = 9) 2

8

9 =

Recall that the main book explained a shortcut formula

for this process which is equivalent to the five-step

procedure you just practiced. The shortcut tells us to take

the denominator of the fraction, and multiply it by the

integer. Add the numerator of the fraction to that, and

that result becomes the numerator of the improper

fraction. That numerator it placed over the denominator

of fractional part of the mixed number.

OTHER TOPICS IN FRACTIONS

67

For practice, repeat the previous exercises using the

shortcut, and convince yourself that it works.

𝑎 +𝑏

𝑐=

(𝑎 × 𝑐) + 𝑏

𝑐

CONVERTING IMPROPER FRACTIONS INTO MIXED

NUMBERS

Remember: To convert an improper fraction (or division

problem) into a mixed number, we perform the five-step

procedure in reverse, but again there is a shortcut.

Remember that a fraction is a division problem—top

divided by bottom. To convert an improper fraction to a

mixed number, perform this division. Note the number

of times the denominator "goes into" the numerator. That

becomes the integer part of the mixed number. The

remainder is placed over the denominator of the original

fraction which becomes the fractional part of the mixed

number. Review the main book for more details.

For practice, convert these improper fractions into mixed

numbers. As an example 38/5 = 38 ÷ 5 = 7 ⅗.

10) 11/8 12) 37/5 14) 44/3

11) 23/7 13) 79/6 15) 31/4


68

WORKING WITH EQUIVALENT FRACTIONS

16) How can we use the process of simplifying ("reduc-

ing") to determine if two fractions are equivalent?

17) How can we use the concept of "cross products" to

determine if two fractions are equivalent?

18) What is the difference, if any, between "cross multip-

lying" and "multiplying across"?

“CROSS CANCELING” BEFORE MULTIPLYING

FRACTIONS TO AVOID REDUCING THE PRODUCT

Remember: Before multiplying two fractions, we can do

what is informally referred to as “cross canceling” in an

effort to simplify the arithmetic. We can optionally “pull

out” a common factor from the tops and bottoms of the

two involved fractions so we won't later have to reduce

the answer after multiplying. Two examples from the

main book are shown below:

Remember: Do not utilize this shortcut when dividing a

fraction by another fraction. Recall that to divide two

OTHER TOPICS IN FRACTIONS

69

fractions, we must do several steps to convert the prob-

lem into multiplication. Do not do any type of “cross

canceling” prior to that step, no matter how tempting the

numbers might make it seem.

For practice, compute the products and quotients below

using the "cross-canceling" shortcut where appropriate.

Reduce your answers to lowest terms.

19) 1

2÷

2

7 21)

3

8×

4

9 23)

4

5÷

5

4

20) 5

6÷

12

13 22)

1

2÷

3

4 24)

3

10×

15

16

NEGATIVE FRACTIONS

25) Is the statement below true or false?

– 3

4 =

3

– 4= –

3

4


– 3

– 4 ≠ –

3

4


– 3

– 4 =

3

4


70

DIVISION PROBLEMS INVOLVING 0

28) What is 0 divided by any number (except 0)?

29) What is any number divided by 0?

SO NOW WHAT?



this one. The next chapter introduces some new topics

involving fractions. If you don’t fully understand this

chapter, the next chapter will probably be confusing and

difficult for you.

Take time to review the material. See the last page of the

book for the companion website that you can use to

contact me for additional information or help.

71

CHAPTER EIGHT

The Metric System, Unit

Conversion, Proportions,

Rates, Ratios, Scale

COMMON UNITS OF MEASURE IN THE CUSTOMARY

OR IMPERIAL SYSTEM

If an exam or coursework requires you to memorize facts

about measurements in the customary/imperial system,

quiz yourself on the questions below, reviewing the

corresponding chapter in the main book if necessary.

1) How many inches are in a foot?

2) How many feet are in a yard?

3) How many feet are in a mile?

4) How many ounces are in a pound?

5) How many pounds are in a (US) ton?

6) How many fluid ounces are in a cup?

7) How many cups are in a pint?

8) How many pints are in a quart?

9) How many quarts are in a gallon?


72

PROBLEMS INVOLVING TIME SPANS

If an exam or coursework requires you to memorize facts

about time measurement, quiz yourself on the questions

below, reviewing the main book if necessary.

What fraction of an hour is represented by each of these

durations of time?

10) 5 minutes 14) 20 minutes 18) 45 minutes



13) 15 minutes 17) 36 minutes 21) 30 seconds

COMMON UNITS OF MEASURE IN THE

METRIC SYSTEM

If an exam or coursework requires you to work with and

memorize facts about measurement in the Metric System,

quiz yourself on the questions below, reviewing the

corresponding chapter in the main book if necessary.

22) What is the basic Metric unit of length? What custo-

mary measurement is it roughly equal to?

23) What is the basic Metric unit of volume (capacity)?

What customary measurement is it roughly equal to?

24) What is the basic Metric unit of mass (weight)? What

fraction of an ounce is that unit roughly equal to?

THE METRIC SYSTEM, UNIT CONVERSION, PROPORTIONS, RATES, RATIOS, SCALE

73

COMMON METRIC SYSTEM PREFIXES

25) What does the prefix "kilo-" mean?

26) What does the prefix "milli-" mean?

27) What does the prefix "centi-" mean?

If required for exams or coursework:

28) A kilogram is equal to roughly how many pounds?

29) About how many drops of water fill a milliliter?

30) About how many inches is a centimeter?

THREE DIFFERENT WAYS OF WRITING A RATIO

31) Define or explain what a ratio is.

32) What are the three different ways that we can express

the ratio of some quantity a to some quantity b?

WORKING WITH UNIT RATIOS AND RATES

Remember: A rate is basically the same as a ratio, but it

involves the comparison of two quantities that have

different units. We often “reduce” ratios and rates so that

the denominator (or second value) is 1. We use the word

“unit” to describe such ratios and rates since they are

based on one single unit of whatever is being compared.


74

For practice, convert these ratios and rates into unit

ratios and rates so that the denominators or second

values are equal to 1:

33) 1768 students to 52 teachers 34) 21:3 35) 30/5

COMPUTING THE COST PER UNIT

A common problem on the topic of unit rates is to

compute the cost of one unit of something when we are

provided with the total cost of many such items. All we

do is divide the dollar amount by the number of items.

Refer to the main book for a more detailed explanation.

In each of the problems below, compute the cost of one

item based on the cost of many of each item:

36) 34 apples cost $27.

37) $8 is the cost of 5 apples.

38) Apples are on sale for $5 for 6.

CONVERTING MEASUREMENTS WITH UNIT RATIOS

Remember: Unit ratios such as 3 ft. / 1 yd. make it easy

for us to convert measurements from one unit to another

such as feet to inches, or milliliters to liters. The general

idea is that we multiply the given value by a unit ratio


75

such that the unit we are converting from will be “can-

celed out,” and we will be left with the unit to which we

want to convert. Study the examples below, and review

the main book for details if necessary.

17 𝑓𝑡. =17 𝑓𝑡.

1×

1 𝑦𝑑.

3 𝑓𝑡.=

17

3 𝑦𝑑𝑠. = 5

2

3 𝑦𝑑𝑠.

2500 𝑚𝑔 =2500 𝑚𝑔

1×

1 𝑔

1000 𝑚𝑔=

2500

1000 𝑔 = 2.5 𝑔

For practice, multiply each of the measurements below

by a unit ratio to convert to the unit indicated:

39) How many inches are in 6 feet?

40) Convert 15 feet to yards.

41) How many centimeters are in 7 meters?

42) Convert 9500 kilometers to meters.

INTRODUCTION TO PROPORTIONS

Remember: A proportion is a way of showing that two

ratios (effectively fractions) are equivalent. Many propor-

tion problems involve solving for an unknown value.

Recall from the main book that in a proportion problem

such as 1/4 = 5/?, we can get the answer of 20 by noting


76

that the denominator of the first fraction is 4 times the

size of the numerator. We can also get the answer by

noting that the numerator of the second fraction is 5

times the size of the numerator of the first fraction.

For practice, use one of those techniques to solve each of

the basic proportion problems below:

43) 2

5=

?

30 45)

1

3=

10

? 47)

5

100=

?

20

44) 1

6=

7

? 46)

7

8=

?

24 48)

1

2=

13

?

