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Title: Pre-Algebra 365pg
Description: This will help you in pre-algebra and math in general. This PDF is aimed at people who want to learn or review pre-algebra/math.
Description: This will help you in pre-algebra and math in general. This PDF is aimed at people who want to learn or review pre-algebra/math.
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Basic Math &
Pre-Algebra
FOR
DUMmIES
by Mark Zegarelli
‰
Basic Math &
Pre-Algebra
FOR
DUMmIES
by Mark Zegarelli
‰
Basic Math & Pre-Algebra For Dummies®
Published by
Wiley Publishing, Inc
...
Hoboken, NJ 07030-5774
www
...
com
Copyright © 2007 by Wiley Publishing, Inc
...
, Indianapolis, Indiana
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or
by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written
permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the
Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600
...
, 10475 Crosspoint Blvd
...
wiley
...
Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for the
Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies
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and/or its affiliates in the United
States and other countries, and may not be used without written permission
...
Wiley Publishing, Inc
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LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE
CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT
LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE
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INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY
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Wiley also publishes its books in a variety of electronic formats
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Library of Congress Control Number: 2007931551
ISBN: 978-0-470-13537-2
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 1
About the Author
Mark Zegarelli is the author of Logic For Dummies (Wiley)
...
He has earned his living for
many years writing vast quantities of logic puzzles, a hefty chunk of software
documentation, and the occasional book or film review
...
He likes writing best, though
...
Dedication
I dedicate this book to the memory of my mother, Sally Ann Zegarelli
(Joan Bernice Hanley)
...
Many
thanks for the editorial guidance and wisdom of Lindsay Lefevere, Natalie
Harris, Danielle Voirol, and Sarah Faulkner of Wiley Publications
...
Beardsley of Wanamassa School and Mr
...
Thanks to Martin Gardner and his
“Mathematical Games” column in Scientific American for showing me what
amazing things numbers really are
...
And thanks to all my professors at Rutgers
University, especially Holly Carley, Zheng-Chao Han, Richard Lyons, and
David Nacin for their support and encouragement
...
And thanks to Maxfield’s House of Caffeine in San Francisco for brewing the
coffee
...
dummies
...
Some of the people who helped bring this book to market include the following:
Acquisitions, Editorial, and
Media Development
Project Editor: Natalie Faye Harris
Acquisitions Editor: Lindsay Lefevere
Composition Services
Project Coordinator: Heather Kolter,
Kristie Rees
Technical Editor: Carol Spilker
Layout and Graphics: Carrie A
...
Jumper,
Christine Williams
Editorial Manager: Christine Meloy Beck
Anniversary Logo Design: Richard Pacifico
Editorial Assistants: Leeann Harney,
David Lutton, Erin Calligan Mooney,
Joe Niesen
Proofreaders: Laura L
...
the5thwave
...
Cocks, Product Development Director, Consumer Dummies
Michael Spring, Vice President and Publisher, Travel
Kelly Regan, Editorial Director, Travel
Publishing for Technology Dummies
Andy Cummings, Vice President and Publisher, Dummies Technology/General User
Composition Services
Gerry Fahey, Vice President of Production Services
Debbie Stailey, Director of Composition Services
Contents at a Glance
Introduction
...
9
Chapter 1: Playing the Numbers Game
...
29
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
...
55
Chapter 4: Putting the Big Four Operations to Work
...
71
Chapter 6: Say What? Turning Words into Numbers
...
97
Chapter 8: Fabulous Factors and Marvelous Multiples
...
119
Chapter 9: Fooling with Fractions
...
133
Chapter 11: Dallying with Decimals
...
179
Chapter 13: Word Problems with Fractions, Decimals, and Percents
...
205
Chapter 14: A Perfect Ten: Condensing Numbers with Scientific Notation
...
217
Chapter 16: Picture This: Basic Geometry
...
251
Chapter 18: Turning Geometry and Measurements into Word Problems
...
275
Chapter 20: Setting Things Up with Basic Set Theory
...
295
Chapter 21: Enter Mr
...
297
Chapter 22: Unmasking Mr
...
315
Chapter 23: Putting Mr
...
329
Part VI: The Part of Tens
...
341
Chapter 25: Ten Important Number Sets You Should Know
...
355
Table of Contents
Introduction
...
1
Conventions Used in This Book
...
2
Foolish Assumptions
...
3
Part I: Arming Yourself with the Basics of Basic Math
...
4
Part III: Parts of the Whole: Fractions, Decimals, and Percents
...
5
Part V: The X-Files: Introduction to Algebra
...
6
Icons Used in This Book
...
7
Part I: Arming Yourself with the Basics of Basic Math
...
11
Inventing Numbers
...
12
Evening the odds
...
13
Getting square with square numbers
...
15
Stepping out of the box with prime numbers
...
17
Looking at the Number Line
...
18
Getting a handle on nothing, or zero
...
21
Multiplying the possibilities
...
23
Discovering the space in between: Fractions
...
25
Counting on the counting numbers
...
26
Staying rational
...
27
viii
Basic Math & Pre-Algebra For Dummies
Chapter 2: It’s All in the Fingers — Numbers and Digits
...
30
Counting to ten and beyond
...
31
Reading long numbers
...
33
Rounding numbers
...
34
Chapter 3: The Big Four: Addition, Subtraction,
Multiplication, and Division
...
37
In line: Adding larger numbers in columns
...
38
Take It Away: Subtracting
...
40
Can you spare a ten? Borrowing to subtract
...
44
Signs of the times
...
45
Double digits: Multiplying larger numbers
...
51
Making short work of long division
...
54
Part II: Getting a Handle on Whole Numbers
...
57
Knowing Properties of the Big Four Operations
...
58
Commutative operations
...
60
Distributing to lighten the load
...
61
Addition and subtraction with negative numbers
...
64
Understanding Units
...
65
Multiplying and dividing units
...
66
Doesn’t equal (≠)
...
67
Approximately equals (≈)
...
68
Understanding exponents
...
69
Figuring out absolute value
...
71
The Three E’s of Math: Equations, Expressions, and Evaluations
...
72
Hey, it’s just an expression
...
73
Putting the Three E’s together
...
74
Applying order of operationsto Big Four expressions
...
78
Understanding order of precedence in
expressions with parentheses
...
83
Dispelling Two Myths about Word Problems
...
84
Word problems are useful
...
85
Turning word problems into word equations
...
88
Solving More-Complex Word Problems
...
90
Too much information
...
93
Chapter 7: Divisibility
...
97
Counting everyone in: Numbers you can divide everything by
...
98
Add it up: Checking divisibility by adding up digits
...
102
Chapter 8: Fabulous Factors and Marvelous Multiples
...
105
Connecting Factors and Multiples
...
107
Deciding when one number is a factor of another
...
108
Prime factors
...
114
ix
x
Basic Math & Pre-Algebra For Dummies
Marvelous Multiples
...
116
Finding the least common multiple (LCM)
...
119
Chapter 9: Fooling with Fractions
...
122
Knowing the Fraction Facts of Life
...
123
Flipping for reciprocals
...
124
Mixing things up
...
125
Increasing and Reducing Terms of Fractions
...
126
Reducing fractions to lowest terms
...
129
Knowing the parts of a mixed number
...
130
Converting an improper fraction to a mixed number
...
131
Chapter 10: Parting Ways: Fractions and the
Big Four Operations
...
133
Multiplying numerators and denominators straight across
...
136
All Together Now: Adding Fractions
...
137
Adding fractions with different denominators
...
143
Subtracting fractions with the same denominator
...
144
Working Properly with Mixed Numbers
...
147
Adding and subtracting mixed numbers
...
155
Basic Decimal Stuff
...
156
Place value of decimals
...
159
Table of Contents
Performing the Big Four with Decimals
...
164
Subtracting decimals
...
166
Dividing decimals
...
171
Making simple conversions
...
172
Changing fractions to decimals
...
179
Making Sense of Percents
...
180
Converting to and from Percents, Decimals, and Fractions
...
181
Changing decimals into percents
...
182
Turning fractions into percents
...
183
Figuring out simple percent problems
...
185
Deciphering more-difficult percent problems
...
187
Identifying the three types of percent problems
...
188
Chapter 13: Word Problems with Fractions, Decimals,
and Percents
...
193
Sharing a pizza: Fractions
...
194
Splitting the vote: Percents
...
196
Renegade grocery shopping: Buying less than they tell you to
...
197
Multiplying Decimals and Percents in Word Problems
...
198
Finding out how much you started with
...
201
Raking in the dough: Finding salary increases
...
203
Getting a deal: Calculating discounts
...
205
Chapter 14: A Perfect Ten: Condensing Numbers with
Scientific Notation
...
208
Counting up zeros and writing exponents
...
210
Working with Scientific Notation
...
211
Why scientific notation works
...
213
Multiplying with scientific notation
...
217
Examining Differences between the English and Metric Systems
...
218
Looking at the metric system
...
222
Estimating in the metric system
...
225
Chapter 16: Picture This: Basic Geometry
...
230
Making some points
...
230
Figuring the angles
...
232
Closed Encounters: Shaping Up Your Understanding of 2-D Shapes
...
233
Polygons
...
236
The many faces of polyhedrons
...
238
Measuring Shapes: Perimeter, Area, Surface Area, and Volume
...
239
Spacing out: Measuring in three dimensions
...
251
Looking at Three Important Graph Styles
...
252
Pie chart
...
254
Cartesian Coordinates
...
256
Drawing lines on a Cartesian graph
...
259
Chapter 18: Solving Geometry and Measurement
Word Problems
...
261
Setting up a short chain
...
263
Pulling equations out of the text
...
265
Solving Geometry Word Problems
...
267
Breaking out those sketching skills
...
272
Chapter 19: Figuring Your Chances: Statistics and Probability
...
275
Understanding differences between
qualitative and quantitative data
...
277
Working with quantitative data
...
282
Figuring the probability
...
284
Chapter 20: Setting Things Up with Basic Set Theory
...
287
Elementary, my dear: Considering what’s inside sets
...
291
Operations on Sets
...
292
Intersection: Elements in common
...
293
Complement: Feeling left out
...
295
Chapter 21: Enter Mr
...
297
X Marks the Spot
...
298
Evaluating algebraic expressions
...
301
Making the commute: Rearranging your terms
...
304
Identifying similar terms
...
305
Simplifying Algebraic Expressions
...
309
Removing parentheses from an algebraic expression
...
X: Algebraic Equations
...
316
Using x in equations
...
317
The Balancing Act: Solving for X
...
319
Using the balance scale to isolate x
...
321
Rearranging terms on one side of an equation
...
322
Removing parentheses from equations
...
326
Chapter 23: Putting Mr
...
329
Solving Algebra Word Problems in Five Steps
...
330
Setting up the equation
...
332
Answering the question
...
333
Choosing Your Variable Wisely
...
334
Charting four people
...
336
Table of Contents
Part VI: The Part of Tens
...
341
Getting Set with Sets
...
342
Zero: Much Ado about Nothing
...
343
On the Level: Equal Signs and Equations
...
344
In and Out: Relying on Functions
...
345
The Real Number Line
...
346
Chapter 25: Ten Important Number Sets You Should Know
...
348
Identifying Integers
...
349
Making Sense of Irrational Numbers
...
350
Moving through Transcendental Numbers
...
351
Trying to Imagine Imaginary Numbers
...
352
Going beyond the Infinite with Transfinite Numbers
...
355
xv
xvi
Basic Math & Pre-Algebra For Dummies
Introduction
O
nce upon a time, you loved numbers
...
Once upon a time, you really did love numbers
...
You sat next to them
on the couch and recited the numbers from 1 to 10
...
Or maybe you were
5 and discovering how to write numbers, trying hard not to get your bs and
as backwards
...
Numbers were fun
...
Or sorting out how to change fractions to decimals
...
Why do people often enter preschool excited about learning how to count
and leave high school as young adults convinced that they can’t do math?
The answer to this question would probably take 20 books this size, but
solving the problem can begin right here
...
Remember, just for a moment, an
innocent time — a time before math inspired panic attacks or, at best, induced
irresistible drowsiness
...
About This Book
Somewhere along the road from counting to algebra, most people experience
the Great Math Breakdown
...
Stranded on the interstate, you may feel frustrated by circumstances and betrayed by your vehicle, but for the guy holding the toolbox, it’s all in a day’s work
...
2
Basic Math & Pre-Algebra For Dummies
Not only does this book help you with the basics of math, but it also helps you
get past any aversion you may feel toward math in general! I’ve broken down
the concepts into easy-to-understand sections
...
So feel free
to jump around
...
Here are two pieces of advice I give all the time — remember them as you
work your way through the concepts in this book:
ߜ Take frequent study breaks
...
Then feed the cat, do the dishes, take a walk, juggle tennis balls,
try on last year’s Halloween costume — do something to distract yourself
for a few minutes
...
ߜ After you’ve read through an example and think you understand it,
copy the problem, close the book, and try to work it through
...
(Remember
that on any tests you’re preparing for, peeking is probably not allowed!)
Conventions Used in This Book
To help you navigate your way through this book, I use the following
conventions:
ߜ Italicized text highlights new words and defined terms
...
ߜ Monofont text highlights Web addresses
...
What You’re Not to Read
Although every author secretly (or not-so-secretly) believes that each word
he pens is pure gold, you don’t have to read every word in this book unless
you really want to
...
Paragraphs labeled with the Technical Stuff icon are also nonessential
...
So to find out whether you’re ready for this book, take
this simple test:
5 + 6 = __
10 – 7 = __
3 × 5 = __
20 ÷ 4 = __
If you can answer these four questions, you’re ready to begin
...
Part I: Arming Yourself with
the Basics of Basic Math
In Part I, I take what you already know about math and put it in perspective
...
I discuss how number sequences arise
...
I also show you how to use
the number line to perform basic arithmetic
...
I show you how the number
system you use every day — the Hindu-Arabic number system (also called
decimal numbers) — uses place value based on the number 10 to build digits
into numbers
...
I give you a refresher on how to do
column addition with carrying, subtraction with borrowing, multiplication of
large numbers, and the ever-dreaded long division
...
In Chapter 4, I cover inverse operations, the
commutative, associative, and distributive properties, and working with negative numbers
...
I also introduce you to more-advanced operations, such as
powers (exponents), square roots, and absolute value
...
The rest of the chapter focuses on an all-important skill: evaluating mathematical expressions using the order of operations
...
Chapter 7 takes a detailed look at divisibility
...
I also discuss prime
numbers and composite numbers
...
I show you
how to decompose a number to its prime factors
...
Part III: Parts of the Whole: Fractions,
Decimals, and Percents
Part III focuses on how mathematics represents parts of the whole as fractions,
decimals, and percents and on how these three ideas are connected
...
From there, I show you how to multiply and divide fractions,
Introduction
plus a variety of ways to add and subtract fractions
...
In Chapter 11, the topic is decimals
...
I also give you an understanding of repeating decimals
...
I show you how to convert percentages
to both fractions and decimals, and vice versa
...
Finally, Chapter 13 focuses on solving word problems that
involve fractions, decimals, and percents
...
In Chapter 14, I show you how scientific notation makes very large and very
small numbers more manageable by combining decimals and powers of ten
...
I give you a variety of conversion equations and
show you how to convert units of measurement
...
Chapter 16 discusses geometry, giving you a variety of formulas to find the
perimeter and area of basic shapes and the surface area and volume of a few
important solids
...