Remember: Some proportion problems come in the form

of words or diagrams. Read or examine such problems

very carefully, and convert them into a proportion like

the ones above. You’ll need to determine what piece of

information is missing.

Remember: Any comparisons that we do between

numbers in a proportion must be done by way of multip-

lication or division, and not by addition or subtraction.

For example, 5/6 ≠ (does not equal) 7/8.

Remember: Many proportion problems are not as simple

as the ones that we’ve been working with. The numbers

may not be such that the problem can be solved using


77

basic computation, and the missing piece of information

may end up being a decimal number. Such problems

must be solved using basic algebra techniques which you

will learn about in later math, and in the next main book

in this series on basic algebra and geometry. For now just

be sure to understand the general concept. Many exam

questions are based on nothing more than that.

SO NOW WHAT?



this one and in the previous chapters on fractions. In

later math such as algebra you will do much more work

with fractions, but the problems will be abstract in

nature. This means that you must fully understand all of

the concepts at this point while we are still working with

simple numbers.

The next chapter introduces the concept of decimals

which are very much related to fractions. If you don’t

fully understand fractions, the next chapter will probably

be confusing and difficult for you.

79

CHAPTER NINE

Working with Decimals

EXTENDING THE PLACE VALUE CHART TO

DECIMAL PLACES

1) Define or explain what a decimal number is.

2) As we move to the right in the place value chart

(including the decimal places), what is the value of

each place relative to the place on its left?

3) What is the significance of a decimal point?

4) As we move from left to right, what are the names of

the first three places to the right of the decimal point?

5) Where is the "oneths" place?

6) What letters do all decimal place names end in?

THE DIFFERENCE BETWEEN DECIMAL VALUES

7) What is the difference between 2.3 and 2.03? Why?

8) What is the difference between 2.3 and 2.30? Why?

9) What is the difference between .5 and 0.5? Why?

10) What is the difference between 2 and 2.0? Why?


80

WRITING AND SAYING DECIMAL NUMBERS

Review the main book for details on reading and writing

decimal numbers with words. For practice, convert these

written numbers to their numeric form:

11) Twelve and three tenths.

12) Seven hundred one and seven hundredths.

13) Two hundred seventeen thousandths.

14) Five hundred sixty-two and one thousandth.

COMPARING DECIMAL NUMBERS

Review the main book for details on how to compare

decimals. For practice, insert a "<" or ">" symbol into each

statement below to indicate which value is greater:

15) 0.003 0.0004 17) 0.9876 1.002 19) 0.2349 0.238

16) 4.2 3.99999 18) 12.3 4.9999 20) 0.7391 0.739

CONVERTING DECIMAL NUMBERS TO FRACTIONS

Remember: To convert a decimal to a fraction, look to

see how far to the right the decimal digits reach, and use

that place to represent the fraction's denominator. The

decimal digits themselves become the numerator of the

fraction. For example, 0.89 = 89/100, and 0.306 = 306/1000.

W ORKING W ITH DECIMALS

81

For practice, convert these decimals into fractions, and

reduce the fractions to lowest terms:

21) 0.24 24) 0.101 27) 0.09

22) 0.007 25) 0.004 28) 0.06

23) 0.3 26) 0.5 29) 0.018

CONVERTING FRACTIONS TO DECIMAL NUMBERS

Remember: To convert a fraction to a decimal, just

compute what a fraction literally means—top divided by

bottom. This book assumes you do most of your compu-

tations on a calculator. Be careful to enter the numbers in

the proper order. The top number must be keyed in first.

For practice, convert these fractions into decimals.

Round your answers to the nearest hundredth. Refer to

the next chapter if you need help with rounding.

30) 7/11 32) 5/24 34) 1/2

31) 12/13 33) 1/7 35) 9/21

ANOTHER METHOD FOR CONVERTING FRACTIONS

TO DECIMALS

Remember: We can always multiply the top or bottom of

a fraction by a chosen number as long we do the same

thing to the other part of the fraction. If we can multiply

both parts of a fraction so that its denominator is a power


82

of 10 (10, 100, 1000, etc.), it will be very easy to convert it

to a fraction without doing any division as described

above. Of course it is even easier if the fraction started

out with such a denominator.

For example, 92/1000 already has a denominator which is

a power of 10. We can convert it to a decimal by writing

92 such that it extends to the thousandths place, that is to

say 0.092. To convert the fraction 8/25 to a decimal, we

could multiply top and bottom by 4 to get the equivalent

32/100, which then becomes 0.32.

For practice, convert these fractions into decimals by

multiplying top and bottom by the same value such that

the denominator is a power of 10 if it isn't already:

36) 7/500 38) 3/250 40) 11/20

37) 13/1000 39) 2/5 41) 9/50

SO NOW WHAT?



this one. The next chapter introduces some more ad-

vanced decimal topics that are very important. If you

don’t fully understand this chapter, the next chapter will

probably be confusing and difficult for you.

83

CHAPTER NINE

AND FIVE-TENTHS

More Topics in Decimals

REPEATING DECIMAL NUMBERS

Remember: When we convert a fraction to a decimal by

computing top divided by bottom, we sometimes end up

with a repeating decimal.

1) What is a common example of a fraction which is

equivalent to a repeating decimal?

2) (True/False): Repeating decimal digits stop repeating

when the calculator display runs out of room.

3) (True/False): If your calculator computes 2 ÷ 3 as

0.66666667, it means that the 6 doesn't repeat forever.

4) What notation do we usually use to represent a

repeating decimal?

5) (True/False): Repeating decimals can sometimes have

many digits which repeat.

6) (True/False): 0.101001000100001... is an example of a

repeating decimal.


84

For practice, convert these fractions into repeating

decimals using the bar notation as appropriate.

7) 2/3 10) 5/9 13) 123/999

8) 4/7 11) 7/11 14) 7/33

9) 8/15 12) 37/99 15) 12/39

TERMINATING DECIMALS

Remember: Many decimal numbers just stop. If we were

to keep computing more digits using long division, all

we would get is more and more zeroes. For example,

we’ve seen that 0.7 is the same as 0.70, 0.700, 0.7000, and

so on, so we could say that the decimal number termi-

nates at the 7. We say that 0.7 is a terminating decimal.

For practice, indicate if each fraction below is equivalent

to a repeating or a terminating decimal:

16) 9/13 18) 7/100 20) 1234/10000

17) 3/8 19) 1/6 21) 17/300

NON-REPEATING DECIMALS

Remember: A third type of decimal is known as a non-

repeating decimal. Some decimal numbers don't termi-

nate, but they don't have any digits which repeat in a

pattern like the examples we’ve seen.

MORE TOPICS IN DECIMALS

85

COMPARISONS USING REFERENCE FRACTIONS

Remember: When comparing two fractions to determine

which is larger, we sometimes must convert the fractions

into decimals. But, in some cases we can determine

which of two fractions is larger by using common

reference fractions such as ½ or ¼ as guideposts. Review

the main book for a detailed explanation if necessary.

For practice, insert the "<" and ">" symbols as appropriate

in each comparison below. Do this exercise "visually"

based on reference fractions. Avoid doing any computa-

tions on a calculator or by hand.

22) 12/26 21/40 24) 11/100 6/70 26) 20/81 16/60

23) 2/7 461/899 25) 142/199 7/8 27) 34/100 9/30

ARRANGING “NON-OBVIOUS” FRACTIONS IN

ORDER FROM LEAST TO GREATEST

Remember: Some fractions seem so close in value that

using reference fractions to compare them doesn’t help.

In these cases we must examine their decimal equiva-

lents to accurately compare them. We compare decimal

values by starting the comparison in the tenths place and

moving to the right as needed in order to “break ties.”

For practice, insert the "<" and ">" symbols as appropriate

in each comparison below. Do this exercise by using a


86

calculator to convert each fraction into a decimal by

computing top divided by bottom. Then compare the

decimals as practiced in the last chapter.

28) 4/7 30/52 30) 74/99 261/532 32) 9/91 29/293

29) 9/13 55/79 31) 31/39 307/987 33) 27/68 71/179

SHORTCUT FOR MULTIPLYING BY POWERS OF 10

Remember: To multiply a number times 10, move the

decimal point one place to the right, remembering that

whole numbers have an “invisible” decimal point on the

right. Tack on zeroes as needed to hold places. To

multiply by 100, move the decimal point two places to

the right. The pattern continues for higher powers of 10.

For practice, compute all of the products in the following

exercise using the shortcut.

34) 12.34 × 1000 36) 53 × 10,000 38) 1.0001 × 10

35) 0.0043 × 100 37) 27.1234 × 10 39) 0.00009 × 100

SHORTCUT FOR DIVIDING BY POWERS OF 10

Remember: To divide a number times 10, move the

decimal point one place to the left, remembering that

whole numbers have an “invisible” decimal point on the

right. Tack on zeroes as needed to hold places. To

multiply by 100, move the decimal point two places to

the left. The pattern continues for higher powers of 10.

MORE TOPICS IN DECIMALS

87

For practice, compute all of the quotients in the following

exercise using the shortcut.

40) 495 ÷ 1000 42) 94 ÷ 10 44) 41.0003 ÷ 10

41) 23.45 ÷ 10,000 43) 0.0004 ÷ 100 45) 7 ÷ 100

ROUNDING DECIMAL NUMBERS

In Chapter Three we practiced rounding whole numbers

to various places. Review that section if you don’t fully

remember the procedure. Rounding decimal numbers

works in exactly the same way, but review the corres-

ponding section in the main book if you need to.

For practice, round these numbers to the specified places,

making sure to read the place value name carefully.

46) 123.495 to the nearest hundredth

47) 456.7891 to the nearest thousandth

48) 9876.1234 to the nearest ten

49) 12345.678912 to the nearest ten thousandth

50) 3952.46317 to the nearest tenth

SCIENTIFIC NOTATION

Remember: Scientific notation format (𝑎 × 10𝑏) is used

to represent numbers that are either extremely large or

extremely small. The “a” portion is a value that is greater

than or equal to 1, but less than 10. It contains the signifi-


88

cant (non-zero) digits of the very large or very small

number that we are representing. The value of a is

multiplied by the 10 raised to a given power, which we’ll

denote “b.” The exponent “b” can either be positive or

negative. It tells us how many places to move the decim-

al point in the value a. If b is positive, we move the

decimal the given number of spaces to the right, and if

it’s negative we move it to the left.

As an example, 3,850,000,000 is represented in scientific

notation as 3.85 × 109, and 0.00000006401 can be written

in scientific notation as 6.401 × 10−8.

For practice, convert the scientific notation numbers

below into standard notation, and convert the standard

notation numbers into scientific notation:

51) 67,800,000,000 53) 2.17 × 10−5 55) 95,340,000

52) 4.3 × 107 54) 0.0000023 56) 9.843 × 10−6

SO NOW WHAT?



next chapter introduces percents which are yet another

form that fractions and decimals can take. If you don’t

fully understand this chapter, the next chapter will

probably be very confusing and difficult for you.

89

CHAPTER TEN

Working with

Percents

WHAT IS A PERCENT?

1) How are percents, decimals, and fractions related?

2) What does the % symbol literally mean?

3) When we do a computation involving a percent,

what do we usually first convert the percent into?

4) How do we convert from a percent to a fraction?

5) How do we convert from a percent to a decimal?

6) How do we convert from a decimal to a percent?

CONVERTING FROM A PERCENT TO A FRACTION

For practice, convert these percents to reduced fractions:

7) 14% 9) 7% 11) 0.4%

8) 25% 10) 200% 12) 0.09%

CONVERTING FROM A PERCENT TO A DECIMAL

For practice, convert these percents to decimals:


90

13) 87% 15) 0.5% 17) 6%

14) 150% 16) 0.04% 18) 99.44%

CONVERTING FROM A DECIMAL TO A PERCENT

For practice, convert these decimals to percents:

19) 0.27 21) 0.0034 23) 0.5

20) 0.06 22) 1.5 24) 2.0

CONVERTING FROM A FRACTION TO A PERCENT

25) How do we convert from a fraction to a percent if the

denominator can easily be "converted" into 100?

For practice, convert these fractions to percents in the

manner that you described above:

26) 4/25 28) 13/50 30) 230/1000

27) 17/20 29) 7/200 31) 7/10

32) How do we convert from a fraction to a percent if the

denominator cannot easily be "converted" into 100?

For practice, convert each of these fractions to a percent.

Round your answers to the nearest tenth of a percent.

You may use a calculator unless your coursework or

exam requires otherwise.

33) 35/94 35) 7/598 37) 41/99

34) 284/107 36) 92/1234 38) 2/3

W ORKING W ITH PERCENTS

91

IS IT A DECIMAL OR A PERCENT?

39) Is 0.5% a decimal or a percent? Why? Elaborate on

what that value actually represents.

40) Is there such a thing as 110%? Why or why not?

COMPUTING PERCENT OF INCREASE/DECREASE

Remember: The main book outlined a three-step proce-

dure for computing the percent of increase/decrease

when an item goes up/down in price. Review the main

book for a more detailed explanation and examples.

Step 1: Compute the change in price.

Step 2: Divide it by the original price, regardless of

whether the price increased or decreased.

Step 3: Convert the resulting decimal into a percent.

For practice, compute the percent of increase or decrease

for each of the following price changes. Round your

answers to the nearest tenth of a percent:

41) $37 $42 43) $195 $136 45) $7 $21

42) $205 $317 44) $4 $8 46) $95 $94

EQUIVALENT PERCENTS / DECIMALS / FRACTIONS

There are some percents that occur very frequently. It’s

best if you can memorize their decimal and reduced


92

fractional equivalents. For practice, complete the follow-

ing chart using the techniques you have been practicing:

Percent Dec. Fract. Percent Dec. Fract.

0% 25%

0.5% 33 ⅓%

1% 50%

2% 66 ⅔%

5% 75%

10% 100%

12 ½% 150%

20% 200%

PROBLEMS INVOLVING “PERCENT OF”

Remember: Many word problems involve computing a

“percent of” or a “percent off” a number. These two

things are not at all the same, although they look and

sound very similar.

47) What operation does the word "of" translate into

when it appears in between two values?

48) To review, what should we do with a percent when-

ever it is involved in a computation?

For practice, compute the following problems in which

we must calculate a percent of a value. Try to estimate

your answers to see if they are reasonable. Round your

answers to the nearest hundredth.


93

49) 51% of 417 51) 27% of 3093 53) 0.01% of 500

50) 100% of 53 52) 6.2% of 87 54) 0.5% of 200

PROBLEMS INVOLVING “PERCENT OFF”

Remember: Problems involving “percent off” usually

involve computing a discount or price reduction. It is

important to read such problems carefully to determine

if you are being asked to compute the amount of the

discount, or the price after the discount has been de-

ducted. In problems that involve money, we typically

round our answers to the nearest cent (hundredth).

55) If a $599 item is on sale for "38% Off," what percent of

the original price will you actually pay?

56) How much money will you save on the above item

during the sale?

57) How much will the item actually cost during the sale?

For practice, use any method to compute what you will

pay for the items below at the specified discounts.

58) A $395 item on sale for 50% off





94

INCREASING/DECREASING VALUES BY A PERCENT

Remember: Sometimes instead of being asked to com-

pute the percent increase or decrease between two values

like we practiced earlier, we are instead given an amount

and a percent, and are told to increase or decrease the

given amount by the given percent. A typical problem

might be to compute the results of a rent increase.

Remember: First compute the amount of increase, and

then add that increase to the original amount to get the

new amount including the increase. For example, to

increase $1250 by 4.5%, compute the increase as 0.045 ×

$1250 to get $56.25, then add that to the original amount

to get $1306.25. Review the main book for more details.

Remember: For problems involving a percent decrease,

we subtract the amount of decrease instead of adding it,

but we compute the amount of decrease in the same way.

For practice, compute the results of the given percent

increase/decrease in each problem below.

62) A person's $845 monthly rent has increased by 2.3%.

What is the person's new monthly rent?

63) A person's $249 monthly social security benefits have

been decreased by 6.93%. What is the person's new

monthly benefit?


95

64) A person's $307 monthly health care premium has

increased by 19%. What is the new premium?

Remember: Sometimes a problem will ask us to only

compute what the increase/decrease will be based on a

given percent change. For such a problem it is wrong to

add/subtract the percent change to/from the original

amount. Always read problems carefully.

Remember: In this section we practiced solving prob-

lems in which we were given a value and a percent, and

were asked to increase or decrease the amount by that

percent. That is different than the problems that we

practiced earlier in the chapter in which we were given

two values, and were asked to compute the percent of

change between them.