I also give
you the basics of the most common graphing method used in math, the
Cartesian graph
...
Chapter 18 gives you still more practice solving word problems, especially those focusing on geometry and on weights and measures
...
You discover the difference between qualitative data and quantitative data and how to calculate
both the mean average and the median average
...
In Chapter 20, I give you the basics of set theory, including how to define a
set, identify elements and subsets, and understand the empty set
...
5
6
Basic Math & Pre-Algebra For Dummies
Part V: The X-Files: Introduction to Algebra
Part V is your introduction to algebra
...
Chapter 22 gives you a variety of ways to solve algebraic equations
...
Part VI: The Part of Tens
Just for fun, this part of the book includes a few top-ten lists on a variety of
topics, including key math concepts and number sets
...
Make sure you understand before reading on! Remember this info even after you close the book
...
Try them out, especially if you’re taking a math course
...
Get clear about where
these little traps are hiding so you don’t fall in
...
Introduction
Where to Go from Here
You can use this book in a few ways
...
The advantage to
this method is that you realize how much math you do know — the first few
chapters go very quickly
...
Or how about this: When you’re ready to work, read up on the topic you’re
studying
...
You’d be surprised how
a little refresher on simple stuff can suddenly cause more-advanced concepts
to click
...
Wherever you open the book, you can find a clear
explanation of the topic at hand, as well as a variety of hints and tricks
...
Here’s a short list of topics that tend to back students up:
ߜ Negative numbers (Chapter 4)
ߜ Order of operations (Chapter 5)
ߜ Word problems (Chapters 6, 13, 18, and 23)
ߜ Factoring numbers (Chapter 8)
ߜ Fractions (Chapters 9 and 10)
Most of these topics are in Parts II and III, but they’re foundational to what’s covered later in the book
...
As soon as you feel comfortable adding negative numbers or multiplying fractions, your confidence soars
...
If you get stuck along the way, take a break and come back to the problem; you
may find that the answer suddenly clicks for you as soon as you read it again
with a refreshed mind
...
Sometimes, working through a few easier examples is the best way to prepare for when the
going gets tough
...
ou already know more about math than you think
you know
...
I also reintroduce you to what I call the Big
Four operations (adding, subtracting, multiplying, and
dividing)
...
(This fact,
however, probably won’t get you out of having to know about them —
nice try!)
For example, you can picture three of anything: three cats, three baseballs,
three cannibals, three planets
...
Oh, sure, you can picture the numeral
3, but the threeness itself — much like love or beauty or honor — is beyond
direct understanding
...
In this chapter, I give you a brief history of how numbers came into being
...
After that, I describe how some of these ideas come together with a simple
yet powerful tool – the number line
...
Finally, I describe how the counting numbers (1, 2, 3,
...
I also show you how these sets of numbers are nested — that
is, how one set of numbers fits inside another, which fits inside another
...
Before that, people in prehistoric,
hunter-gatherer societies were pretty much content to identify bunches of
things as “a lot” or “a little
...
So people began using stones, clay tokens, and similar
objects to keep track of their goats, sheep, oil, grain, or whatever commodity
they had
...
Eventually, traders realized that they could draw pictures instead of using
tokens
...
Whether they realized it or not, their attempts to keep track of commodities had led these early humans to invent something entirely new: numbers
...
Although Roman numerals gained wide currency as the Roman Empire
expanded throughout Europe and parts of Asia and Africa, the more advanced
system that the Arabs invented turned out to be more useful
...
Understanding Number Sequences
Although numbers were invented for counting commodities, as I explain in
the preceding section, they were soon put to a wide range of applications
...
But beyond their many uses for understanding the external world, numbers
also have an internal order all their own
...
One path into this new and often strange world is the number sequence: an
arrangement of numbers according to a rule
...
Chapter 1: Playing the Numbers Game
Evening the odds
One of the first things you probably heard about numbers is that all of them
are either even or odd
...
But when you try to divide an odd number of
marbles the same way, you always have one odd, leftover marble
...
You can easily keep the sequence of even numbers going as long as you like
...
Similarly, here are the first few odd numbers:
1
3
5
7
9
11
13
15
...
Starting with the
number 1, keep adding 2 to get the next number
...
Counting by threes, fours, fives, and so on
After you get used to the concept of counting by numbers greater than one,
you can run with it
...
This time, the pattern is generated by starting with 3 and continuing to add 3
...
And here’s how to count by fives:
5
10
15
20
25
30
35
40
...
(In general, people seem to have the most trouble multiplying by 7, but 8
and 9 are also unpopular
...
These types of sequences are also useful for understanding factors and multiples, which you get a look at in Chapter 8
...
(Later in this book, I show you
how one picture can be worth a thousand numbers when I discuss geometry
in Chapter 16 and graphing in Chapter 17
...
(You probably have a box sitting somewhere in the pantry
...
) Shake a bunch
out of a box and place the little squares together to make bigger squares
...
1
2
3
4
7
8
9 10
1
Figure 1-1:
Square
numbers
...
You get a square number by multiplying a number by itself, so knowing the
square numbers is another handy way to remember part of the multiplication
table
...
Knowing the
square numbers gives you another way to etch that multiplication table forever into your brain, as I show you in Chapter 3
...
Chapter 1: Playing the Numbers Game
Composing yourself with composite numbers
Some numbers can be placed in rectangular patterns
...
For example, 12 is a composite number
because you can place 12 objects in rectangles of two different shapes, as
shown in Figure 1-2
...
As with square numbers, arranging numbers in visual patterns like this tells
you something about how multiplication works
...
Figure 1-3:
Composite
numbers,
such as 8
and 15,
can form
rectangles
...
And these visual patterns show this:
2×4=8
3 × 5 = 15
15
16
Part I: Arming Yourself with the Basics of Basic Math
The word composite means that these numbers are composed of smaller numbers
...
Here are all the composite
numbers between 1 and 16:
4
6
8
9
10
12
14
15
16
Notice that all the square numbers (see “Getting square with square numbers”) also count as composite numbers because you can arrange them in
boxes with at least two rows and two columns
...
Stepping out of the box with prime numbers
Some numbers are stubborn
...
Look at how the number 13 is depicted in Figure 1-4, for example
...
Try as you may, you just can’t make a rectangle out of 13 objects
...
) Here are all
the prime numbers less than 20:
2
3
5
7
11
13
17
19
As you can see, the list of prime numbers fills the gaps left by the composite
numbers (see the preceding section)
...
The only exception is the number 1, which is neither prime nor composite
...
Chapter 1: Playing the Numbers Game
Multiplying quickly with exponents
Here’s an old question that still causes surprises: Suppose you took a job that
paid you just 1 penny the first day, 2 pennies the second day, 4 pennies the
third day, and so on, doubling the amount every day, like this:
1
2
4
8
16
32
64
128
256
512
...
23 — but who’s counting?)
...
” At first glance, this looks like a good answer, but
here’s a glimpse at your second ten days’ earnings:
...
65,536
By the end of the second 10 days, your total earnings would be over $10,000
...
Each new number in the sequence is obtained by multiplying
the previous number by 2:
21 = 2 = 2
22 = 2 × 2 = 4
23 = 2 × 2 × 2 = 8
24 = 2 × 2 × 2 × 2 = 16
As you can see, the notation 24 means multiply 2 by itself 4 times
...
Here’s another sequence
you may be familiar with:
1
10
100
1,000
10,000
100,000
1,000,000
...
You can also generate these numbers using exponents:
101 = 10 = 10
102 = 10 × 10 = 100
103 = 10 × 10 × 10 = 1,000
104 = 10 × 10 × 10 × 10 = 10,000
17
18
Part I: Arming Yourself with the Basics of Basic Math
This sequence is important for defining place value, the basis of the decimal
number system, which I discuss in Chapter 2
...
You find out
more about exponents in Chapter 5
...
Figure 1-5:
Basic
number line
...
People often see their
first number line — usually made of brightly colored construction paper —
pasted above the blackboard in school
...
You can use it to show how numbers get bigger in one direction
and smaller in the other
...
Adding and subtracting on the number line
You can use the number line to demonstrate simple addition and subtraction
...
Here’s
the main thing to remember:
ߜ As you go right, the numbers go up, which is addition (+)
...
For example, 2 + 3 means you start at 2 and jump up 3 spaces to 5, as illustrated in Figure 1-6
...
1
2
3
4
5
6
7
8
9
10
As another example, 6 – 4 means start at 6 and jump down 4 spaces to 2
...
See Figure 1-7
...
1
2
3
4
5
6
7
8
9
10
You can use these simple up and down rules repeatedly to solve a longer
string of added and subtracted numbers
...
In this case, the number line
would show you that 3 + 1 – 2 + 4 – 3 – 2 = 1
...
Getting a handle on nothing, or zero
An important addition to the number line is the number 0, which means
nothing, zilch, nada
...
For one thing — as more than one philosopher has pointed out — by
definition, nothing doesn’t exist! Yet, we routinely label it with the number 0,
as shown in Figure 1-8
...
It’s called the empty set, which is sort of the mathematical version of a
box containing nothing
...
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Part I: Arming Yourself with the Basics of Basic Math
Figure 1-8:
The number
line starting
at 0 and
continuing
with 1, 2,
3,
...
0
1
2
3
4
5
6
7
8
9
10
Nothing sure is a heavy trip to lay on little kids, but they don’t seem to mind
...
That is, 3 – 3 =
0
...
Figure 1-9:
Starting at 3
and moving
down three
...
Infinity: Imagining a never-ending story
The arrows at the ends of the number line point
onward to a place called infinity, which isn’t
really a place at all, just the idea of foreverness,
because the numbers go on forever
...
The wacky symbol ∞ represents infinity
...
Because ∞ isn’t a number, you can’t technically
add the number 1 to it, any more than you can
add the number 1 to a cup of coffee or your Aunt
Louise
...
Chapter 1: Playing the Numbers Game
Taking a negative turn: Negative numbers
When people first find out about subtraction, they often hear that you can’t
take away more than you have
...
It isn’t long, though, before you find out what any credit card holder knows
only too well: You can, indeed, take away more than you have — the result is
a negative number
...
That is, 4 – 7 = –3
...
Figure 1-10 shows
how you place negative whole numbers on the number line
...
−5 −4 −3 −2 −1
0
1
2
3
4
5
Adding and subtracting on the number line works pretty much the same with
negative numbers as with positive numbers
...
Figure 1-11:
Subtracting
4 – 7 on the
number line
...
Placing 0 and the negative counting numbers on the number line expands the
set of counting numbers to the set of integers
...
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Part I: Arming Yourself with the Basics of Basic Math
Multiplying the possibilities
Suppose you start at 0 and circle every other number on a number line, as
shown in Figure 1-12
...
In
other words, you’ve circled all the multiples of two
...
) You can now use this number line to multiply
any number by two
...
Just
start at 0 and jump 5 circled spaces to the right
...
0
1
2
3
4
5
6
7
8
9
10
This number line shows you that 5 × 2 = 10
...
Figure 1-13 shows you that –3 × 2 = –6
...
(I talk about multiplying by
negative numbers in Chapter 4
...
−6 −5 −4 −3 −2 −1
0
1
2
3
4
Multiplying on the number line works no matter what number you count off
by
...
Figure 1-14:
Number line
counted off
by 5s
...
For example, Figure 1-15 shows
how to multiply 2 × 5
...
0
5
10 15 20 25 30 35 40 45 50
So 2 × 5 = 10, the same result as when you multiply 5 × 2
...
(I discuss
the commutative property in Chapter 4
...
For example, suppose you want
to divide 6 by some other number
...
Figure 1-16:
Number line
from 0 to 6
...
This split (or division) occurs at 3, showing
you that 6 ÷ 2 = 3
...
0
1
2
3
4
5
6
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Part I: Arming Yourself with the Basics of Basic Math
Similarly, to divide 6 ÷ 3, split the same number line into three equal parts, as
in Figure 1-18
...
This
number line shows you that 6 ÷ 3 = 2
...
0
1
2
3
4
5
6
But suppose you want to use the number line to divide a small number by a
larger number
...
Following the method I show you earlier, first draw a number line from 0 to 3
...
Unfortunately, none of these splits has
landed on a number
...
You just have to add some new
numbers to the number line, as you can see in Figure 1-19
...
0
¾1
1½
2 2º
3
Welcome to the world of fractions
...
This image tells you that 3 ÷ 4 = 3⁄4
...
Division
and fractions are closely related
...
(I explain the connection between division and fractions in more detail in Chapters 9 and 10
...
For example, Figure 1-20 shows a close-up of
a number line from 0 to 1
...
0
¼
½
¾
1
This number line may remind you of a ruler or a tape measure, with lots of tiny
fractions filled in
...
The addition of fractions to the number line expands the set of integers to
the set of rational numbers
...
In fact, no matter how small things get in the real world, you can always find a
tiny fraction to approximate it as closely as you need
...
Mathematicians
call this trait the density of fractions on the real number line, and this type of
density is a topic in a very advanced area of math called real analysis
...
In
this section, I provide a quick tour of how numbers fit together as a set of
nested systems, one inside the other
...
You can use the number line to deal with four important sets
of numbers:
ߜ Counting numbers (also called natural numbers): The set of numbers
beginning 1, 2, 3, 4
...
This nesting of one set inside another is similar to the way
that a city (for example, Boston) is inside a state (Massachusetts), which is
inside a country (the United States), which is inside a continent (North
America)
...
Counting on the counting numbers
The set of counting numbers is the set of numbers you first count with, starting with 1
...
The counting numbers are infinite, which means they go on forever
...
Similarly, when you multiply two counting numbers, the answer is
always a counting number
...
Introducing integers
The set of integers arises when you try to subtract a larger number from a
smaller one
...
The set of integers includes the following:
ߜ The counting numbers
ߜ Zero
ߜ The negative counting numbers
Here’s a partial list of the integers:
...
Like the counting numbers, the integers are closed under addition and multiplication
...
That is, the integers are also closed under subtraction
...
Furthermore, when you divide one rational number by
another, the answer is always a rational number
...
Getting real
Even if you filled in all the rational numbers, you’d still have points left unlabeled on the number line
...
An irrational number is a number that’s neither a whole number nor a fraction
...
In other words, no matter how many decimal places you write down, you
can always write down more; furthermore, the digits in this decimal never
become repetitive or fall into any pattern
...
)
The most famous irrational number is π (you find out more about π when I
discuss the geometry of circles in Chapter 17
...
14159265358979323846264338327950288419716939937510
...
In this book, I don’t spend
too much time on irrational numbers, but just remember that they’re there
for future reference
...
The fact that our ten fingers are matched up
so nicely with numbers may seem like a happy accident
...
Fingers were the first calculator that humans possessed
...
In fact, the very
word digit has two meanings: numerical symbol and finger
...
I also
show you when 0 is an important placeholder in a number and why leading
zeros don’t change the value of a number
...
After that, I discuss two important skills: rounding numbers and estimating values
...
30
Part I: Arming Yourself with the Basics of Basic Math
Telling the difference between numbers and digits
Sometimes, people confuse numbers and digits
...
ߜ A number is a string of one or more digits
...
In
fact, it’s a one-digit number
...
And 426 is a three-digit number
...
In a sense, a digit is like a letter of the alphabet
...
(How much can you do with a
single letter such as K or W?) Only when you
begin using strings of letters as building blocks
to spell words does the power of letters become
apparent
...