PROBLEMS INVOLVING SALES TAX

Remember: Problems involving sales tax are solved in

exactly the same way as any problem involving a percent

increase. The inclusion of sales tax is just a percent

increase on an amount. Just follow the steps in the

previous section. Always read problems carefully to

determine if you are being asked to compute an item's

price including sales tax, or just the sales tax itself.


96

For practice, solve the sales tax problems below

65) What is the tax on a $2000 item if the tax rate is 7.5%?

66) What is the price, including tax, of a $29.95 item

taxed at 6.75%?

67) What is the tax on a $795 item if the tax rate is 8.875%?

68) What is the final price of a 99¢ item taxed at 5%?

A COMMON MODEL OF WORD PROBLEMS

INVOLVING PERCENTS

Remember: A typical word problem involving percents

is of the form, “61 is what percent of 80?” That problem

could also be stated equivalently as, “What percent of 80

is 61?” Instead of being given a particular percent, we are

asked to compute one. This problem is effectively asking

us to do a comparison of 61 and 80 by division. It wants

us to determine what portion of 80 is represented by 61.

Remember: To solve problems of this form, we must

arrange the given numbers into a fraction. It’s easy to

then convert that fraction into a decimal and then into a

percent, both of which we practiced. In this problem, we

must arrange the given values into 61/80 to do our

comparison. Remember that a fraction is really a division

problem (top divided by bottom). We must compute 61 ÷

80 to get 0.76 (rounded). Move the decimal two places to

the right to convert that to 76% which is our answer.


97

“x is what percent of y?”

𝑥

𝑦 or x ÷ y

“What percent of y is x?”

For practice, solve the percentage problems below, being

very careful in determining the values which correspond

to x and y in the chart above. Round your answers to the

nearest hundredth of a percent:

69) 17 is what percent of 67?

70) What percent of 421 is 3?

71) 60 is what percent of 30?

72) What percent of 489 is 489?

SO NOW WHAT?


essential that you fully understand all of the concepts

presented up to this point. They form the foundation of

all the math that you will study from this point forward.

If you don’t fully understand everything that has been

presented, your study of later math including algebra

will probably be very confusing and difficult for you.

The next chapter introduces basic concepts in probability

and statistics. Most students find those topics to be fun

and interesting, but most standardized exams only

include a few token questions on those topics, instead

favoring the material presented in the earlier chapters.

99

CHAPTER ELEVEN

Basic Probability

and Statistics

COMPUTING THE MEAN (AVERAGE)

1) Define or explain the term "mean."

2) (True/False) When computing the mean of a list of

scores, zeroes don't count because they're just 0.

3) What is the actual formula for computing the mean?

For practice, compute the average (mean) of each of these

lists of values. Round to the nearest tenth.

4) 17, 34, 0, 45, 109, 98 6) 81, 78, 82, 83, 0

5) 98, 98, 98, 98, 98 7) 59, 71, 63, 105, 68

COMPUTING THE MEDIAN

8) Define or explain the term "median."

9) (True/False) When computing the median of a list of

values, the order of the entries doesn't matter.

10) How does the procedure to find the median change if

a list has an even number of values?


100

For practice, compute the median of these lists of values.

Round to the nearest tenth.

11) 103, 46, 53, 64, 77 13) 87, 35, 43, 99, 55, 55

12) 98, 98, 98, 98, 98 14) 0, 30, 20, 40, 50, 999

MEAN VERSUS MEDIAN

15) Give an example of when it makes more sense to find

the median of a list of values instead of the mean.

FINDING THE MODE

16) Define or explain the term "mode."

17) (True/False): A list could have no mode. Why?

18) (True/False): A list could have several modes. Why?

19) (True/False): When computing the mode of a list of

values, the order of the entries doesn't matter.

For practice, compute the mode of these lists of values:

20) 17, 34, 0, 45, 109, 98 22) 54, 74, 36, 54, 46, 63

21) 98, 98, 98, 98, 98 23) 43, 64, 999, 64, 43

FINDING THE RANGE OF A LIST OF NUMBERS

24) Define "range" as it applies to a list of values. Then

compute the range of these lists:

25) 17, 34, 45, 109, 98 26) 98, 98, 98, 98, 98

BASIC PROBABILITY AND STATISTICS

101

BASIC CONCEPTS IN PROBABILITY

27) What is the probability of an "impossible" event?

Can your answer be expressed in another way?

28) What is the probability of a "guaranteed" event? Can

you answer be expressed in another way?

29) What is the probability of an event that is equally

likely to occur as it is to not occur? Can your answer

be expressed in another way?

30) Define or explain the general probability formula.

For practice, compute the probabilities of these events:

31) Rolling 8 on a single roll of one die.

32) Rolling a prime number on a single roll of one die.

33) Drawing a red or a blue marble from an urn that only

contains red and blue marbles.

34) A fair coin landing on heads.

THE CHANCE OF SOMETHING NOT HAPPENING

Remember: We determine the chance of an event not

happening by computing 100% (or 1) minus the chance

of it happening. We usually express the chance of an

event not happening using the form presented in the

problem (fraction, decimal, percent). For practice, com-

pute the probabilities of these events NOT happening:


102

35) There is a 2/5 chance of rain.

36) There is a 72.9% chance of rain.

37) There is a 0.41 chance of rain.

38) There is a 0. 3 chance of rain.

TRICK QUESTIONS AND PROBABILITY MYTHS

39) What does it mean if a probability problem includes

the word "fair"?

40) (True/False): If a fair coin lands on heads 3 times in a

row, the chances of it landing on heads the next time

are good because heads are coming up frequently.

41) (True/False): If a fair coin lands on heads 5 times in a

row, the chances of it landing on tails the next time

are quite good because tails are overdue.

THE PROBABILITY OF COMPOUND INDEPENDENT

EVENTS

42) What does it mean if two events are independent?

43) What operation do we use when solving the proba-

bility of independent events occurring together?

For practice, compute the probabilities of these com-

pound independent events:

44) A coin landing on heads, and a single die landing on

either 1 or 2.

BASIC PROBABILITY AND STATISTICS

103

45) A coin landing on heads, followed by the same coin

landing on heads on the next flip.

46) Three flipped coins all landing on tails.

47) A red die landing on an even number, and a blue die

landing on 4.

PROBABILITY WITH AND WITHOUT REPLACEMENT


the term "with replacement?"


the term "without replacement?

50) An urn has 6 red marbles and 7 blue marbles. What

is the probability of drawing a red marble followed

by a blue marble, with replacement?

51) An urn has 8 red marbles and 11 blue marbles. What

is the probability of drawing a blue marble followed

by a blue marble, without replacement?

PROBLEMS OF THE FORM “HOW MANY WAYS...?”

Remember: If a problem asks you for the number of

combinations that can be made by choosing items from

different categories, just multiply the numbers involved.

Read the problem carefully to ensure that you are not

being tricked in any way. For practice, compute the

number of combinations for each of these problems:


104

52) An ice cream sundae is comprised of a scoop of ice

cream, a wet topping, and a dry topping. How many

possible sundae combinations can be made if cus-

tomers can choose from fourteen flavors of ice cream,

twelve wet toppings, and twenty dry toppings?

53) A man has 4 shirts, 5 pairs of pants, 6 cats, and 7 ties.

How many possible outfits comprised of a shirt, a

pair of pants, and a tie can he create?

SO NOW WHAT?

Be certain to always read probability questions slowly

and carefully. Misreading one word can totally change

the entire problem, and of course most probability

questions are in the form of word problems.

By far, your time is best spent ensuring that you are fully

comfortable with the material on basic arithmetic, as well

as fractions, decimals, and percents. You will work with

those topics again and again as you progress to more

advanced math such as algebra. If you don’t master

those topics now, you will simply have to master them

later when you are busy with other work.

105

End-of-Book Self-Test

Take this self-test after you've worked through the

exercises in this book, but not immediately after. Use it to

determine how much of the material you're retaining,

and what concepts you haven't fully internalized. Don't

be concerned about how many questions you get right or

wrong. Just make sure you understand why the right

answers are right and the wrong answers are wrong.

After completing this test and checking your answers,

review the exercises in this book with an emphasis on the

topics that you either forgot or had trouble with. If you

have questions or need help, contact me via my website.

1) Multiply: 7 ×3

27

2) Add: 2

5+

3

7

3) True/False: 2

3=

10

19

4) True/False: 101

102=

19

20

5) True/False: −3

−8=

12

32

6) True/False: −9

4=

9

−4

7) Add: −7

11+

5

11

8) Multiply: −3

8×

4

7

9) Divide: −3

7÷

7

−3

10) Multiply: −17

39×

39

−17

11) Compute 58 + 99

12) Compute 72 - 35

13) Define: Integer

14) Evaluate (−8)1

15) Evaluate 81

16) Evaluate 7 + 5 × 3


106

17) Evaluate -18 – 5 + 1

18) Compute: 5 + (-2)

19) Compute: (-1) + (-5)

20) Compute: 2 – 11

21) Compute: (-7) – 3

22) Compute: (-9) – (-5)

23) Compute: (-8) × 7

24) Compute (-2) × (-13)

25) Compute: 55 ÷ (-11)

26) Compute: (-8) ÷ (-4)

27) Evaluate: (−7)2

28) Evaluate: −143

29) Evaluate 45

30) Evaluate: |-13|

31) Evaluate: |(-5) – 9|



34) Is 8 prime? Why?

35) Compute: 0 ÷ -1

36) Compute: -1 ÷ 0

37) What is 12 squared?

38) What is 62% of 482?

(round to the near-

est whole number)

39) Which basic operations are commutative?

40) Find the product of -3 and -7

41) Find the sum of 7 and -8

42) List the first ten multiples of 9

43) Compute 61 ÷ 11 in mixed number format

44) Is the number 682,403 even or odd?

45) What is the result of adding an even plus an odd

number? (Even or Odd)

46) Insert "<" or ">": 630,001 99,999

47) Evaluate -7 × [-40 ÷ (-4 + 2)]

48) Is 4 a composite number? Why?

49) Write "Seven million, fifty-five thousand, seventeen"

as a number

END-OF-BOOK SELF -TEST

107

50) Round 35,012 to the nearest ten thousand

51) Round 986,749 to the nearest hundred

52) Evaluate: 4. Include both roots

53) Write "Three thousand twenty-four and eighteen

hundredths" as a number

54) Insert "<" or ">": 4.09999 4.13

55) Insert "<" or ">": 0.375 0.37499

56) Convert to a fraction: 0.0087

57) Convert 3/11 to a decimal (round to the nearest

hundredth):

58) Convert to a decimal (no calculator): 13/50

59) Convert to a decimal: 2/3

60) True/False: 0.8 = .8 = 0.080

61) True/False: 0. 1750 is a repeating decimal

62) Insert "<" or ">" (no calculator): 31/99 24/47

63) Insert "<" or ">" (use calculator): 498/998 349/701

64) Multiply (no calculator): 37.194 × 10,000

65) Divide (no calculator): 23.4 ÷ 100,000

66) Round 784.1234 to the nearest ten

67) Round 61.23495 to the nearest ten thousandth

68) Express 7,890,000,000 in scientific notation

69) Express 2.123 × 10−7 in standard notation

70) Convert 22% to a reduced fraction

71) Convert 209% to a decimal

72) Convert 0.1234 to a percent

73) Convert 13/21 to a percent (round to nearest tenth)


108

74) Convert 9/25 to a percent (no calculator)

75) Convert 0.005% to a decimal

76) Compute the percent of change from 69 to 61 (round

to the nearest tenth of a percent)

77) How much money will you save on a $139.95 item

during a "25% Off" sale?

78) What will a person's monthly rent be after a 1.95%

increase if it is currently $1043?

79) What is the cost including tax on a $275.95 item if

the tax rate is 7.5%?

80) What percent of 23 is 8.2? Round to nearest tenth.

81) What is the reciprocal of -5/13?

82) Convert 8 ⅔ to an improper fraction

83) How is a kilogram related to a gram?

84) Apples are being sold at the rate of 72 apples for

$39.95. How much does one apple cost at that rate?

85) Convert 275 centimeters to meters

86) Solve for the unknown value: 1

7=

?

56

87) Simplify to a single fraction: (2

7) / 9

88) Find the mean of this list (rounded to the nearest

integer): 237, 0, 391, 0, 62, 93

89) Find the median of this list: 36, 7, 52, 247, 12

90) Find the median of this list: 2, 8, 3, 6, 7, 1

91) Find the mode of this list: 1, 2, 1, 2, 3, 4

92) Find the mode of this list: 1, 1, 1, 1, 1, 1, 1

END-OF-BOOK SELF -TEST

109

93) Find the range of this list: 64, 49, 21, 20, 43, 29

94) Find the probability of rolling a 2 or 5 on a single

roll of one standard die

95) Find the probability of rolling an 8 on a single roll of

one standard die

96) What is the chance that a fair coin will land on heads

after having landed on heads three times in a row?

97) If there is an 82.1% chance that it will rain tomor-

row, what is the chance that it will not rain?

98) In an experiment comprised of a coin toss followed

by a roll of a single die, what is the probability of

flipping tails and rolling either 3 or 6?


the probability of drawing a red marble followed by

a blue marble, with replacement.


the probability of drawing a blue marble followed

by a blue marble, without replacement.

101) A woman has four blouses, five skirts, and six hats.

How many different outfits comprised of a blouse, a

skirt, and a hat can she create?

102) Convert -29 to a fraction

103) What is the GCF of 12 and 24?

104) What is the LCM of 1 and 7?

105) Reduce 4/21 to lowest terms

111

CHAPTER TWELVE

How to Study and Learn

Math, and Improve

Scores on Exams

This chapter is intended to review and supplement the

material presented in the corresponding chapter in the

main book. Be sure to read or review that material first.

This chapter offers some practical exercises which will

hopefully allow you to study math more efficiently and

perform better on exams.

PREPARING FOR “WHAT-IF” SCENARIOS

There is a fine line between attempting to outguess what

will be on an exam, and being outright obsessive. If you

have started studying for an exam well in advance, make

it a point to do at least some preparation for "what-if"

exam scenarios, as long as you are not making yourself

nervous in the process. Try to think realistically about

what type of problems are likely to be on your exam, and


112

make sure that you know how to solve them. Don't

approach your exam with the attitude of, "Well, I just

hope that there won't be any negative numbers involved,

and if there are, I'll just get those questions wrong and

hope to get the others right, and I'll probably still pass."

With all of that said, if you have left your studying for

the last minute, just utilize your remaining time to

remain calm and focused. Any last minute studying or

worrying about "what-if" scenarios will almost certainly

do more harm than good.

AVOIDING A “PASS OR FAIL” MINDSET

When preparing for an exam, make it a point to com-

pletely remove the words "pass" and "fail" from your

vocabulary. If you believe that you will fail, you proba-

bly will, even if just to self-fulfill your own prophecy. If

your goal is to just barely pass, you will probably either

fail by just a few points, or you will pass by the "skin of

your teeth," resulting in stress during your exam and

while you are waiting for the results.

Always aim for a perfect score. The point is not whether

you achieve it, or whether it is necessary to do so. Even

without studying any additional math, your scores will

improve if you maintain a positive attitude, and set

HOW TO STUDY AND LEARN MATH, AND IMPROVE SCORES ON EXAMS

113

higher goals for yourself. Remember, there is no such

thing as over-studying for an exam as long as you are not

making yourself anxious in the process.

PRACTICING MEDITATION TO CULTIVATE THE

OPTIMAL MINDSET FOR TAKING EXAMS

Meditation is just the practice of placing your mind on a

single object of focus, as opposed to what we usually do

which is try to think about countless things all at the

same time. In moments of silence we can become aware

of the endless chatter that we usually have in our heads.

Practicing meditation can help you cultivate a relaxed

yet alert mindset that is optimal for taking math exams.

It is the complete opposite of being "in a trance," al-

though an onlooker might not be able to make the

distinction. Sit in a position that is comfortable but that

will not result in slouching or dozing off. Pick something

to be the object of your focus, and just practice keeping

your focus on that object. The object can be something

tangible, or a meaningful phrase, or even what is taking

place in your own mind. If you lose focus, just practicing

bringing your focus back to the task at hand without

analyzing or being concerned about why you lost focus.