Knowing Your Place Value
The number system you’re most familiar with — Hindu-Arabic numbers —
has ten familiar digits:
0
1
2
3
4
5
6
7
8
9
Yet with only ten digits, you can express numbers as high as you care to go
...
Counting to ten and beyond
The ten digits in our number system allow you to count from 0 to 9
...
Place value assigns a digit a greater
or lesser value depending on where it appears in a number
...
To understand how a whole number gets its value, suppose you write the
number 45,019 all the way to the right in Table 2-1, one digit per cell, and add
up the numbers you get
...
The
chart shows you that the number breaks down as follows:
45,019 = 40,000 + 5,000 + 0 + 10 + 9
In this example, notice that the presence of the digit 0 in the hundreds place
means that zero hundreds are added to the number
...
For example, the number
5,001,000 breaks down into 5,000,000 + 1,000
...
Table 2-2 shows what you’d get
...
Clearly, this answer is wrong!
As a rule, when a 0 appears to the right of at least one digit other than 0, it’s a
placeholder
...
However, when a 0 appears to the left of every digit
other than 0, it’s a leading zero
...
For example, place the number 003,040,070
on the chart (see Table 2-3)
...
You can drop these 0s from the number, leaving you with 3,040,070
...
Reading long numbers
When you write a long number, you use commas to separate periods
...
They help make long numbers more
readable
...
Table 2-4
A Place-Value Chart Separated into Periods
Quintillions
Quadrillions
Trillions
Billions
Millions
Thousands
Ones
234
845
021
349
230
467
304
This version of the chart helps you read the number
...
”
When you read and write whole numbers, don’t say the word and
...
That’s why when you write a
check, you save the word and for the number of cents, which is expressed as
a decimal or fraction
...
)
Chapter 2: It’s All in the Fingers — Numbers and Digits
Close Enough for Rock ’n’ Roll:
Rounding and Estimating
As numbers get longer, calculations become tedious, and you’re more likely
to make a mistake or just give up
...
At other times, you may want to round off an answer after you
do your calculations
...
Rounding numbers allows you to change a difficult number to an easier
one
...
In this section, you build both skills
...
Rounding numbers
Rounding numbers makes long numbers easier to work with
...
Rounding numbers to the nearest ten
The simplest kind of rounding you can do is with two-digit numbers
...
For example,
39 → 40
51 → 50
73 → 70
Even though numbers ending in 5 are in the middle, always round them up to
the next-highest number that ends in 0:
15 → 20
35 → 40
85 → 90
Numbers in upper 90s get rounded up to 100:
99 → 100
95 → 100
94 → 90
After you know how to round a two-digit number, you can round just about
any number
...
(This is a lot like when the odometer in your car rolls a bunch of 9s
over to 0s
...
Change all other digits to the right of these two digits to 0s
...
Focus
on the hundreds digit (6) and the digit to its immediate right (4):
642
I’ve underlined these two digits
...
Round the number by focusing only
on the two underlined digits and, when you’re done, change all digits to the
right of these to 0s:
4,984 → 5,000
78,521 → 79,000
1,099,304 → 1,099,000
Even when rounding to the nearest million, the same rules apply:
1,234,567 → 1,000,000
78,883,958 → 79,000,000
Estimating value to make problems easier
After you know how to round numbers, you can use this skill in estimating
values
...
When you get an approximate answer, you don’t use an equal sign; instead,
you use this wavy symbol, which means is approximately equal to: ≈
...
This computation is tedious, and you may make a mistake
...
This answer isn’t very far off from the exact
answer, which is 3,792
...
Again, this computation
doesn’t look like a lot of fun
...
Again, not a bad estimate
...
You’re most likely to arrive at a bad estimate when you
ߜ Round numbers that are close to the middle of the range
ߜ Round too many numbers in the same direction (either up or down)
ߜ Multiply or divide rounded numbers
For example, suppose you want to multiply 349 × 243
...
What happened? First, notice that 349 is very close to the
middle of the range between 300 and 400
...
So when you rounded these numbers, you
changed their values a lot
...
That’s why the estimate ended up so much lower than the real
answer
...
In general, multiplying and dividing throw off
estimates more than addition and subtraction do
...
I call these operations the Big Four all through the book
...
Although I assume you’re already familiar with the Big Four, this chapter
reviews these operations, taking you from what you may have missed to what
you need to succeed as you move onward and upward in math
...
It’s simple, friendly, and straightforward
...
Addition is all about bringing things together, which is a positive thing
...
I have $15 and you have only $5
...
Or instead, you and I can
38
Part I: Arming Yourself with the Basics of Basic Math
join forces, adding together my $15 and your $5 to make $20
...
Addition uses only one sign — the plus sign (+): Your equation may read
2 + 3 = 5 or 12 + 2 = 14 or 27 + 44 = 71, but the plus sign always means the
same thing
...
So in the first example, the addends are 2 and
3, and the sum is 5
...
(Chapter 1 has the scoop on digits and place value
...
Not surprisingly, this method is called column addition
...
First add the ones column:
55
31
+ 12
8
Next, move to the tens column:
55
31
+ 12
98
This problem shows you that 55 + 31 + 12 = 98
...
In
that case, you need to write down the ones digit of that number and carry
the tens digit over to the next column to the left — that is, write this digit
above the column so you can add it with the rest of the numbers in that
column
...
In the ones
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
column, 6 + 9 + 8 = 23, so write down the 3 and carry the 2 over to the top
of the tens column:
2
376
49
+ 18
3
Now continue by adding the tens column
...
Take It Away: Subtracting
Subtraction is usually the second operation you discover, and it’s not much
harder than addition
...
Suppose you and I have been
running on treadmills at the gym
...
You subtract and tell me that I
should be very impressed that you ran 7 miles farther than I did
...
You end
up with equations such as 4 – 1 = 3 and 14 – 13 = 1 and 93 – 74 = 19
...
This term makes sense when you think about it: When you subtract, you find
out the difference between a higher number and a lower one
...
But almost nobody ever remembers which is
which, so when I talk about subtraction, I prefer to say the first number and
the second number
...
In that case, the second number can’t
be larger than the first
...
For example, 3 – 3 = 0 and 11 – 11 = 0 and 1776 – 1776 = 0
...
When
you do, though, you need to place a minus sign in front of the difference to
show that you have a negative number, a number below 0:
4 – 5 = –1
10 – 13 = –3
88 – 99 = –11
When subtracting a larger number from a smaller number, remember the
words switch and negate
...
For example, to find 10 – 13, you switch the order of
these two numbers, giving you 13 – 10, which equals 3; then negate this result
to get –3
...
The minus sign gets double duty, so don’t get confused
...
But when you attach it to the front of a number, it means that this
number is a negative number
...
I also go into more detail on negative numbers and the Big Four operations
in Chapter 4
...
(For subtraction, however, don’t stack more than two numbers —
put the larger number on top and the smaller under it
...
To start out, stack the two numbers and begin
subtracting in the ones column: 6 – 4 = 2:
386
– 54
2
Next, move to the tens column and subtract 8 – 5 to get 3:
386
– 54
32
Finally, move to the hundreds column
...
Can you spare a ten? Borrowing
to subtract
Sometimes, the top digit in a column is smaller than the bottom digit in that
column
...
Borrowing is a two-step process:
1
...
Cross out the number you’re borrowing from, subtract 1, and write the
answer above the number you crossed out
...
Add 10 to the top number in the column you were working in
...
The first step is to subtract 4 from 6 in the ones column, which gives you 2:
386
– 94
2
41
42
Part I: Arming Yourself with the Basics of Basic Math
When you move to the tens column, however, you find that you need to subtract 8 – 9
...
First, cross out the 3 and replace it with a 2, because 3 – 1 = 2:
2
386
– 94
2
Next, place a 1 in front of the 8, changing it to an 18, because 8 + 10 = 18:
2
3 18 6
– 94
2
Now you can subtract in the tens column: 18 – 9 = 9:
2 18 6
– 94
92
The final step is simple: 2 – 0 = 2:
2 18 6
– 94
2 92
Therefore, 386 – 94 = 292
...
Suppose, for instance, you want to subtract 1,002 – 398
...
Because 2 is smaller than 8,
you need to borrow from the next column to the left
...
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
In this example, the column you need to borrow from is the thousands
column
...
Then place a 1 in front
of the 0 in the hundreds column:
0
1 10 0 2
–398
Now, cross out the 10 and replace it with a 9
...
Then place
a 1 in front of the 2:
0 9 9
1 10 10 12
–3 9 8
At last, you can begin subtracting in the ones column: 12 – 8 = 4:
0 9 9
1 10 10 12
–3 9 8
4
Then subtract in the tens column: 9 – 9 = 0:
0 9 9
1 10 10 12
–3 9 8
0 4
Then subtract in the hundreds column: 9 – 3 = 9:
0 9 9
1 10 10 12
–3 9 8
6 0 4
Because nothing is left in the thousands column, you don’t need to subtract
anything else
...
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Part I: Arming Yourself with the Basics of Basic Math
Multiplying
Multiplication is often described as a sort of shorthand for repeated addition
...
For example, suppose you coach a Little League baseball team and
you’ve just won a game against the toughest team in the league
...
To find out how many hot dogs you need, you could add 3 together 9 times
...
Therefore,
you need 27 hot dogs (plus a whole lot of mustard and sauerkraut)
...
In multiplication, the first number is also called the multiplicand and the
second number is the multiplier
...
Signs of the times
When you’re first introduced to multiplication, you use the times sign (×)
...
As you
move onwards and upwards on your math journey, you should be aware of
the conventions I discuss in the following sections
...
For example,
4⋅2=8
means 4 × 2 = 8
6 ⋅ 7 = 42
means 6 × 7 = 42
53 ⋅ 11 = 583
means 53 × 11 = 583
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
That’s all there is to it: Just use the ⋅ symbol anywhere you would’ve used the
standard times sign (×)
...
The parentheses can enclose the first number,
the second number, or both numbers
...
For example,
3 + (5) = 8
means 3 + 5 = 8
(8) – 7 = 1
means 8 – 7 = 1
(9) ⋅ (10) = 90 means 9 × 10 = 90
Note: In the third example, you don’t really need the ⋅, but it isn’t doing any
harm, either
...
That is,
you consider being called upon to remember 9 × 7 a tad less appealing than
being dropped from an airplane while clutching a parachute purchased from
the trunk of some guy’s car
...
Looking at the old multiplication table
One glance at the old multiplication table, Table 3-1, reveals the problem
...
” Well, in my humble opinion, the
multiplication table has too many numbers
...
Just looking at it
makes my eyes glaze over
...
Introducing the short multiplication table
If the multiplication table from Table 3-1 were smaller and a little more manageable, I’d like it a lot more
...
Table 3-2
The Short Multiplication Table
3
3
4
5
6
7
8
9
4
5
6
7
8
9
9
12
15
18
21
24
27
16
20
24
28
32
36
25
30
35
40
45
36
42
48
54
49
56
63
64
72
81
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
As you can see, I’ve gotten rid of a bunch of numbers
...
I’ve also shaded 11 of the numbers I’ve kept
...
If a
hammer’s too heavy to pick up, then you ought to buy a lighter one
...
Besides, I’ve removed only the numbers you don’t need
...
Here’s why:
ߜ Any number multiplied by 0 is 0 (people call this trait the zero property
of multiplication)
...
ߜ Multiplying by 2 is fairly easy; if you can count by 2s — 2, 4, 6, 8, 10, and
so forth — you can multiply by 2
...
(And not just redundant, but also repeated, extraneous, and unnecessary!) For example, any way
you slice it, 3 × 5 and 5 × 3 are both 15 (you can switch the order of the factors
because multiplication is commutative — see Chapter 4 for details)
...
So what’s left? Just the numbers you need
...
The gray row is the 5 times table, which you probably
know pretty well
...
)
The numbers on the gray diagonal are the square numbers
...
You probably know these numbers better than you think
...
To start out, make a set of flash cards that give a multiplication problem on the front and the answer on the back
...
47
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Part I: Arming Yourself with the Basics of Basic Math
7
×6
Figure 3-1:
Both sides
of a flash
card, with
7 × 6 on the
front and 42
on the back
...
Split these 28 into two piles — a “gray” pile with 11 cards and a
“white” pile with 17
...
) Then
begin:
1
...
If
you get the answer right, put that card on the bottom of the pile
...
2
...
3
...
Now take a break
...
Come
back later in the day and do the same thing
...
At this point, feel free to
make cards for the rest of the standard times table — you know, the cards
with all the 0, 1, 2 times tables on them and the redundant problems — mix
all 100 cards together, and amaze your family and friends
...
To multiply any one-digit number by 9,
1
...
For example, suppose you want to multiply
7 × 9
...
2
...
You’ve just written the answer you were
looking for
...
This trick works for every one-digit
number except 0 (but you already know that
0 × 9 = 0)
...
So 7 × 9 = 63
...
For example, suppose you want to multiply 53 × 7
...
Because 3 × 7 = 21, write down the 1 and
carry the 2:
2
53
× 7
1
Next, multiply 7 by 5
...
But you also need to add the 2
that you carried over, which makes the result 37
...
For example, suppose
you want to multiply 53 by 47
...
) Now you’re ready to
multiply by the 4 in 47
...
So to begin, put a 0 directly under the 1 in 371:
53
× 47
371
0
This 0 acts as a placeholder so that this row is arranged properly
...
)
When multiplying by larger numbers with two digits or more, use one placeholding zero when multiplying by the tens digit, two placeholding zeros when
multiplying the hundreds digit, three zeros when multiplying by the thousands
digit, and so forth
...
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
Doing Division Lickety-Split
The last of the Big Four operations is division
...
For example, suppose you’re a parent on a picnic with your
three children
...
Each child gets four pretzel sticks
...
So some other ways to write
the same information are
⁄3 = 4
12
and
12 = 4
3
Whichever way you write it, the idea is the same: When you divide 12 pretzel
sticks equally among three people, each person gets four of them
...
For example, in the
division from the earlier example, the dividend is 12, the divisor is 3, and the
quotient is 4
...
For one
thing, the multiplication table focuses on multiplying all the one-digit numbers by each other
...
Besides, you can use the multiplication table for
division, too, by reversing the way you normally
use the table
...
You can reverse this
equation to give you these two division problems:
42 ÷ 6 = 7
42 ÷ 7 = 6
Using the multiplication table in this way takes
advantage of the fact that multiplication and
division are inverse operations
...
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Part I: Arming Yourself with the Basics of Basic Math
Making short work of long division
In the olden days, knowing how to divide large numbers — for example,
62,997 ÷ 843 — was important
...
The process involved
dividing, multiplying, subtracting, and dropping numbers down
...
Having said that, I need to add that your teacher and math-crazy friends may
not agree
...
But if do you get stuck doing page after
page of long division against your will, you have my deepest sympathy
...
In this section, I give you a good
start with long division, and with luck, your instructor won’t hammer it home
with a vengeance!
One-digit divisors
Recall that the divisor in a division problem is the number that you’re dividing
by
...
So
I start with a nice, small, one-digit divisor
...
Start off by writing the problem like this:
5 g 860
Unlike the other Big Four operations, long division moves from left to right
...
To begin,
ask how many times 5 goes into 8 — that is, what’s 8 ÷ 5? The answer is 1
(with a little bit left over), so write 1 directly above the 8
...
(Note: After you subtract, the result should always be
smaller than the divisor
...
) Then bring down the 6 to make the new number 36:
1
5 g 860
-5
36
Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division
These steps are one complete cycle, and to complete the problem you just
need to repeat them
...
Write 7 just above the 6,
and then multiply 7 × 5 to get 35; write the answer under 36:
17
5 g 860
-5
36
35
Now subtract to get 36 – 35 = 1; bring down the 0 next to the 1 to make the
new number 10:
17
5 g 860
-5
36
- 35
10
Another cycle is complete, so begin the next cycle by asking how many times
5 goes into 10 — that is, 10 ÷ 5
...
Write down the 2 in
the answer above the 0
...
Because you have no more numbers to bring down,
you’re finished, and here’s the answer (that is, the quotient):
172 ! Quotient
5 g 860
-5
36
35
10
-10
0
So 860 ÷ 5 = 172
...
The following sections tell you
what to do when you run out of numbers to bring down, and Chapter 11
explains how to get a decimal answer
...
A remainder is simply a portion left over from
the division
...
For example, suppose you want to divide seven candy bars between two
people without breaking any candy bars into pieces (too messy)
...
This problem shows you the following:
7 ÷ 2 = 3 with a remainder of 1, or 3 r 1
In long division, the remainder is the number that’s left when you no longer
have numbers to bring down
...
introduce a whole host of new concepts related to the
Big Four operations (adding, subtracting, multiplying,
and dividing)
...
I tell you all about the three E’s of math —
expressions, equations, and evaluation
...
You discover how
to solve word problems (also called story problems) by
writing word equations that make sense of what you’re
reading
...
Chapter 4
Putting the Big Four
Operations to Work
In This Chapter
ᮣ Identifying which operations are inverses of each other
ᮣ Knowing the operations that are commutative, associative, and distributive
ᮣ Performing the Big Four operations on negative numbers
ᮣ Using four symbols for inequality
ᮣ Understanding exponents, roots, and absolute values
W
hen you understand the Big Four operations that I cover in Chapter 3 —
adding, subtracting, multiplying, and dividing — you can begin to look
at math on a whole new level
...
I begin by focusing on
four important properties of the Big Four operations: inverse operations,
commutative operations, associative operations, and distribution
...
I continue by introducing you to some important symbols for inequality
...
Knowing Properties of the
Big Four Operations
When you know how to do the Big Four operations — add, subtract, multiply,
and divide — you’re ready to grasp a few important properties of these important operations
...
58
Part II: Getting a Handle on Whole Numbers
In this section, I introduce you to four important ideas: inverse operations,
commutative operations, associative operations, and the distributive property
...
Inverse operations
Each of the Big Four operations has an inverse — an operation that undoes it
...
For example, here are two inverse equations:
1+2=3
3–2=1
In the first equation, you start with 1 and add 2 to it, which gives you 3
...
The main idea here is that you’re given a starting number — in
this case, 1 — and when you add a number and then subtract the same
number, you end up again with the starting number
...
Similarly, addition undoes subtraction — that is, if you subtract a number
and then add the same number, you end up where you started
...
In the second equation, you have 174 and add 10 to it,
which brings you back to 184
...
In the same way, multiplication and division are inverse operations
...
And
then you divide 20 by 5 to return to where you started at 4
...
Similarly,
30 ÷ 10 = 3
3 ⋅ 10 = 30
Chapter 4: Putting the Big Four Operations to Work
Here, you start with 30, divide by 10, and multiply by 10 to end up back at 30
...
Commutative operations
Addition and multiplication are both commutative operations
...
This property of addition and multiplication is called the commutative property
...
In each case, you end up with 8 books
...
In both cases, someone buys 14 flowers
...
When
you switch around the order of the numbers, the result changes
...
In the first case, you still have $2
left over
...
That is, switching the numbers
around turns the result into a negative number
...
)
And here’s an example of how division is noncommutative:
5÷2=2r1
but
2÷5=0r2
For example, when you have five dog biscuits to divide between two dogs,
each dog gets two biscuits and you have one biscuit left over
...
59
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Part II: Getting a Handle on Whole Numbers
Associative operations
Addition and multiplication are both associative operations, which means that
you can group them differently without changing the result
...
Here’s an
example of how addition is associative
...
You can solve this problem in two ways:
(3 + 6) + 2
3 + (6 + 2)
=9+2
=3+8
= 11
= 11
In the first case, I start by adding 3 + 6 and then add 2
...
Either way, the sum is 11
...
Suppose you
want to multiply 5 ⋅ 2 ⋅ 4
...
In the
second case, I start by multiplying 2 ⋅ 4 and then multiply by 5
...
In contrast, subtraction and division are nonassociative
operations
...
Don’t confuse the commutative property with the associative property
...
The associative property tells you that
it’s okay to regroup three numbers using parentheses
...
You’ll find that the freedom to rearrange
expressions as you like to be very useful as you move on to algebra in Part V
...
This same
concept also works for multiplication
...
For example, suppose you want to multiply these two numbers:
17 × 101
You can multiply them out, but distribution provides a different way to think
about the problem that you may find easier
...
At this point,
you may be able to solve the two multiplications in your head and then add
them up easily:
= 1,700 + 17 = 1,717
Distribution becomes even more useful when you get to algebra in Part VI
...
In this section, I give you a closer look at how to perform
the Big Four operations with negative numbers
...
For example,
5 – 8 = –3
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Part II: Getting a Handle on Whole Numbers
In real-world applications, negative numbers are used to represent debt
...
Even though you may have trouble picturing –3 chairs, you still need to account for this debt, and negative numbers
are the right tool for the job
...
After you know
how to do this, you find that all these problems are quite simple
...
Don’t worry about memorizing every little bit of this procedure
...
(If you need a quick refresher on how the number line
works, see Chapter 1
...
For example, suppose you want to solve –3 + 4
...
Similarly, suppose you want to solve –2 – 5
...
You’re subtracting, so move to the left:
Start at –2, down 5
–9 –8 –7 –6 –5 –4 –3 –2 –1
So –2 – 5 = –7
...
You already know to start at –2, but where
do you go from there? Here’s the up and down rule for adding a negative
number:
Adding a negative number is the same as subtracting a positive number — that
is, go down on the number line
...
If you rewrite a subtraction problem as an addition problem — for instance,
rewriting 3 – 7 as 3 + (–7) — you can use the commutative and associative
properties of addition, which I discuss earlier in this chapter
...
Subtracting a negative number
The last rule you need to know is how to subtract a negative number
...
Here’s the up and down rule:
Subtracting a negative number is the same as adding a positive number —
that is, up on the number line
...
When subtracting negative numbers, you can think of the two minus signs
canceling each other out to create a positive
...
The presence of one or more minus signs (–)
doesn’t change the numerical part of the answer
...
ߜ If the numbers have opposite signs, the result is always negative
...
The only
question is whether the complete answer is 6 or –6
...
Another way of thinking of this rule is that the two negatives cancel each
other out to make a positive
...
When the signs
are the same, the result is positive, and when the signs are different, the
result is negative
...
That’s a pretty large category, because
almost anything that you can name can be counted
...
For now, just understand that all units can
be counted, which means that you can apply the Big Four operations to units
...
Just remember that you can only add or subtract when the units
are the same
...
For example, suppose
you have four chairs and but find that you need twice as many for a party
...
Here’s how you represent this idea:
20 cherries ÷ 4 = 5 cherries
But you have to be careful when multiplying or dividing units by units
...
In these cases, multiplying or
dividing by units is meaningless
...
For example,
multiplying units of length (such as inches, miles, or meters) results in square
units
...
Similarly, here are
some examples of when dividing units makes sense:
12 slices of pizza ÷ 4 people = 3 slices of pizza/person
140 miles ÷ 2 hours = 70 miles/hour
In these cases, you read the fraction slash (/) as per: slices of pizza per person
or miles per hour
...
Understanding Inequalities
Sometimes, you want to talk about when two quantities are different
...
In this section, I discuss four types of
inequalities: ≠ (doesn’t equal), < (less than), > (greater than), and ≈ (approximately equals)
...
For example,
2+2≠5
3 × 4 ≠ 34
999,999 ≠ 1,000,000
You can read ≠ as “doesn’t equal” or “is not equal to
...
”
Chapter 4: Putting the Big Four Operations to Work
Less than (<) and greater than (>)
The symbol < means less than
...
For example,
5>4
100 > 99
2+2>3
The two symbols < and > are similar and easily confused
...
This L should remind you that it
means less than
...
Approximately equals (≈)
In Chapter 2, I show you how rounding numbers makes large numbers easier
to work with
...
For example,
49 ≈ 50
1,024 ≈ 1,000
999,999 ≈ 1,000,000
You can also use ≈ when you estimate the answer to a problem:
1,000,487 + 2,001,932 + 5,000,032
≈ 1,000,000 + 2,000,000 + 5,000,000 = 8,000,000
67
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Part II: Getting a Handle on Whole Numbers
Beyond the Big Four: Exponents, Square
Roots, and Absolute Value
In this section, I introduce you to three new operations that you need as you
move on with math: exponents, square roots, and absolute value
...
To tell the truth, these three operations have fewer everyday applications
than the Big Four
...
Fortunately, they aren’t difficult, so this is a good time to
become familiar with them
...
For
example, 23 means to multiply 2 by itself 3 times
...
You can read 23 as
“2 to the third power” or “2 to the power of 3” (or even “2 cubed,” which has
to do with the formula for finding the value of a cube — see Chapter 16 for
details)
...
Read 105 as “10 to the
fifth power” or “10 to the power of 5
...
Just write
down a 1 and that many 0s after it:
102 = 100
(1 with two 0s)
7
(1 with seven 0s)
20
(1 with twenty 0s)
10 = 10,000,000
10 = 100,000,000,000,000,000,000
Chapter 4: Putting the Big Four Operations to Work
Exponents with a base number of 10 are very important in scientific notation,
which I cover in Chapter 14
...
When you take any whole
number to the power of 2, the result is a square number
...
) For this reason, taking a number to
the power of 2 is called squaring that number
...
Here are some squared numbers:
32 = 3 ⋅ 3 = 9
42 = 4 ⋅ 4 = 16
52 = 5 ⋅ 5 = 25
Any number raised to the 0 power equals 1
...
Discovering your roots
Earlier in this chapter, in “Knowing Properties of the Big Four Operations,”
I show you how addition and subtraction are inverse operations
...
In a similar way,
roots are the inverse operation of exponents
...
A square root undoes an exponent
of 2
...
” So,
You can read the symbol
read 9 as either “the square root of 9” or “radical 9
...
For example, to find 100, you ask the question, “What
number when multiplied by itself equals 100?” The answer in this case is 10,
because
10 ⋅ 10 = 100, so 100 = 10
You probably won’t use square roots too much until you get to algebra, but at
that point they become very handy
...
It tells
you how far away from 0 a number is on the number line
...
Taking the absolute value of a positive number doesn’t change that number’s
value
...
The Three E’s of Math: Equations,
Expressions, and Evaluations
You should find the Three E’s of math very familiar because whether you realize it or not, you’ve been using them for a long time
...
In this section, I shed light on this stuff and give you a new way to look at it
...
An expression is a string of
mathematical symbols that can be placed on one side of an equation — for
example, 1 + 1
...
72
Part II: Getting a Handle on Whole Numbers
Throughout the rest of the chapter, I show you how to turn expressions into
numbers using a set of rules called the order of operations (or order of precedence)
...
Equality for All: Equations
An equation is a mathematical statement that tells you that two things have
the same value — in other words, it’s a statement with an equal sign
...
Mathematical equations come in lots of varieties: arithmetic equations,
algebraic equations, differential equations, partial differential equations,
Diophantine equations, and many more
...
In this chapter, I discuss only arithmetic equations, which are equations
involving numbers, the Big Four operations, and the other basic operations I
introduce in Chapter 4 (absolute values, exponents, and roots)
...
Here are a few examples of simple
arithmetic equations:
2+2=4
3 ⋅ 4 = 12
20 ÷ 2 = 10
Three properties of equality
Three properties of equality are called reflexivity, symmetry, and transitivity:
ߜ Reflexivity says that everything is equal to
itself
...
For
example,
4 ⋅ 5 = 20, so 20 = 4 ⋅ 5
ߜ Transitivity says that if something is equal
to two other things, then those two other
things are equal to each other
...
The inequalities that I introduce
in Chapter 4 (≠, >, <, and ≈) don’t necessarily
share all these properties
...
Mathematical expressions, just like equations, come
in a lot of varieties
...
In Part V, I introduce you to algebraic expressions
...
In other words, when
you evaluate something, you find its value
...
The
words may change, but the idea is the same — boiling a string of numbers
and math symbols down to a single number
...
For example,
evaluate the following arithmetic expression:
7⋅5
73
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Part II: Getting a Handle on Whole Numbers
How? Simplify it to a single number:
35
Putting the Three E’s together
I’m sure you’re dying to know how the Three E’s — equality, expressions, and
evaluation — are all connected
...
Then, you can make an equation, using an equal sign to connect the expression
and the number
...
Put on socks
...
Put on shoes
...
A simple rule to follow, right?
In this section, I outline a similar set of rules for evaluating expressions called
the order of operations (sometimes called order of precedence)
...
Order of operations is just a set of rules to make sure
you get your socks and shoes on in the right order, mathematically speaking,
so you always get the right answer
...
But order of operations is a
bit too confusing to present that way
...
Don’t let the complexity of these rules scare you off before you work through them!
Evaluate arithmetic expressions from left to right according to the following
order of operations:
1
...
Exponents
3
...
Addition and subtraction
Don’t worry about memorizing this list right now
...
ߜ In “Using order of operations in expressions with exponents,” I show you
how Step 2 fits in — how to evaluate expressions with Big Four operations
plus exponents, square roots, and absolute value
...
Applying order of operations
to Big Four expressions
As I explain earlier in this chapter, evaluating an expression is just simplifying
it down to a single number
...
(For more on the Big Four,
see Chapter 3
...
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Part II: Getting a Handle on Whole Numbers
Table 5-1
The Three Types of Big Four Expressions
Expression
Example
Rule
Contains only addition
and subtraction
12 + 7 – 6 – 3 + 8
Evaluate left to right
...
Mixed-operator expressions:
contains a combination of
addition/subtraction and
multiplication/division
9+6÷3
1
...
2
...
In this section, I show you how to identify and evaluate all three types of
expressions
...
When this is the
case, the rule for evaluating the expression is simple
...
For example, suppose you want to evaluate this
expression:
17 – 5 + 3 – 8
Because the only operations are addition and subtraction, you can evaluate
from left to right, starting with 17 – 5:
= 12 + 3 – 8
As you can see, the number 12 replaces 17 – 5
...
Next, evaluate 12 + 3:
= 15 – 8
This breaks the expression down to two numbers, which you can evaluate
easily:
=7
So 17 – 5 + 3 – 8 = 7
...
When this is the
case, the rule for evaluating the expression is pretty straightforward
...
Suppose you want to evaluate this
expression:
9⋅2÷6÷3⋅2
Again, the expression contains only multiplication and division, so you can
move from left to right, starting with 9 ⋅ 2:
= 18 ÷ 6 ÷ 3 ⋅ 2
=3÷3⋅2
=1⋅2
=2
Notice that the expression shrinks one number at a time until all that’s left is 2
...
Here’s another quick example:
–2 ⋅ 6 ÷ –4
Even though this expression has some negative numbers, the only operations
it contains are multiplication and division
...