114

Over time, this skill will carry over into all facets of your

life, including taking exams. Certainly it is important to

maintain focus throughout an exam. If you lose focus,

you want to be aware of such as quickly as possible, and

be able to bring your focus back to the exam without

getting flustered. This skill is precisely what is developed

during meditation. Any time that you invest in practic-

ing meditation will be returned to you many times over

in the form of a more relaxed and alert existence.

MORE PRACTICAL TIPS FOR REDUCING ANXIETY

Try to avoid interacting with other students before an

exam who will only serve to wind you up in various

ways. Instead, use that time to get into the state of mind

that you've been practicing. Avoid last minute studying

which is much more likely to make you nervous and

discouraged than help you. By far you are better of using

your pre-exam time to breathe deeply and relax.

During the exam itself, just place all of your focus on the

exam. If you are relaxed and alert you should have no

trouble determining when you have lost focus, and it

should be easy to bring your focus back. If at any time

you are feeling flustered or overwhelmed, look up from

your exam, take a slow deep breath, and then refocus


115

yourself. The few moments that you lose while taking a

deep breath will be returned to you in the form of a more

relaxed and focused mind for the remainder of your

exam time.

LEARNING TO FOLLOW INSTRUCTIONS

Make an agreement with yourself to not lose any points

on exams due to a simple lack of following instructions.

Remember, some instructions may be applicable to the

entire test such as where and how to write your answers,

and some may apply to individual problems or sets of

problems. Read every word on the page slowly and

carefully, and don't make any assumptions about any-

thing. Even if some instructions seem generic, read them

carefully anyway. Don’t approach any written words on

the page with the attitude of, "Yeah, yeah, whatever."

CHECKING ANSWERS FOR REASONABLENESS

Plan ahead of time to handle each question by first

estimating the answer, and then checking to see if the

answer that you get is reasonable. For some problems

this will not be practical or applicable so just do the best

you can with this tip. Just don't lose points by submitting

an answer that couldn't possibly be right.


116

CAN YOU TEACH THE TOPICS TO SOMEONE ELSE?

The best way to know if you are prepared for an exam is

to see if you can teach the topics to someone else. Ideally

this should be done in a study group, but if no one is

around you could even try teaching an imaginary

person, even if doing so is effectively talking to yourself.

The point is that if you can effectively explain a concept

to someone else, you should be able to demonstrate

mastery of that concept on your exam. That is all exams

are actually designed to do. They just check to see the

extent to which you have internalized the various

concepts that are being tested.

BASIC LOGISTICAL ISSUES OF TAKING EXAMS

Make a conscious effort to not lose points due to logistic-

al test-taking issues. Make sure that you have extra pens

and pencils. Wear a watch so you can keep track of the

time. Make sure that your calculator has batteries and

that you know how to use it. If you are going to a special

testing location that you haven't been to before, plan

your trip in advance, and allow extra travel time.

Make sure you're not overly hungry or thirsty, but don't

nervously eat or drink to the point where you'll have to


117

worry about using the bathroom. If any ID or special

pass is required to take an exam, be sure that you have it.

Don't bring any items to your test that are going to be a

point of a concern such as a large or unusual bag, or any

type of electronic device that could result in you being

suspected of cheating. Certainly don't cheat or do any-

thing that appears as though it could involve cheating.

This might include making unusual noises or gestures.

THE FINAL WORD

Just the fact that you even purchased this book proves

that you want to achieve your math goals, and that at

least at some level, you believe that you can. You proba-

bly already read through the first book in the series, and

recognized the importance of getting additional practice

with the material presented.

I truly believe that virtually everyone can succeed in

their math goals. You can achieve whatever math goals

you have set for yourself, but doing so will certainly take

time and effort. Don't let anyone tell you otherwise.

Contact me via my website if you have questions about

the material, or would like to discuss your academic

situation. Study hard and believe in yourself! ☺

119

Answers to Exercises

and Self-Tests

Please read the section in the Introduction on typos and

errors. Remember that you can contact me with any

questions, and visit my website for help and information.

On request, I can provide additional practice exercises

for any topic, although you should also try making up

your own practice exercises.

Make sure that you understand why your right answers

are right, and why your wrong answers are wrong.

Answers involving words do not need to match these

answers exactly as long as you understand the concept.

ASSESSMENT SELF-TEST

1) Subtraction,

Division

2) 24

3) 15

4) 7, 14, 21, 28, 35, 42,

49, 56, 63, 70

5) 7 8

9

6) Even

7) Even

8) 73,001 > 72,999

9) 12


120

10) No. It is prime. It

has two unique fac-

tors: 1 and itself.

11) 9,503,040,017

12) 27,800

13) 140,000

14) ±11

15) 203.59

16) 5.99999 < 6.18

17) 0.29 > 0.2876

18) 21/1000

19) 0.29

20) 0.12

21) 0. 3

22) True

23) False: It terminates.

24) 6/25 < 251/500

25) 425/639 < 541/803

26) 7230

27) 0.00123

28) 12.35

29) 80.0

30) 1.234 × 109

31) 0.0000789

32) 7/50

33) 1.07

34) 56.7%

35) 64.7%

36) 6%

37) 0.0007

38) 60.9%

39) $8.99

40) $839.06

41) $211.09

42) 40.2%

43) 11/7

44) 29/4

45) It is one-thousandth

the size

46) 56¢ or $0.56

47) 8.5 or 8 ½ feet

48) 28

49) 3/56

50) 53.6

51) 12

52) 85

53) 12

54) No mode

55) 59

56) 2/6 or 1/3

57) 0 or 0%

58) 1 or 100%

ANSW ERS TO EXERCISES AND SELF -TESTS

121

59) 17/20

60) 1/12

61) 15/64

62) 1/3

63) 60

64) 17/1

65) 2

66) 60

67) 3/17

68) 8/9

69) 5/6

70) True

71) False

72) True

73) True

74) 12/13

75) 6/35

76) 16/25

77) 1

78) 126

79) 56

80) A whole number

that is positive,

negative, or zero.

81) 9

82) 8

83) 11

84) 4

85) -3

86) -14

87) -7

88) -6

89) -4

90) -54

91) 40

92) -4

93) 4

94) 16

95) Undefined

96) 81

97) 8

98) 7

99) 1, 2, 24, 46

100) 1, 23

101) No. It has factors

besides 1 and itself.

102) 0

103) Undefined

104) 9

105) 383


122

CHAPTER TWO

1) Sum

2) Yes. 3 + 5 = 5 + 3

3) Nothing. It says the

same.

4) 15

5) 12

6) 11

7) 5

8) 9

9) 10

10) 11

11) 10

12) 16

13) 16

14) 15

15) 12

16) 9

17) 13

18) 11

19) 10

20) 18

21) 17

22) Difference

23) No. 5 – 3 ≠ 3 – 5

24) Nothing. It says the

same.

25) No. We're not com-

bining groups.

26) 2

27) 0

28) 2

29) 0

30) 1

31) 4

32) 1

33) 3

34) 9

35) Product

36) Yes. 8 × 7 = 7 × 8

37) Repeated addition

38) 0

39) The number itself

40) Multiply the num-

ber times 1, times 2,

times 3, etc.

41) 56

42) 36

43) 64


123

Below is a completed 12 by 12 multiplication table. The

exercise of listing the first 12 multiples of each number can be

checked by reading across each row.

44) 55

45) 36

46) 63

47) 54

48) 42

49) 0

50) Quotient

51) No. 10 ÷ 1 ≠ 1 ÷ 10

52) 7

53) 6

54) 1

55) 9

56) 8

57) 8

58) 8

59) 12

60) 7

61) 7 R 1

62) 4 R 5

63) 1 R 4


124

64) 8 R 6

65) 8 R 2

66) 8 R 8

67) 7 R 1

68) 6 R 2

69) 12 R 2

70) Ones, Tens, Hun-

dreds, Thousands

71) Each place value

must be aligned.

72) Right to left.