Mixed-operator expressions
Often, an expression contains
ߜ At least one addition or subtraction operator
ߜ At least one multiplication or division operator
I call these mixed-operator expressions
...
Here’s the rule you want to follow
...
Evaluate the multiplication and division from left to right
...
Evaluate the addition and subtraction from left to right
...
To evaluate it, start out by
underlining the multiplication and division in the expression:
5+3⋅2+8÷4
Now, evaluate what you underlined from left to right:
=5+6+8÷4
=5+6+2
At this point, you’re left with an expression that contains only addition, so
you can evaluate it from left to right:
= 11 + 2
= 13
Thus, 5 + 3 ⋅ 2 + 8 ÷ 4 = 13
...
Evaluate exponents from left to right before you begin evaluating Big Four
operations (adding, subtracting, multiplying, and dividing)
...
” For example, suppose you want to evaluate the
following:
3 + 52 – 6
Chapter 5: A Question of Values: Evaluating Arithmetic Expressions
First, evaluate the exponent:
= 3 + 25 – 6
At this point, the expression contains only addition and subtraction, so you
can evaluate it from left to right in two steps:
= 28 – 6
= 22
So 3 + 52 – 6 = 22
...
When it comes to evaluating expressions, here’s what you need
to know about parentheses
...
Evaluate the contents of parentheses, from the inside out
...
Evaluate the rest of the expression
...
This expression contains two sets of parentheses, so evaluate these from left to right
...
Expressions with exponents and parentheses
As another example, try this out:
1 + (3 – 62 ÷ 9) ⋅ 22
Start out by working only with what’s inside the parentheses
...
This expression
takes three steps, starting with the exponent:
= 1 + –1 ⋅ 4
= 1 + –4
= –3
So 1 + (3 – 62 ÷ 9) ⋅ 22 = –3
...
In this case, evaluate the contents of the parentheses before evaluating
the exponent, as usual
...
As always,
evaluate what’s in the parentheses first
...
I underlined the part that you need to evaluate first:
= 21(19 + –18)
Now you can finish off what’s inside the parentheses:
= 211
At this point, all that’s left is a very simple exponent:
= 21
So 21(19 + 3 ⋅ –6) = 21
...
If
you see an expression in the exponent, treat it as though it had parentheses
around it
...
Expressions with nested parentheses
Occasionally, an expression has nested parentheses: one or more sets
of parentheses inside another set
...
When evaluating an expression with nested parentheses, evaluate what’s
inside the innermost set of parentheses first and work your way toward the
outermost parentheses
...
To start you off, I underlined what’s deep inside this third set of
parentheses
...
Again, work
from the inside out
...
As I say earlier in this section, this problem is about as hard as they come at
this stage of math
...
Chapter 6
Say What? Turning Words
into Numbers
In This Chapter
ᮣ Dispelling myths about word problems
ᮣ Knowing the four steps to solving a word problem
ᮣ Jotting down simple word equations that condense the important information
ᮣ Writing more complex word equations
ᮣ Plugging numbers into the word equations to solve the problem
ᮣ Attacking more-complex word problems with confidence
T
he very mention of word problems — or story problems, as they’re sometimes called — is enough to send a cold shiver of terror into the bones
of the average math student
...
” But word
problems help you understand the logic behind setting up equations in reallife situations, making math actually useful — even if the scenarios in the
word problems you practice on are pretty far-fetched
...
Then, I show you
how to solve a word problem in four simple steps
...
Some of these problems have longer numbers to calculate, and others may have more complicated stories
...
84
Part II: Getting a Handle on Whole Numbers
Dispelling Two Myths about
Word Problems
Here are two common myths about word problems:
ߜ Word problems are always hard
...
Both of these ideas are untrue
...
Word problems aren’t always hard
Word problems don’t have to be hard
...
Then Brenda gave him 5 more apples
...
” (Of course, if you were
the class clown, you probably wrote, “Adam doesn’t have any apples because
he ate them all
...
In this chapter, I give you a
system and show you how to apply it to problems of increasing difficulty
...
Word problems are useful
In the real world, math rarely comes in the form of equations
...
Chapter 6: Say What? Turning Words into Numbers
Whenever you paint a room, prepare a budget, bake a double batch of oatmeal cookies, estimate the cost of a vacation, buy wood to build a shelf, do
your taxes, or weigh out the pros and cons of buying a car or leasing one, you
need math
...
Word problems give you practice turning situations — that is, stories — into
numbers
...
Read through the problem and set up word equations — that is,
equations that contain words as well as numbers
...
Plug in numbers in place of words wherever possible to set up a regular math equation
...
Use math to solve the equation
...
Answer the question that the problem asks
...
This chapter and Chapters 13, 18, and 24
are all about Steps 1 and 2
...
When you know how to turn a word problem into an equation, the hard part is
done
...
From there, Step 4 is usually pretty easy,
though at the end of each example, I make sure you understand how to do it
...
In this section, I show you how to squeeze
the juice out of a word problem and leave the pits behind!
85
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Part II: Getting a Handle on Whole Numbers
Jotting down information as word equations
Most word problems give you information about numbers, telling you exactly
how much, how many, how fast, how big, and so forth
...
The width of the house is 80 feet
...
You need this information to solve the problem
...
(If you’re concerned about trees, write on the back of all
that junk mail you get
...
For example, here’s how you can jot down “Nunu is spinning 17 plates”:
Nunu = 17
Here’s how to note that “
...
”
So you can jot down the following:
Local = 25
Don’t let the word if confuse you
...
” and then asks you a question, assume it is true and use this information to answer the question
...
A word equation has an equal sign
like a math equation, but it contains both words and numbers
...
For example,
Bobo is spinning five fewer plates than Nunu
...
The express train is moving three times faster than the local train
...
Statements like these look like English, but they’re really
math, so spotting them is important
...
Look again
at the first example:
Bobo is spinning five fewer plates than Nunu
...
But you know that these two numbers are related
...
And, as you
see in the next section, word equations are easy to turn into the math that
you need to solve the problem
...
You don’t know the width or height of the house, but you know that these
numbers are connected
...
Figuring out what the problem’s asking
The end of a word problem usually contains the question that you need to
answer to solve the problem
...
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Part II: Getting a Handle on Whole Numbers
For example, you can write the question, “All together, how many plates are
Bobo and Nunu spinning?” as
Bobo + Nunu = ?
You can write the question “How tall is the house” as:
Height = ?
Finally, you can rephrase the question “What’s the difference in speed
between the express train and the local train?” in this way:
Express – Local = ?
Plugging in numbers for words
After you’ve written out a bunch of word equations, you have the facts you
need in a form you can use
...
In this section, I show
you how to use the word equations you built in the last section to solve three
problems
...
Here’s an example:
Bobo is spinning five fewer plates than Nunu (Bobo dropped a few)
...
All together, how many plates are Bobo and Nunu
spinning?
Here’s what you have already, just from reading the problem:
Nunu = 17
Bobo + 5 = Nunu
Plugging in the information gives you the following:
Bobo + 5 = Nunu 17
If you see how many plates Bobo is spinning, feel free to jump ahead
...
That is, you need to find out the following:
Bobo + Nunu = ?
Just plug in the numbers, substituting 12 for Bobo and 17 for Nunu:
Bobo 12 + Nunu 17 = 29
So Bobo and Nunu are spinning 29 plates
...
Here’s an example:
The height of a house is half as long as its width, and the width of the
house is 80 feet
...
Example: I hear the train a comin’
Pay careful attention to what the question is asking
...
Here’s an example:
The express train is moving three times faster than the local train
...
Finding the difference
between two numbers is subtraction, so here’s what you want to find:
Express – Local = ?
You can get what you need to know by plugging in the information you’ve
already found:
Express 75 – Local 25 = 50
Therefore, the difference in speed between the express train and the local
train is 50 miles per hour
...
And what’s more, you can use those same skills to find your
way through more complex problems
...
(For example, instead of a dress costing $30, now it costs $29
...
)
ߜ The amount of information in the problem increases
...
)
Don’t let problems like these scare you
...
When numbers get serious
A lot of problems that look tough aren’t much more difficult than the problems I show you in the previous sections
...
84 hidden in her pillowcase, and Aunt Jezebel has
$234
...
How much money do the two women
have all together?
Chapter 6: Say What? Turning Words into Numbers
One question you may have is how these women ever get any sleep with all
that change clinking around under their heads
...
Start reading from the beginning: “Aunt Effie
has $732
...
” This text is just information to jot down as a simple word
equation:
Effie = $732
...
Aunt Jezebel has $234
...
”
It’s another statement you can write as a word equation:
Jezebel = Effie – $234
...
84 where you see Aunt Effie’s name in
the equation:
Jezebel = Effie $732
...
19
So far, the big numbers haven’t been any trouble
...
84
– $234
...
65
Now you can jot this information down as always:
Jezebel = $498
...
Here’s how to represent this question as an equation:
Effie + Jezebel = ?
You can plug information into this equation:
Effie $732
...
65 = ?
Again, because the numbers are large, you probably have to stop to do the
math:
$732
...
65
$1,231
...
49
...
The only difference is that
you have to stop to do some addition and subtraction
...
Here’s a word problem that’s designed to scare you
off — but with your new skills, you’re ready for it:
Four women collected money to save the endangered Salt Creek tiger
beetle
...
How much money did the four women collect all together?
If you try to do this problem all in your head, you’ll probably get confused
...
First, “Keisha collected $160
...
Keisha collected $160, so you can plug in 160 anywhere you
find Keisha’s name:
Brie = Keisha 160 + 50 = 210
Chapter 6: Say What? Turning Words into Numbers
Now you know how much Brie collected, so you can plug this information
into the next equation:
Amy = Brie 210 ⋅ 2 = 420
This equation tells you how much Amy collected, so you can plug this
number into the last equation:
Amy 420 + Sophia = 700
To solve this problem, change it from addition to subtraction using inverse
operations, as I show you in Chapter 4:
Sophia = 700 – 420 = 280
Now that you know how much money each woman collected, you can answer
the question at the end of the problem:
Keisha + Brie + Amy + Sophia = ?
You can plug in this information easily:
Keisha 160 + Brie 210 + Amy 420 + Sophia 280 = 1,070
So you can conclude that the four women collected $1,070 all together
...
Try
writing down this problem and working it through step by step on your own
...
When you can solve it from beginning to end
with the book closed, you’ll have a good grasp of how to solve word problems:
On a recent shopping trip, Travis bought six shirts for $19
...
60 each
...
08 less than he paid for both pairs of pants
...
Believe me — it was quite a challenge
...
You can jot down the following word equations:
Shirts = $19
...
60 ⋅ 2
Jacket = Pants – $37
...
95
×
6
$119
...
60
×
2
$69
...
70
Pants = $69
...
08
Now you can plug in $69
...
20 – $37
...
20
– $37
...
12
This equation gives you the price of the jacket:
Jacket = $32
...
70 + Pants $69
...
12
Again, you have another equation to solve:
$119
...
20
+ $ 32
...
02
Chapter 6: Say What? Turning Words into Numbers
So you can jot down the following:
Amount Travis spent = $221
...
02
And do just one more equation:
$300
...
02
$ 78
...
98
Therefore, Travis received $78
...
95
96
Part II: Getting a Handle on Whole Numbers
Chapter 7
Divisibility
In This Chapter
ᮣ Finding out whether a number is divisible by 2, 3, 5, 9, 10, or 11
ᮣ Seeing the difference between prime numbers and composite numbers
W
hen one number is divisible by another, you can divide the first
number by the second number without getting a remainder (see
Chapter 3 for details on division)
...
To start out, I show you a bunch of handy tricks for discovering whether one
number is divisible by another without actually doing the division
...
This discussion, plus what follows in Chapter 8, can help make your
encounter with fractions in Part III a lot friendlier
...
In this section, I give you a
bunch of time-saving tricks for finding out whether one number is divisible by
another without actually making you do the division
...
As you can see, when you divide any number
by 1, the answer is the number itself, with no remainder:
98
Part II: Getting a Handle on Whole Numbers
2÷1=2
17 ÷ 1 = 17
431 ÷ 1 = 431
Similarly, every number (except 0) is divisible by itself
...
Mathematicians say that dividing by 0 is
undefined
...
No calculations required
...
For example, the following bolded numbers are divisible by 2:
6÷2=3
538 ÷ 2 = 269
77,144 ÷ 2 = 38,572
22 ÷ 2 = 11
6,790 ÷ 2 = 3,395
212,116 ÷ 2 = 106,058
Divisible by 5
Every number that ends in either 5 or 0 is divisible by 5
...
The following bolded numbers
are divisible by 10:
20 ÷ 10 = 2
170 ÷ 10 = 17
56,720 ÷ 10 = 5,672
Chapter 7: Divisibility
Every number that ends in 00 is divisible by 100:
300 ÷ 100 = 3
8,300 ÷ 100 = 83
634,900 ÷ 100 = 6,349
And every number that ends in 000 is divisible by 1,000:
6,000 ÷ 1,000 = 6
99,000 ÷ 1,000 = 99
1,234,000 ÷ 1,000 = 1,234
In general, every number that ends with a string of 0s is divisible by the
number you get when you write 1 followed by that many 0s
...
In Chapter 14, I show you all
about how to work with scientific notation
...
The sum of a number’s digits is called its digital roots
...
To find the digital root of a number, just add up the digits and repeat this
process until you get a one-digit number
...
The digital root of 143 is 8 because 1 + 4 + 3 = 8
...
Sometimes, you need to do this process more than once
...
You have to repeat the process three times,
but eventually you find that the digital root of 87,482 is 2:
8 + 7 + 4 + 8 + 2 = 29
2 + 9 = 11
1+1=2
99
100
Part II: Getting a Handle on Whole Numbers
Read on to find out how sums of digits can help you check for divisibility by
3, 9, or 11
...
First find the digital root of a number by adding its digits until you get a
single-digit number
...
With 975, when you add up the digits, you first get
21, so you then add up the digits in 21 to get the digital root 3
...
If you do the actual division, you find
that 18 ÷ 3 = 6, 51 ÷ 3 =17, and 975 ÷ 3 = 325, so the method checks out
...
If you try to
divide by 3, you end up with 345 r 2
...
To test whether a number is divisible by 9, find its digital root by adding up
its digits until you get a one-digit number
...
With 7,587, however, when you add up the digits, you
get 27, so you then add up the digits in 27 to get the digital root 9
...
You can verify this by doing the
division: 36 ÷ 9 = 4, 243 ÷ 9 = 27, and 7,857 ÷ 9 = 873
...
Here’s an example:
706:
7 + 0 + 6 = 13; 1 + 3 = 4
Because the digital root of 706 is 4, 706 isn’t divisible by 9
...
Divisible by 11
Two-digit numbers that are divisible by 11 are hard to miss because they
simply repeat the same digit twice
...
For
example, suppose you want to decide whether the number 154 is divisible
by 11
...
If you divide, you get 154 ÷ 11 = 14, a whole number
...
Add the
first and third digits:
1+6=7
Because the first and third digits add up to 7 instead of 3, the number 136
isn’t divisible by 11
...
For numbers of any length, the rule is slightly more complicated, but it’s still
often easier than doing long division
...
To start out, underline alternate digits (every other digit):
15,983
101
102
Part II: Getting a Handle on Whole Numbers
Now add up the underlined digits and the non-underlined digits:
1 + 9 + 3 = 13
‰
15,983
Ê
5 + 8 = 13
Because these two sets of digits both add up to 13, the number 15,983 is
divisible by 11
...
Now suppose you want to find out whether 9,181,909 is divisible by 11
...
But notice that 35 – 2 = 33
...
The actual
answer is 9,181,909 ÷ 11 = 834,719
Identifying Prime and Composite Numbers
In the earlier section titled “Counting everyone in: Numbers you can divide
everything by,” I show you that every number (except 0 and 1) is divisible by
at least two numbers: 1 and itself
...
In Chapter 8, you need to know how to tell prime numbers from composite in
order to break a number down into its prime factors
...
A prime number is divisible by exactly two positive whole numbers: 1 and the
number itself
...
For example, 2 is a prime number because when you divide it by any number
but 1 and 2, you get a remainder
...
So the only way to multiply two numbers together and
get 3 as a product is the following:
1⋅3=3
On the other hand, 4 is a composite number because it’s divisible by three
numbers: 1, 2, and 4
...
Here’s the only
way to multiply two counting numbers together and get 5 as a product:
1⋅5=5
And 6 is a composite number because it’s divisible by 1, 2, 3, and 6
...
The reason 1 is
neither is that it’s divisible by only one number, which is 1
...
Every composite
number less than 100 is divisible by at least one of these numbers
...
If it’s divisible by any of these numbers, it’s
composite — if not, it’s prime
...
Here’s how you think it out,
using the tricks I show you earlier in “Knowing the Divisibility Tricks”:
ߜ 79 is an odd number, so it isn’t divisible by 2
...
103
104
Part II: Getting a Handle on Whole Numbers
ߜ 79 doesn’t end in 5 or 0, so it isn’t divisible by 5
...
So 79 ÷ 7 would leave remainder of 2, which tells you that
79 isn’t divisible by 7
...
Now test whether 93 is prime or composite:
ߜ 93 is an odd number, so it isn’t divisible by 2
...
You don’t need to look further
...
Chapter 8
Fabulous Factors and
Marvelous Multiples
In This Chapter
ᮣ Understanding how factors and multiples are related
ᮣ Listing all the factors of a number
ᮣ Breaking a number down into its prime factors
ᮣ Generating multiples of a number
ᮣ Finding the greatest common factor (GCF) and least common multiple (LCM)
I
n Chapter 2, I introduce you to sequences of numbers based on the multiplication table
...
Factors and multiples are really two sides of the same coin
...
For starters, I show you how to decompose (split up) any number into its
prime factors
...
To finish up on factors, I show
you how to find the greatest common factor (GCF) of any set of numbers
...
Knowing Six Ways to Say
the Same Thing
In this section, I introduce you to factors and multiples, and I show you how
these two important concepts are connected
...
For example, the following
equation is true:
5 ⋅ 4 = 20
So this inverse equation is also true:
20 ÷ 4 = 5
You may have noticed that in math, you tend to run into the same ideas over
and over again
...
The following three statements all focus on the relationship between 5 and 20
from the perspective of multiplication:
ߜ 5 multiplied by some number is 20
ߜ 5 is a factor of 20
ߜ 20 is a multiple of 5
In two of the examples, you can see this relationship reflected in the words
multiplied and multiple
...
Similarly, the following three statements all focus on the relationship between
5 and 20 from the perspective of division:
ߜ 20 divided by some number is 5
ߜ 20 is divisible by 5
ߜ 5 is a divisor of 20
Why do mathematicians need all these words for the same thing? Maybe for
the same reason that Eskimos need a bunch of words for snow
...
When you understand the concepts, which word you choose doesn’t matter a whole lot
...
For example, 20 is divisible by 5, so
ߜ 5 is a factor of 20
ߜ 20 is a multiple of 5
Chapter 8: Fabulous Factors and Marvelous Multiples
Don’t mix which number is the factor and which is the multiple
...
If you have trouble remembering which number is the factor and which is the
multiple, jot them down in order from lowest to highest and write the letters
F and M in alphabetical order under them
...
Fabulous Factors
In this section, I introduce you to factors
...
Then I show you how to list a
number’s factors
...
This information all leads up to an essential skill: finding the greatest
common factor (GCF) of a set of numbers
...
If it divides evenly (with no remainder),
the number is a factor; otherwise, it’s not a factor
...
Here’s
how you find out:
56 ÷ 7 = 8
Because 7 divides 56 without leaving a remainder, 7 is a factor of 56
...
This method works no matter how large the numbers are
...
For a
refresher on how to do long division, see Chapter 3
...
Here’s how
to list all the factors of a number:
1
...
2
...
If it is, add 2 to the list, along with the original number divided by 2 as
the second-to-last number on the list
...
Test the number 3 in the same way
...
Continue testing numbers until the beginning of the list meets the end
of the list
...
Suppose you want to list all the
factors of the number 18
...
18
Remember from Chapter 7 that every number — whether prime or
composite — is divisible by itself and 1
...
Next, test whether the number 2 is a factor of 18:
18 ÷ 2 = 9
Because 2 divides 18 without a remainder, 2 is a factor of 18
...
) So both 2 and 9 are factors of 18,
and you can add them both to the list:
1
2
...
Doing this
reminds you that you don’t have to check any number higher than 9
...
6
9
18
At this point, you’re almost done
...
A prime
number is divisible only by 1 and itself — for example, the number 7 is divisible only by 1 and 7
...
A number’s prime factors are the set of prime numbers (including repeats)
that equal that number when multiplied together
...
109
110
Part II: Getting a Handle on Whole Numbers
The best way to break a composite number down into its prime factors is
using a factorization tree
...
Split the number into any two factors and check off the original
number
...
If either of these factors is prime, circle it
...
Repeat steps 1 and 2 for any number that is neither circled nor checked
...
When every number in the tree is either checked or circled, the tree
is finished, and the circled numbers are the prime factors of the original number
...
In this case, remember that 7 ⋅ 8 = 56
...
56 ✓
Figure 8-1:
Finding two
factors of 56;
7 is prime
...
I also circle 7
because it’s a prime number
...
56 ✓
Figure 8-2:
Continuing
the number
breakdown
with 8
...
This time, 2 is
prime, so I circle it
...
At this point, every number in the tree is either circled or checked, so the
tree is finished
...
To check this result, just multiply the prime factors together:
2 ⋅ 2 ⋅ 2 ⋅ 7 = 56
Chapter 8: Fabulous Factors and Marvelous Multiples
56 ✓
7
Figure 8-3:
The finished
tree,
completed
from
Figure 8-1
...
What happens when you try to build a tree starting with a prime number —
for example, 7? Well, you don’t have to go very far (see Figure 8-4)
...
7
That’s it — you’re done! This example shows you that every prime number is
its own prime factor
...
(As you
find out in Chapter 2, 1 is neither prime nor composite, so it doesn’t have a
prime factorization
...
The remaining numbers are composite, so they can all be
broken down into smaller prime factors
...
This fact is important — so
important that it’s called the Fundamental Theorem of Arithmetic
...
Knowing how to break a number down to its prime factorization is a handy
skill to have
...
Finding prime factorizations for numbers 100 or less
When you build a factorization tree, the first step is usually the hardest
...
With fairly small numbers, the factorization tree is usually easy to use
...
This is especially true when you don’t recognize the
number from the multiplication table
...
Whenever possible, factor out 5s and 2s first
...
For example, suppose you want the prime factorization of the number 84
...
Figure 8-5:
Factoring
out 2
from 84
...
This tree is now easy to complete (see Figure 8-6)
...
42 ✓
6
2
✓
3
7
Chapter 8: Fabulous Factors and Marvelous Multiples
The resulting prime factorization for 84 is as follows:
84 = 2 ⋅ 7 ⋅ 2 ⋅ 3
If you like, though, you can rearrange the factors from lowest to highest:
84 = 2 ⋅ 2 ⋅ 3 ⋅ 7
By far, the most difficult situation occurs when you’re trying to find the prime
factors of a prime number but don’t know it
...
This time, you don’t recognize the number from the multiplication tables and it isn’t divisible by 2 or 5
...
Testing for divisibility by 3 by finding the digital root of 71 (that is, by adding
the digits) is easy
...
7+1=8
Because the digital root of 71 is 8, 71 isn’t divisible by 3
...
Therefore, 71 is a
prime number, so you’re done
...
Just in case, though, here’s what you need
to know
...
A special
case is when the number you’re factoring ends in one or more 0s
...
For example, Figure 8-7 shows the first step
...
700 ✓
10
10
7
113
114
Part II: Getting a Handle on Whole Numbers
After you do the first step, the rest of the tree becomes very easy (see
Figure 8-8):
700 ✓
Figure 8-8:
Completing
the factoring
of 700
...
Then factor out 7s if
possible (sorry, I don’t have a trick for 7s) and finally 11s
...
As always, every prime number is its own prime factorization, so when you
know that a number is prime, you’re done
...
Finding the greatest common factor (GCF)
After you understand how to find the factors of a number (see “Generating a
number’s factors”), you’re ready to move on to the main event: finding the
greatest common factor of several numbers
...
For example, the GCF of the numbers 4
and 6 is 2 because 2 is the greatest number that’s a factor of both 4 and 6
...
Using a list of factors to find the GCF
The first method for finding the GCF is quicker when you’re dealing with
smaller numbers
...
” The greatest factor appearing
on every list is the GCF
...
Factors of 6: 1, 2, 3, 6
Factors of 15: 1, 3, 5, 15
Because 3 is the greatest factor that appears on both lists, 3 is the GCF of 6
and 15
...
Start
by listing the factors of each:
Factors of 9: 1, 3, 9
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 25: 1, 5, 25
In this case, the only factor that appears on all three lists is 1, so 1 is the GCF
of 9, 20, and 25
...
This often
works better for large numbers, where generating lists of all factors can be
time-consuming
...
List the prime factors of each number (see the earlier “Prime factors”
section)
...
Circle every common prime factor — that is, every prime factor that’s
a factor of every number in the set
...
Multiply all the circled numbers
...
For example, suppose you want to find the GCF of 28, 42, and 70
...
Step 2 says to circle every prime factor
that’s common to all three numbers (as shown in Figure 8-9)
...
28 = 2 ⋅ 2 ⋅ 7
42 = 2 ⋅ 3 ⋅ 7
70 = 2 ⋅ 5 ⋅ 7
115
116
Part II: Getting a Handle on Whole Numbers
As you can see, the numbers 2 and 7 are common factors of all three numbers
...
Knowing how to find the GCF of a set of numbers is important when you
begin reducing fractions to lowest terms
...
)
Marvelous Multiples
Even though multiples tend to be larger numbers than factors, most students
find them easier to work with
...
Generating multiples
The preceding section, “Fabulous Factors,” tells you how to find all the factors of a number
...
So no matter how
large a number is, it always has a finite (limited) number of factors
...
(The only exception to this is 0, which is a multiple of every number
...
Nevertheless, generating a partial list of multiples for any number is simple
...
For example, here are the first few positive multiples of 7:
7
14
21
28
35
42
As you can see, this list of multiples is simply part of the multiplication table
for the number 7
...
)
Finding the least common multiple (LCM)
The least common multiple (LCM) of a set of numbers is the lowest positive
number that’s a multiple of every number in that set
...
Using the multiplication table to find the LCM
To find the LCM of a set of numbers, take each number in the set and jot
down a list of the first several multiples in order
...
When looking for the LCM of two numbers, start by listing multiples of the
higher number, but stop this list when the number of multiples you’ve written down equals the lower number
...
For example, suppose you want to find the LCM of 4 and 6
...
In this case, list only four of these
multiples because the lower number is 4
...
Now, start listing multiples of 4:
Multiples of 4:
4, 8, 12,
...
This method works especially well when you want to find the LCM of two
numbers, but it may take longer if you have more numbers
...
For the secondhighest number, find the product of the other two numbers and list that many
multiples
...
Suppose, for instance, you want to find the LCM of 2, 3, and 5
...
117
118
Part II: Getting a Handle on Whole Numbers
Next, list multiples of 3, listing ten of them (because 2 ⋅ 5 = 10):
Multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30,
...
In this case, you can
save yourself the trouble of making the last list because 30 is obviously a multiple of 2, and 15 isn’t
...
Using prime factorization to find the LCM
A second method for finding the LCM of a set of numbers is to use the prime
factorizations of those numbers
...
List the prime factors of each number
...
”
Suppose you want to find the LCM of 18 and 24
...
For each prime number listed, underline the most repeated occurrence
of this number in any prime factorization
...
Multiply all the underlined numbers
...
This checks out because
18 ⋅ 4 = 72
24 ⋅ 3 = 72
Part III
Parts of the
Whole: Fractions,
Decimals, and
Percents
M
In this part
...
Even though they may look
different, all three of these concepts are closely related
to division
...
I discuss
proper and improper fractions, how to reduce and increase
the terms of fractions, and how to work with mixed numbers
...
I show you how to use the percent
circle to solve three common types of percent problems
...
Chapter 9
Fooling with Fractions
In This Chapter
ᮣ Looking at basic fractions
ᮣ Knowing the numerator from the denominator
ᮣ Understanding proper fractions, improper fractions, and mixed numbers
ᮣ Increasing and reducing the terms of fractions
ᮣ Converting between improper fractions and mixed numbers
ᮣ Using cross-multiplication to compare fractions
S
uppose that today is your birthday and your friends are throwing you a
surprise party
...
Several solutions are proposed:
ߜ You could all go into the kitchen and bake seven more cakes
...
ߜ Because it’s your birthday, you could eat the whole cake and everyone
else could eat celery sticks
...
)
ߜ You could cut the cake into eight equal slices so that everyone can
enjoy it
...
With that decision,
you’ve opened the door into the exciting world of fractions
...
In this chapter, I give you
some basic information about fractions that you need to know, including the
three basic types of fractions: proper fractions, improper fractions, and mixed
numbers
...
I also show you how to convert between improper fractions and mixed numbers
...
By the time you’re done with this chapter, you’ll see how fractions really can
be a piece of cake!
122
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Slicing a Cake into Fractions
Here’s a simple fact: When you cut a cake into two equal pieces, each piece is
half of the cake
...
In Figure 9-1, the shaded
piece is half of the cake
...
Every fraction is made up of two numbers separated by a line, or a fraction
bar
...
The numerator tells you
how many pieces you have
...
The number below the line is called the denominator
...
In this case,
the denominator is 2
...
Figure 9-2:
Cake cut
into thirds
...
Again, the numerator
tells you how many pieces you have, and the denominator tells you how
many equal pieces the whole cake has been cut up into
...
A
...
Figure 9-3:
Cakes cut
and shaded B
...
D
...
The fraction bar can also mean a division sign
...
If you take three cakes and divide them among four people, each
person gets 3⁄4 of a cake
...
When you know them, you find
working with fractions a lot easier
...
For example, look at the following fraction:
3
4
123
124
Part III: Parts of the Whole: Fractions, Decimals, and Percents
In this example, the number 3 is the numerator, and the number 4 is the
denominator
...
Flipping for reciprocals
When you flip a fraction over, you get its reciprocal
...
Or conversely, you can turn any whole
number into a fraction by drawing a line and placing the number 1 under it
...
That’s
because if you cut a cake into eight pieces and you keep all eight of them, you
have the entire cake
...
For example,
0=0
1
0 =0
12
0 =0
113
Chapter 9: Fooling with Fractions
The denominator of a fraction can never be 0
...
Remember from earlier in this chapter that placing a number in the denominator is similar to cutting a cake up into that number of pieces
...