73) 121

74) 198

75) 117

76) 131

77) 112

78) 70

79) 150

80) 153

81) 100

82) 68

83) 9

84) 40

85) 43

86) 26

87) 45

88) 57

89) 20

90) 21

91) 891

92) 510

93) 343

94) 424

95) 522

96) 152

97) 400

98) 414

99) 0

CHAPTER THREE

1) A whole number:

Positive, negative, or 0

2) Ex. 17, 0, -3

3) Ex. ½, 0.7

4) Its rightmost digit

ends with 0, 2, 4, 6, or 8

5) Its rightmost digit

ends with 1,3, 5, 7, or 9


125

6) Even

7) Odd

8) Even

9) Even

10) Even

11) Odd

12) >

13) <

14) <

15) The number of times

to multiply the base

times itself

16) 81

17) 1024

18) 1

19) 0

20) 256

21) 4,782,969

22) Squared

23) Base × Base

24) Cubed

25) Base × Base × Base

26) Just the base itself

27) 1

28) 49

29) 1

30) 9

31) 225

32) 27

33) 125

34) 1

35) 8

36) 144

37) Product of a whole

number times itself

Below are the completed charts of squares and square roots.

12 = 1 52 = 25 92 = 81 132 = 169 252 = 625

22 = 4 62 = 36 102 = 100 142 = 196 302 = 900

32 = 9 72 = 49 112 = 121 152 = 225 402 = 1600

42 = 16 82 = 64 122 = 144 202 = 400 502 = 2500


126

1 = 1 36 = 6 121 = 11 400 = 20

4 = 2 49 = 7 144 = 12 625 = 25

9 = 3 64 = 8 169 = 13 900 = 30

16 = 4 81 = 9 196 = 14 1600 = 40

25 = 5 100 = 10 225 = 15 2500 = 50

38) A number which

when squared equals

the number under

the symbol

39)

40) They're inverse ops.

41) No, e.g., 16 = 4.

Just plain 4.

42) Parentheses, Expo-

nents, ×, ÷, +, –

43) Work inner to outer

44) Evaluate the pairs

from left to right

45) No. Handle × and ÷

in order from L to R

46) No. Handle + and –

in order from L to R

47) Yes. × and ÷ have

priority over + and –

48) 9

49) 42

50) 53

51) 2001

52) 20

53) 38

54) 100

55) 14

56) 14

57) Determine the num-

bers which divide

into that number

evenly

58) 1

59) 1 and itself

60) 1, 2, 3, 4, 6, 9, 12, 18, 36

61) 1, 2, 4, 5, 10, 20, 25,

50, 100

62) 1, 2


127

63) 1, 2, 3, 4, 6, 8, 12, 16,

24, 48

64) 1, 41

65) 1, 3, 9, 27

66) A number that has

two unique factors: 1

and itself

67) A number that has

other factors besides

1 and itself

68) Only the number 2.

Any other even num-

ber has 2 as a factor so

it can't be prime

69) Neither. It's a special

case because 1 and

itself are not unique.

70) Prime

71) Prime

72) Composite

73) Prime

74) Neither

75) Composite

76) Prime

77) Composite

78) Composite

79) Ones, tens, hundreds,

thousands, ten thou-

sands, hundred thou-

sands, millions, ten

millions, hundred mil-

lions, billions

80) No such thing

81) We use a comma to

the left of every third

place starting on the

right.

82) 304,000

83) 101

84) 27,000,030

85) 2,000,048,000

86) To estimate, to stop a

repeating decimal,

when dealing with $.

87) The place on its right

88) 5 to 9

89) 0 to 4

90) 23,600

91) 4,570,000

92) 6,357,000

93) 10,000,000,000


128

CHAPTER FOUR

1) Think of positives as

assets and negatives

as debts

2) Positive. You are

adding your assets.

3) Do you have more

than you owe?

Compute the differ-

ence between them.

4) Negative. Just add

the involved num-

bers. You're adding

your debts.

5) 3

6) -14

7) -5

8) 1

9) -19

10) 0

11) -15

12) -6

13) 0

14) False

15) False

16) -3

17) 1

18) -12

19) -2

20) -7

21) -12

22) 20

23) -4

24) 16

25) -3

26) Positive

27) Negative

28) Negative

29) Positive

30) No

31) Matching is good

(positive). Mismatched

is bad (negative).

32) -56

33) 42

34) 90

35) 30

36) 0

37) 3


129

38) -1

39) -72

40) 4

41) Positive

42) Negative

43) Negative

44) Positive

45) No

46) Multiplication

47) -4

48) -3

49) 5

50) -10

51) -2

52) 1

53) Undefined. We can't

square a number and

get a negative result.

54) +4 or -4. Squaring

either yields 16.

55) ±4. Plus or minus 4.

56) We give the positive

version (principal root.

57) It tells us a number's

distance from 0 which

is always positive.

58) Vertical bars: | |

59) 18

60) 27

61) 5

62) 8

63) 0

64) 3

CHAPTER FIVE

1) Part of a whole

2) False

3) False

4) Numerator

5) How many parts are

of concern

6) Denominator

7) How many parts we

have in total


130

8) False (upside-down)

9) Increases

10) Decreases

11) Decreases

12) Increases

13) Just add the numera-

tors and keep the

denominator

14) 9/15

15) 5/51

16) 8/21

17) 0

18) 1

19) 38/39

20) Multiply the top

straight across and the

bottom straight across.

It doesn’t matter if the

denominators match.

21) 1/6

22) 10/63

23) 9/64

24) 6/30

25) 20/20

26) 60/70

27) A fraction "flipped"

"upside-down"

28) When the numerator

and denom. are equal

29) 2/7

30) 6/(-5)

31) 17/17

32) 3/2

33) 49/64

34) 30/30

35) 21/34

36) 2/3

37) 8/15

CHAPTER SIX

1) The horizontal line

means division: top

divided by bottom

2) Put it over a deno-

minator of 1

3) Anything divided by

1 equals itself


131

4) If an integer is part

of a fraction problem

5) Multiplying both top

and bottom by the

integer

6) The integer is really

over 1

7) Place the integer over

1 and multiply. The

integer only multiplies

the numerator.

8) 6/5

9) 28/7

10) 6/10

11) 16/5

12) 2/3

13) 18/18

14) 88/90

15) 18/35

16) 12/13

17) 15/4

18) 1/42

19) 8/55

20) 4/3

21) When numerator and

denom. are equal

22) The largest number

that divides evenly into

the given numbers

23) When simplifying

fractions

24) 2

25) 12

26) 1

27) 1

28) 7

29) 20

30) 2

31) 40

32) 1

33) When the GCF of top

and bottom is 1

34) 2/3

35) 1/682

36) 1/2

37) 3/17

38) 1

39) 3/8

40) 5/24

41) 2/3

42) 4/5


132

43) Greater than or

equal to

44) Less than or equal to

45) The smallest number

that appears in the

lists of all the given

numbers.

46) When we need to

add/subtract frac-

tions with unlike

denominators

47) For the above, we

use the LCM as our

LCD

48) 12

49) 7

50) 77

51) 10

52) 46

53) 24

54) 21

55) 36

56) 4

57) 5/6

58) 7/12

59) 4/7

60) 5/8

61) 31/40

62) 41/24

63) It will just take some

extra steps to reduce

your answer

CHAPTER SEVEN

1) A fraction whose

numerator is greater

than or equal to its

denominator

2) A value comprised

of an integer plus a

fraction

3) Addition

4) 29/7

5) 17/5

6) 75/8

7) 23/2

8) 9/8

9) 26/9


133

10) 1 3

8

11) 3 2

7

12) 7 2

5

13) 13 1

6

14) 14 2

3

15) 7 3

4

16) See if the reduced

fractions are equal

17) See if the cross

products are equal

18) "Multiply across" to

multiply fractions, and

"cross multiply" in or-

der to compare cross

products

19) 7/4

20) 65/72

21) 1/6

22) 2/3

23) 16/25

24) 9/32

25) True

26) True

27) True

CHAPTER EIGHT

1) 12

2) 3

3) 5280

4) 16

5) 2000

6) 8

7) 2

8) 2

9) 4

10) 1/12

11) 1/6

12) 1/5

13) 1/4

14) 1/3

15) 5/12

16) 1/2

17) 3/5

18) 3/4

19) 5/6

20) 1


134

21) 1/120

22) Meter. Yard.

23) Liter. Quart.

24) Gram. 3/100 oz.

25) 1000 times as big

26) 1/1000 the size

27) 1/100 the size

28) 2.2

29) 20

30) 0.4

31) A comparison of two

values using division

32) a:b, a to b, a/b

33) 34 students to 1

teacher

34) 7:1

35) 6/1

36) $0.79

37) $1.60

38) $0.83

39) 72 in.

40) 5 yds.

41) 700 cm.

42) 9,500,000 m.

43) 12

44) 42

45) 30

46) 21

47) 1

48) 26

CHAPTER NINE

1) A number with a

fractional component

2) One-tenth the value

3) It separates the

whole and fractional

place values

4) Tenths, hundredths,

thousandths

5) There isn't any

6) "-ths"

7) The first number is

bigger. The 3 is in

the tenths place and

not the hundredths.


135

8) They're equal. The 0 at

the end doesn't change

the value.

9) They are equal. Both

have no wholes and

five tenths.

10) They are equal. Both

have 2 wholes and no

fractional component.

11) 12.3

12) 701.07

13) 0.217

14) 562.001

15) >

16) >

17) <

18) >

19) <

20) >

21) 6/25

22) 7/1000

23) 3/10

24) 101/1000

25) 1/250

26) 1/2

27) 9/100

28) 3/50

29) 9/500

30) 0.63

31) 0.92

32) 0.21

33) 0.14

34) 0.5

35) 0.43

36) 0.014

37) 0.013

38) 0.012

39) 0.4

40) 0.55

41) 0.18

CHAPTER NINE AND FIVE-TENTHS

1) Ex. ⅓

2) False

3) False

4) A bar over the

repeating digits

5) True


136

6) False

7) 0. 6

8) 0. 571428

9) 0.53

10) 0. 5

11) 0. 63

12) 0. 37

13) 0. 123

14) 0. 21

15) 0. 307692

16) Repeating

17) Terminating

18) Terminating

19) Repeating

20) Terminating

21) Repeating

22) <

23) <

24) >

25) <

26) <

27) >

28) <

29) <

30) >

31) >

32) <

33) >

34) 12,340

35) 0.43

36) 530,000

37) 271.234

38) 10.001

39) 0.009

40) 0.495

41) 0.002345

42) 9.4

43) 0.000004

44) 4.10003

45) 0.07

46) 123.50

47) 456.789

48) 9880

49) 12345.6789

50) 3952.5

51) 6.78 × 1010

52) 43,000,000

53) 0.0000217

54) 2.3 × 10−6

55) 9.534 × 107

56) 0.000009843


137

CHAPTER TEN

1) They're all ways of

representing part of

a whole

2) Out of 100

3) A decimal

4) Drop the % sign, put

the value over 100

5) Drop the % sign and

move the decimal

two places to the left

6) Move the decimal

two places to the

right, add a % sign

7) 7/50

8) 1/4

9) 7/100

10) 2/1 or 2

11) 1/250

12) 9/10000

13) 0.87

14) 1.5

15) 0.005

16) 0.0004

17) 0.06

18) 0.9944

19) 27%

20) 6%

21) 0.34%

22) 150%

23) 50%

24) 200%

25) Multiply or divide top

and bottom by the

same value to make

the denominator 100.

Take the numerator

and add a % sign.

26) 16%

27) 85%

28) 26%

29) 3.5%

30) 23%

31) 70%

32) Compute numerator

divided by denomi-

nator, multiply by

100, add a % sign

33) 37.2%

34) 265.4%

35) 1.2%


138

36) 7.5%

37) 41.4%

38) 60.7%

39) It is a % because it

has a % sign. It is

half of 1% or 1/200.

40) It represents greater

than a whole, common

in price-increase prob-

lems.

41) 13.5%

42) 54.6%

43) 30.3%

44) 100%

45) 200%

46) 1.1%

Below is the chart of equivalent percents, decimals, and fractions

Percent Dec. Fract. Percent Dec. Fract.

0% 0 0 25% 0.25 1/4

0.5% 0.005 1/200 33 ⅓% 0.333 1/3

1% 0.01 1/100 50% 0.5 1/2

2% 0.02 1/50 66 ⅔% 0.666 2/3

5% 0.05 1/20 75% 0.75 3/4

10% 0.1 1/10 100% 1 1

12 ½% 0.125 1/8 150% 1.5 3/2

20% 0.2 1/5 200% 2 2

47) Multiplication

48) Convert to a decimal

49) 212.7

50) 53.0

51) 835.1

52) 5.4

53) 0.05

54) 1.0

55) 62%

56) $227.62

57) $371.38

58) $197.50

59) $25.65

60) $3996


139

61) $3.60

62) $864.44

63) $231.74

64) $365.33

65) $150

66) $31.97

67) $70.56

68) $1.04

69) 25.37%

70) 0.71%

71) 200%

72) 100%

CHAPTER ELEVEN

1) A value representing

the balance point of

a list of values

2) False

3) Mean = (Sum of

values) ÷ (# of values)

4) 50.5

5) 98

6) 64.8

7) 73.2

8) The middlemost val-

ue in a sorted list

9) False

10) The median is the

mean of the two

middlemost values

11) 64

12) 98

13) 55

14) 35

15) Reporting the income

of a town's residents

16) The value that occurs

the most frequently in

a list of values.

17) True, if no value

occurs more fre-

quently than others

18) True, if more than 1

value is tied for most

frequently occurring

19) True

20) No mode

21) 98

22) 54

23) 43 and 64


140

24) The difference be-

tween the largest

and smallest values

25) 92

26) 0

27) 0 or 0%

28) 1 or 100%

29) ½ or 50% or 0.5

30) The probability of an

event equals the # of

favorable outcomes

divided by the total

# of outcomes

31) 0 or 0%

32) ½ or 50%

33) 1 or 100%

34) ½ or 50%

35) 3/5

36) 27.1%

37) 0.59

38) 0. 6

39) No tricks or biases

40) False

41) False

42) The outcome of one

event doesn't affect

the other

43) Multiplication

44) 1/6

45) 1/4

46) 1/8

47) 1/12

48) Drawn items are put

back after each draw

49) Drawn items are not

put back.

50) 42/169

51) 110/342 or 55/171

52) 3360

53) 140

END-OF-BOOK SELF-TEST

1) 21/27

2) 29/35

3) False

4) False

5) True

6) True

7) -2/11

8) -3/14


141

9) 9/49

10) 1

11) 157

12) 37

13) A whole number:

Positive, negative,

or 0

14) -8

15) 9

16) 22

17) -22

18) 3

19) -6

20) -9

21) -10

22) -4

23) -56

24) 26

25) -5

26) 2

27) 49

28) Undefined

29) 1024

30) 13

31) 14

32) 1, 2, 4, 7, 8, 14, 28, 56

33) 1, 29

34) No. It has factors

other than 1 & itself.

35) 0

36) Undefined

37) 144

38) 299

39) Addition, Multipli-

cation

40) 21

41) -1

42) 9, 18, 27, 36, 45, 54,

63, 72, 81, 90

43) 56

11

44) Odd

45) Odd

46) >

47) -140

48) Yes. It has a factor

besides 1 & itself.

49) 7,053,017

50) 40,000

51) 986,700

52) ±2

53) 3024.18

54) <


142

55) >

56) 87/10000

57) 0.2727

58) 0.26

59) 0. 6

60) False

61) True

62) <

63) >

64) 371,940

65) 0.000234

66) 780

67) 61.2350

68) 7.89 × 109

69) 0.0000002123

70) 11/50

71) 2.09

72) 12.34%

73) 61.9%

74) 36%

75) 0.00005

76) 11.6% decrease

77) $34.99

78) $1063.34

79) $296.65

80) 35.7%

81) 13/-5

82) 26/3

83) It is 1000 times the

mass

84) 55¢

85) 2.75 m

86) 8

87) 2/63

88) 131

89) 36

90) 4.5

91) 1 and 2

92) 1

93) 44

94) 2/6 or 1/3

95) 0 or 0%

96) ½

97) 17.9%

98) 1/6

99) 28/121

100) 14/39

101) 120

102) -29/1

103) 12

104) 7

105) 4/21

143

About the Author

Larry Zafran was born and raised in

Queens, NY where he tutored and

taught math in public and private

schools. He has a Bachelors Degree in

Computer Science from Queens

College where he graduated with

highest honors, and has earned most of the credits

toward a Masters Degree in Secondary Math Education.

He is a dedicated student of the piano, and the leader of

a large and active group of board game players which

focuses on abstract strategy games from Europe.

He presently lives in Cary, NC where he works as an

independent math tutor, writer, and webmaster.

Companion Website

for More Help

For free support related to this or any of the author's

math books, please visit the companion website below.

www.MathWithLarry.com

http://www.mathwithlarry.com/

math made a bit easier workbook: practice exercises, self-tests, and review

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