You can even cut it into one
piece (that is, don’t cut it at all)
...
For
this reason, putting 0 in the denominator — much like lighting an entire book
of matches on fire — is something you should never, never do
...
Here are some examples:
11
2
53
4
99 44
100
A mixed number is always equal to the whole number plus the fraction
attached to it
...
Knowing proper from improper
When the numerator and the denominator are equal, the fraction equals 1:
2=1
2
5=1
5
78 = 1
78
When the numerator (top number) is less than the denominator (bottom
number), the fraction is less than 1:
1 <1
2
3 <1
5
63 < 1
73
Fractions like these are called are called proper fractions
...
However, when the numerator is greater than
the denominator, the fraction is greater than 1
...
It’s customary
to convert an improper fraction to a mixed number, especially when it’s the
final answer to a problem
...
To stabilize it, convert it to a mixed number
...
Later in this chapter, I discuss improper fractions in more detail when I show
you how to convert between improper fractions and mixed numbers
...
(Get it? No matter how you slice it? You may as well laugh at the bad
jokes, too — they’re free
...
The fractions 1⁄2, 2⁄4, and 3⁄6 are all equal in value
...
As long as the numerator is
exactly half the denominator, the fractions are all equal to 1⁄2 — for example,
11
22
100
200
1, 000, 000
2, 000, 000
These fractions are equal to 1⁄2, but their terms (the numerator and denominator) are different
...
Increasing the terms of fractions
To increase the terms of a fraction by a certain number, multiply both the
numerator and the denominator by that number
...
Because you’re
multiplying the numerator and denominator by the same number, you’re
essentially multiplying the fraction by a fraction that equals 1
...
Here’s how you do it:
1
...
To keep the fractions equal, you have to multiply the numerator and
denominator of the old fraction by the same number
...
For example, suppose you want to raise the terms of the fraction 4⁄7 so
that the denominator is 35
...
2
...
You now know how the two denominators are related
...
Multiply 5 by 4, which gives you 20
...
But because you can’t always divide, reducing
takes a bit more finesse
...
For this reason,
if you’re not up on factoring, you may want to review this topic in Chapter 8
...
Then I show you a more informal way that you can use after you’re
more comfortable
...
I discuss this in detail in Chapter 8, so
if you’re shaky on this concept, you may want to review it first
...
Break down both the numerator (top number) and denominator
(bottom number) into their prime factors
...
Break down
both 12 and 30 into their prime factors:
12 = 2 $ 2 $ 3
30 2 $ 3 $ 5
2
...
As you can see, I cross out a 2 and a 3, because they’re common factors —
that is, they appear in both the numerator and denominator:
12 = 2 $ 2 $ 3
30 2 $ 3 $ 5
3
...
This shows you that the fraction 12⁄30 reduces to 2⁄5:
12 = 2 $ 2 $ 3 = 2
30 2 $ 3 $ 5 5
As another example, here’s how you reduce the fraction 32⁄100:
32 = 2 $ 2 $ 2 $ 2 $ 2 = 8
100
25
2$2$5$5
This time, cross out two 2s from both the top and the bottom as common factors
...
So the fraction 32⁄100 reduces to 8⁄25
...
If the numerator (top number) and denominator (bottom number) are
both divisible by 2 — that is, if they’re both even — divide both by 2
...
The numerator and the denominator are both even, so divide them both by 2:
24 = 12
60 30
Chapter 9: Fooling with Fractions
2
...
In the resulting fraction, both numbers are still even, so repeat the first
step again:
12 = 6
30 15
3
...
Now, the numerator and the denominator are both divisible by 3 (see
Chapter 7 for easy ways to tell if one number is divisible by another), so
divide both by 3:
6 =2
15 5
Neither the numerator nor the denominator is divisible by 3, so this step
is complete
...
The numerator is 2, and it obviously isn’t divisible by any larger number, so you know that the fraction
24
⁄60 reduces to 2⁄5
...
Improper
fractions are very useful and easy to work with, but for some reason, people
just don’t like them
...
) Teachers
especially don’t like them, and they really don’t like an improper fraction to
appear as the answer to a problem
...
One
reason they love them is that estimating the approximate size of a mixed
number is easy
...
Although 101⁄3 is the
same as 31⁄3, knowing the mixed number is a lot more helpful in practice
...
129
130
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Knowing the parts of a mixed number
Every mixed number has both a whole number part and a fractional part
...
So this mixed number is made up of three numbers:
the whole number (3), the numerator (1), and the denominator (2)
...
Converting a mixed number
to an improper fraction
To convert a mixed number to an improper fraction,
1
...
For example, suppose you want to convert the mixed number 52⁄3 to an
improper fraction
...
Use this result as your numerator, and place it over the denominator
you already have
...
This method
works for all mixed numbers
...
Chapter 9: Fooling with Fractions
Converting an improper fraction
to a mixed number
To convert an improper fraction to a mixed number, divide the numerator
by the denominator (see Chapter 3)
...
ߜ The remainder is the numerator
...
For example, suppose you want to write the improper fraction 19⁄5 as a mixed
number
...
And, as is true of conversions
in the other direction, if you start with a reduced fraction, you don’t have to
reduce your answer (see “Increasing and reducing terms of fractions”)
...
You can use it in a few different ways, so I explain it here, and then I show you an immediate application
...
Multiply the numerator of the first fraction by the denominator of the
second fraction and jot down the answer
...
Multiply the numerator of the second fraction by the denominator of
the first fraction and jot down the answer
...
When you do so, make sure that you start with the numerator of the
first fraction
...
The larger number is
always under the larger fraction
...
So you
can throw out 5⁄9
...
Pretty straightforward,
right? And that’s all you have to know for now
...
Chapter 10
Parting Ways: Fractions and
the Big Four Operations
In This Chapter
ᮣ Looking at multiplication and division of fractions
ᮣ Adding and subtracting fractions in a bunch of different ways
ᮣ Applying the four operations to mixed numbers
I
n this chapter, the focus is on applying the Big Four operations to fractions
...
Surprisingly, adding and
subtracting fractions is a bit trickier
...
Later in the chapter, I move on to mixed numbers
...
I save adding
and subtracting mixed numbers for the very end
...
Multiplying and Dividing Fractions
One of the odd little ironies of life is that multiplying and dividing fractions
is easier than adding or subtracting them — just two easy steps and you’re
done! For this reason, I show you how to multiply and divide fractions before
I show you how to add or subtract them
...
More good news is that dividing fractions is
nearly as easy as multiplying them
...
All you need for
multiplying fractions is a pen or pencil, something to write on (preferably not
your hand), and a basic knowledge of the multiplication table
...
)
Here’s how to multiply two fractions:
1
...
2
...
For example, here’s how to multiply 2⁄5 ⋅ 3⁄7:
2 3 = 2$3 = 6
5 $ 7 5 $ 7 35
Sometimes, when you multiply fractions, you may have an opportunity to
reduce to lowest terms
...
) As a rule, math people are crazy about reduced fractions, and
teachers sometimes take points off a right answer if you could’ve reduced it
but didn’t
...
Start by dividing both numbers by 2:
28 ' 2 = 14
40 ' 2 20
Again, the numerator and the denominator are both even, so do it again:
14 ' 2 = 7
20 ' 2 10
This fraction is now fully reduced
...
Canceling out equal factors makes the numbers that you’re multiplying smaller and easier to work
with, and it also saves you the trouble of reducing at the end
...
(See the nearby sidebar for why this works
...
In
other words, divide the numerator and denominator by that common
factor
...
)
For example, suppose you want to multiply the following two numbers:
5 13
13 $ 20
You can make this problem easier by canceling out the number 13 as follows:
13 1
5
= 5$1 = 5
1 13 $ 20
1 $ 20 20
You can make it even easier by noticing that 20 = 5 $ 4 , so you can factor out
the number 5 as follows:
15
1
1
1 1
$ 20 4 = 1 $$ 4 = 1
4
One is the easiest number
With fractions, the relationship between the
numbers, not the actual numbers themselves, is
most important
...
When you multiply or divide any number by 1,
the answer is the same number
...
In other words,
the fractions 2⁄2, 3⁄3, and 4⁄4 are all equal to 1
...
But all you’ve done
is multiply the fraction by 1, so the value of the
fraction hasn’t changed
...
Similarly, reducing the fraction 6⁄9 by a factor of
3 is the same as dividing that fraction by 3⁄3
(which is equal to 1):
6 ' 3 = 6'3 = 2
9 3 9'3 3
So 6⁄9 is equal to 2⁄3
...
In fact, when you divide
fractions, you really turn the problem into multiplication
...
(As I discuss in Chapter 9, the reciprocal of a fraction is simply
that fraction turned upside down
...
After that, just multiply the fractions as I
describe in “Multiplying numerators and denominators straight across”:
1 5 =1$5 = 5
3 $ 4 3 $ 4 12
As with multiplication, in some cases you may have to reduce your result at the
end
...
(See
the preceding section
...
If they’re the
same — woohoo! Adding fractions that have the same denominator is a walk
in the park
...
To make matters worse, many teachers make adding fractions even more difficult by requiring you to use a long and complicated method when, in many
cases, a short and easy one will do
...
Then I show you a foolproof method for adding fractions when the
denominators are different
...
After that, I show you a quick method that you can use only for certain
problems
...
Chapter 10: Parting Ways: Fractions and the Big Four Operations
Finding the sum of fractions
with the same denominator
To add two fractions that have the same denominator (bottom number), add
the numerators (top numbers) together and leave the denominator unchanged
...
Why does this work? Chapter 9 tells you that you can think about fractions
as pieces of cake
...
So when you add 1⁄5 + 2⁄5, you’re really adding one
piece plus two pieces
...
Even if you have to add more than two fractions, as long as the denominators
are all the same, you just add the numerators and leave the denominator
unchanged:
1 + 3 + 4 + 6 = 1 + 3 + 4 + 6 = 14
17 17 17 17
17
17
Sometimes, when you add fractions with the same denominator, you may
have to reduce it to lowest terms (to find out more about reducing, flip to
Chapter 9)
...
You
get a numerator that’s larger than the denominator when the two fractions
add up to more than 1, as in this case:
3+5=8
7 7 7
If you have more work to do with this fraction, leave it as an improper fraction so that it’s easier to work with
...
Adding fractions with different
denominators
When the fractions that you want to add have different denominators, adding
them isn’t quite as easy
...
Now, I’m shimmying out onto a brittle limb here, but this needs to be said:
There’s a very simple way to add fractions
...
It makes adding
fractions only a little more difficult than multiplying them
...
So why doesn’t anybody talk about it? I think it’s a clear case of tradition
being stronger than common sense
...
But
generation after generation has been taught that it’s the right way to add fractions
...
But in this book, I’m breaking with tradition
...
Then I show you a quick trick that works in a few special cases
...
Using the easy way
At some point in your life, I bet some teacher somewhere told you these golden
words of wisdom: “You can’t add two fractions with different denominators
...
Cross-multiply the two fractions and add the results together to get
the numerator of the answer
...
To get the numerator of
the answer, cross-multiply
...
Multiply the two denominators together to get the denominator of the
answer
...
3
...
1 + 2 = 11
3 5 15
As you discover in the earlier section “Finding the sum of fractions with the
same denominator,” when you add fractions, you sometimes need to reduce
the answer that you get
...
So try dividing both numbers by 2:
74 ' 2 = 37
80 ' 2 40
This fraction can’t be reduced further, so 37⁄40 is the final answer
...
But if this is your final answer, you may
need to turn it into a mixed number (see Chapter 9 for details)
...
The method is
similar, with one small tweak
...
Start out by multiplying the numerator of the first fraction by the
denominators of all the other fractions
...
Do the same with the second fraction and add this value to the first
...
Do the same with the remaining fraction(s)
...
4
...
In this example, you just need to change to a mixed
number (see Chapter 9 for details):
117 = 117 ' 70 = 1 r 47 = 1 47
70
70
Trying a quick trick
I show you a way to add fractions with different denominators in the preceding
section
...
So why do I want to show you another
way? It feels like déjà vu
...
You can’t always use this method, but you can use it when one
denominator is a multiple of the other
...
)
Look at the following problem:
11 + 19
12 24
First, I solve it the way I show you in the preceding section:
11 + 19 = 11 $ 24 + 19 $ 12 = 264 + 228 = 492
12 24
12 $ 24
288
288
Those are some big numbers, and I’m still not done because the numerator is
larger than the denominator
...
Worse yet,
the numerator and denominator are both even numbers, so the answer still
needs to be reduced
...
The trick is to turn a problem with different denominators into a much
easier problem with the same denominator
...
If it is, you can use the quick trick:
1
...
Look at the earlier problem in this new way:
11 + 19
12 24
As you can see, 12 divides into 24 without a remainder
...
To fill in the
question mark, the trick is to divide 24 by 12 to find out how the denominators are related; then multiply the result by 11:
? = (24 ÷ 12) ⋅ 11 = 22
So ⁄12 = 22⁄24
...
Rewrite the problem, substituting this increased version of the fraction, and add as I show you earlier in “Finding the sum of fractions
with the same denominator
...
The answer here is an improper fraction; changing it into a
mixed number is easy:
41 = 41 ' 24 = 1 r 17 = 1 17
24
24
Relying on the traditional way
In the two preceding sections, I show you two ways to add fractions with different denominators
...
So
why do I want to show you yet a third way? It feels like déjà vu all over again
...
But they’re forcing me to
...
The ones who want to keep you down in the mud, groveling at
their feet
...
But let me impress on you that
you don’t have to add fractions this way unless you really want to (or unless
your teacher insists on it)
...
Find the least common multiple (LCM) of the two denominators (for
more on finding the LCM of two numbers, see Chapter 8)
...
First find the LCM of the
two denominators, 4 and 10
...
2
...
Increase each fraction to higher terms so that the denominator of each
is 20
...
Substitute these two new fractions for the original ones and add as I
show you earlier in “Finding the sum of fractions with the same
denominator
...
1
...
This time, I use the prime factorization method (see Chapter 8 for details
on how to do this)
...
Each prime factor appears only once in any decomposition, so the
LCM of 6, 10, and 15 is
2 ⋅ 3 ⋅ 5 = 30
Chapter 10: Parting Ways: Fractions and the Big Four Operations
2
...
Simply add the three new fractions:
25 + 9 + 4 = 38
30 30 30 30
Again, you need to change this improper fraction to a mixed number:
38 = 38 ' 30 = 1 r 8 = 1 8
30
30
Because both numbers are divisible by 2, you can reduce the fraction:
1 8 =1 4
30
15
Pick your trick: Choosing the best method
As I say earlier in this chapter, I think that the traditional way to add fractions
is more difficult than either the easy way or the quick trick
...
But given the choice, here’s my recommendation:
ߜ Use the easy way when the numerators and denominators are small
(say, 15 or under)
...
ߜ Use the traditional way only when you can’t use either of the other methods (or when you know the LCM just by looking at the denominators)
...
As with
addition, when the denominators are the same, subtraction is easy
...
So to figure out how to subtract fractions, you can read the section “All
Together Now: Adding Fractions” and substitute a minus sign (–) for every
plus sign (+)
...
So in
this section, I show you four ways to subtract fractions that mirror what I discuss earlier in this chapter about adding them
...
When the denominators are the same, you can just think of the fractions as pieces of cake
...
For
example:
3 - 2 = 3-2 = 1
5 5
5
5
Sometimes, as when you add fractions, you may have to reduce:
3 - 1 = 3-1= 2
10 10
10
10
Because the numerator and denominator are both even, you can reduce this
fraction by a factor of 2:
2 = 2'2 = 1
10 10 ' 2 5
Unlike addition, when you subtract one proper fraction from another, you
never get an improper fraction
...
These three methods are similar to the methods I show you for adding
fractions: the easy way, the quick trick, and the traditional way
...
The quick trick is a great timesaver, so use it
when you can
...
Knowing the easy way
This way of subtracting fractions works in all cases, and it’s easy
...
) Here’s the easy way to subtract fractions that have
different denominators:
Chapter 10: Parting Ways: Fractions and the Big Four Operations
1
...
For example, suppose you want to subtract 6⁄7 – 2⁄5
...
(The
first number is the numerator of the first fraction times the denominator
of the second
...
Multiply the two denominators together to get the denominator of the
answer
...
Putting the numerator over the denominator gives you your answer
...
When they’re larger, you may be
able to take a shortcut
...
If it is, you can use the quick trick:
145
146
Part III: Parts of the Whole: Fractions, Decimals, and Percents
1
...
For example, suppose you want to find 17⁄20 – 31⁄80
...
But fortunately, 80 is a multiple of 20, so you can use the quick way
...
2
...
”
Here’s the problem as a subtraction of fractions with the same denominator, which is much easier to solve:
68 - 31 = 37
80 80 80
In this case, you don’t have to reduce to lowest terms, although in other
problems you may have to
...
)
Keeping your teacher happy with the traditional way
As I describe earlier in this chapter in “All Together Now: Adding Fractions,”
you should use the traditional way only as a last resort
...
To use the traditional way to subtract fractions with two different denominators, follow these steps:
1
...
For example, suppose you want to subtract 7⁄8 – 11⁄14
...
So the LCM of 8 and 14 is
2 ⋅ 2 ⋅ 2 ⋅ 7 = 56
Chapter 10: Parting Ways: Fractions and the Big Four Operations
2
...
The denominators of both should be 56:
7 = 7 $ 7 = 49
8 8 $ 7 56
11 = 11 $ 4 = 44
14 14 $ 4 56
3
...
”
49 - 44 = 5
56 56 56
This time, you don’t need to reduce, because 5 is a prime number and 56
isn’t divisible by 5
...
Working Properly with Mixed Numbers
All the methods I describe earlier in this chapter work for both proper and
improper fractions
...
(For
more on mixed numbers, flip to Chapter 9
...
The only way is to convert the mixed numbers to improper fractions and
multiply or divide as usual
...
Convert all mixed numbers to improper fractions (see Chapter 9 for
details)
...
First convert 13⁄5 and
21⁄3 to improper fractions:
13 = 5$1+3 = 8
5
5
5
21 = 3$2+1= 7
3
3
3
147
148
Part III: Parts of the Whole: Fractions, Decimals, and Percents
2
...
8 7 = 8 $ 7 = 56
5 $ 3 5 $ 3 15
3
...
56 = 56 ' 15 = 3 r 11 = 3 11
15
15
In this case, the answer is already in lowest terms, so you don’t have to
reduce it
...
1
...
Divide these improper fractions
...
Convert the answer to a mixed number
...
Doing so is a perfectly valid way of getting the right
answer without learning a new method
...
The good news is that a lot of folks find
this way easier than all the converting stuff
...
For this reason, some
students feel more comfortable adding mixed numbers than adding fractions
...
Add the fractional parts using any method you like, and if necessary,
change this sum to a mixed number and reduce it
...
If the answer you found in Step 1 is an improper fraction, change it to
a mixed number, write down the fractional part, and carry the whole
number part to the whole number column
...
Add the whole number parts (including any number carried)
...
In
the examples that follow, I show you everything you need to know
...
For example, suppose you want to add 31⁄3 + 51⁄3
...
Here’s how you add these two
mixed numbers step by step:
1
...
1+1=2
3 3 3
2
...
Because 2⁄3 is a proper fraction, you don’t have to change it
...
Add the whole number parts
...
In this case, all three steps are
pretty easy
...
For example, suppose you want to add 83⁄5 + 64⁄5
...
Add the fractions
...
Switch improper fractions to mixed numbers, write down the fractional part, and carry over the whole number
...
Write down 2⁄5 and carry the 1 over to the wholenumber column
...
Add the whole number parts, including any whole numbers you carried
over when you switched to a mixed number
...
(Be sure to line up the
whole numbers in one column and the fractions in another
...
The same basic idea works no matter how many mixed numbers you want to
add
...
Add the fractions
...
Switch improper fractions to mixed numbers, write down the fractional part, and carry over the whole number
...
I recommend doing these calculations on a
piece of scrap paper
...
3
...
2 + 5 + 11 + 3 + 1 = 22
Chapter 10: Parting Ways: Fractions and the Big Four Operations
Here’s how the problem looks after you solve it:
2
54
11 7
38
+ 15
22 2
9
9
9
9
3
Summing up mixed numbers when the denominators are different
The most difficult type of mixed number addition is when the denominators
of the fractions are different
...
For example, suppose you want to add 163⁄5 and 77⁄9
...
Add the fractions
...
You can use any method from earlier in this chapter
...
Switch improper fractions to mixed numbers, write down the fractional part, and carry over the whole number
...
Fortunately, the fractional part of this mixed number isn’t reducible
...
3
...
1 + 16 + 7 = 24
Here’s how the completed problem looks:
1
16 3 5
+ 77 9
24 17 45
Subtracting mixed numbers
The basic way to subtract mixed numbers is close to the way you add them
...
Here’s how to subtract two mixed numbers:
1
...
2
...
151
152
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Along the way, though, you may encounter a couple more twists and turns
...
Taking away mixed numbers when the denominators are the same
As with addition, subtraction is much easier when the denominators are the
same
...
Here’s what the
problem looks like in column form:
73 5
- 31 5
42 5
In this problem, I subtract 3⁄5 – 1⁄5 = 2⁄5
...
Not too terrible,
agreed?
One complication arises when you try to subtract a larger fractional part
from a smaller one
...
This time, if you try to
subtract the fractions, you get
⁄6 – 5⁄6 = –4⁄6
1
Obviously, you don’t want to end up with a negative number in your answer
...
This
idea is very similar to the borrowing that you use in regular subtraction, with
one key difference
...
Borrow 1 from the whole-number portion and add it to the fractional
portion, turning the fraction into a mixed number
...
Change this new mixed number into an improper fraction
...
This answer is a weird cross between a mixed number
and an improper fraction, but it’s what you need to handle the job
...
Use the result in your subtraction
...
Fortunately,
though, if you work through this chapter, you have all the skills you need
...
Because the denominators are different, subtracting the fractions becomes more difficult
...
But if 4⁄11 is less than 3⁄7, you do
...
)
In Chapter 9, I show you how to test two fractions to see which is greater by
cross-multiplying:
4
11
4 ⋅ 7 = 28
3
7
3 ⋅ 11 = 33
Because 28 is less than 33, 4⁄11 is less than 3⁄7, so you do have to borrow
...
(They can’t be reduced
because 72 and 77 have no common factors: 72 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3, and 77 = 7 ⋅ 11
...
Take a look over it step by step
...
If you get
stuck, that’s okay
...
That’s why numbers come in ones,
tens, hundreds, thousands, and so on
...
Here’s some lovely news: Decimals are much easier to work with than fractions (which I discuss in Chapters 9 and 10)
...
Performing the Big Four operations — addition, subtraction, multiplication,
and division — on decimals is very close to performing them on whole numbers (which I cover in Part II of the book)
...
As long as you get the decimal point in the right
place, you’re home free
...
I also show you
how to convert fractions to decimals and decimals to fractions
...
156
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Basic Decimal Stuff
The good news about decimals is that they look a lot more like whole numbers
than fractions do
...
In this section, I introduce you to decimals, starting with place value
...
Then I
discuss trailing zeros, and what happens when you move the decimal point
either to the left or to the right
...
And a great way to
begin thinking about decimals is as dollars and cents
...
50 is half of a dollar (see Figure 11-1), so this information tells you:
0
...
5) of a
dollar bill
...
5, I drop the zero at the end
...
You also know that $0
...
25 = 1⁄4
Chapter 11: Dallying with Decimals
1
448B
Figure 11-2:
One-fourth
(0
...
6
Similarly, you know that $0
...
75 = 3⁄4
1
F285937448B
6
Figure 11-3:
Three6
fourths
937448B
(0
...
6
WA S H I N G T O N
Taking this idea even further, you can use the remaining denominations of
coins — dimes, nickels, and pennies — to make further connections between
decimals and fractions
...
10 = 1⁄10 of a dollar, so 1⁄10 = 0
...
05 = 1⁄20 of a dollar, so 1⁄20 = 0
...
01 = 1⁄100 of a dollar, so 1⁄100 = 0
...
1, but that I keep the zeros in
the decimals 0
...
01
...
157
158
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Decimals are just as good for cutting up cake as for cutting up money
...
This time, I give you the decimals that tell you how much cake
you have
...
A
...
Figure 11-4:
Cakes cut
and shaded B
...
75 (A),
...
1 (C),
and
...
D
...
For example, Table 11-1 shows how the whole number 4,672 breaks down in terms
of place value
...
With decimals, this idea is extended
...
Then, more numbers are appended
to the right of the decimal point
...
389 breaks down as shown in Table 11-2
...
389
Thousands Hundreds Tens
Ones
Decimal
Point
Tenths
Hundredths
Thousandths
4
2
...
The connection between fractions and decimals becomes obvious when you
look at place value
...
You can represent any fraction as a decimal
...
Two key ideas are trailing zeros and what happens when you move a decimal
point left or right
...
For example, these three numbers are all
equal in value:
27
027
0,000,027
The reason for this becomes clear when you know about place value of whole
numbers
...
Table 11-3
Example of Attaching Leading Zeros
Millions
Hundred
Ten
Thousands Thousands
Thousands
Hundreds
Tens
Ones
0
0
0
0
2
7
0
159
160
Part III: Parts of the Whole: Fractions, Decimals, and Percents
As you can see, 0,000,027 simply means 0 + 0 + 0 + 0 + 0 + 20 + 7
...
Zeros attached to the beginning of a number in this way are called leading zeros
...
A trailing zero is any zero that appears to the right of both the decimal point
and every digit other than zero
...
8
34
...
8000
All three of these numbers are the same
...
See Table 11-4
...
8
0
0
0
In this example, 34
...
When you understand trailing zeros, you can see that every whole number
can be changed to a decimal easily
...
For example:
4 = 4
...
0
971 = 971
...
For example, look at this number:
0450
...
007
The remaining zeros, however, need to stay where they are as placeholders
between the decimal point and digits other than zero
...
Table 11-5
Example of Zeros as Placeholders
Thousands
Hundreds
Tens
Ones
Decimal Tenths
Point
HunThouTen Thoudredths sandths sandths
0
4
5
0
...
Moving the decimal point
When you’re working with whole numbers, you can multiply any number by
10 just by adding a zero to the end of it
...
Table 11-6
Millions
Example, Decimal Points and Place Value of Digits
Hundred
Thousands
Thousands
Hundreds
Tens
Ones
4
4
Ten
Thousands
5
9
7
1
5
9
7
1
0
Here’s what these two numbers really mean:
45,971 = 40,000 + 5,000 + 900 + 70 + 1
459,710 = 400,000 + 50,000 + 9,000 + 700 + 10 + 0
As you can see, that little zero makes a big difference, because it causes the
rest of the numbers to shift over one place
...
See Table 11-7
...
0
0
5
4
5
9
7
1
0
...
So, for any decimal, when you move the decimal point one
place to the right, you multiply that number by 10
...
0
70
...
0
7,000
...
Similarly, to divide any number by 10, move the decimal point one place to
the left
...
0
0
...
07
0
...
Rounding decimals
Rounding decimals works almost exactly the same as rounding numbers
...
Most
Chapter 11: Dallying with Decimals
commonly, you need to round a decimal either to a whole number or to one
or two decimal places
...
Round the decimal either up or down to the nearest whole number,
dropping the decimal point:
7
...
9 -> 33
184
...
5 -> 84
296
...
5 -> 1,789
If the decimal has other decimal digits, just drop them:
18
...
618 -> 22
3
...
(This
may remind you of when the odometer in your car rolls a bunch of 9s over to
0s):
99
...
5 -> 1,000
99,999
...
For example, to round a decimal to one decimal place, focus on the first and
second decimal places (that is, the tenths and hundredths places):
76
...
5
100
...
7
10
...
1
To round a decimal to two decimal places, focus on the second and third decimal places (that is, the hundredths and thousandths places):
444
...
44
26
...
56
99
...
00
Performing the Big Four with Decimals
Everything you already know about adding, subtracting, multiplying, and
dividing whole numbers (see Chapter 3) carries over when you work with
decimals
...
In this section, I show you how to perform the Big Four math operations with decimals
...
Later in this
book, you find that multiplying and dividing by decimals is useful for calculating percentages (see Chapter 12), using scientific notation (see Chapter 14),
and measuring with the metric system (see Chapter 15)
...
As long as you
set up the problem correctly, you’re in good shape
...
Line up the decimal points
...
Add as usual from right to left, column by column
...
Place the decimal point in the answer in line with the other decimal
points in the problem
...
5 and 1
...
Line up the
decimal points neatly as follows:
14
...
89
Begin adding from the right-hand column
...
5 as
a 0 — you can write this in as a trailing 0 (see earlier in this chapter to see
why adding zeros to the end of a decimal doesn’t change its value)
...
50
+ 1
...
50
+ 1
...
50
+ 1
...
39
When adding more than one decimal, the same rules apply
...
1 + 0
...
2345
...
1
0
...
0
+ 1
...
Because the number 800 isn’t a decimal, I place a decimal point and a 0 at the
end of it to be clear about how to line it up
...
After you properly set up the problem, the addition is
no more difficult than in any other addition problem:
15
...
0050
800
...
2345
816
...
Here’s how you subtract decimals:
1
...
2
...
3
...
165
166
Part III: Parts of the Whole: Fractions, Decimals, and Percents
For example, suppose you want to figure out 144
...
321
...
870
– 0
...
This placeholder
reminds you that in the right-hand column, you need to borrow to get the
answer to 0 – 1:
6
1 4 4
...
3 2 1
4 9
The rest of the problem is very straightforward
...
8 7 10
–
0
...
5 4 9
As with addition, the decimal point in the answer goes directly below where
it appears in the problem
...
In fact, the only difference between multiplying whole numbers
and decimals comes at the very end
...
Perform the multiplication as you would for whole numbers
...
When you’re done, count the number of digits to the right of the decimal point in each factor and add the result
...
Place the decimal point in your answer so that your answer has the
same number of digits after the decimal point
...
Suppose, for instance, you want to multiply
23
...
16
...
5
× 0
...
To do this, notice that 23
...
16 has two digits after the decimal point
...
(You can put your pencil at the 0 at the end of 3760 and
move the decimal point three places to the left
...
5
× 0
...
760
1 + 2 = 3 digits after the decimal point
Even though the last digit in the answer is a 0, you still need to count this as a
digit when placing the decimal point
...
So the answer is 3
...
76
...
Dividing decimals is almost
the same as dividing whole numbers, which is why lots of people don’t particularly like dividing decimals, either
...
The main difference comes at the beginning, before you start
dividing
...
Turn the divisor (the number you’re dividing by) into a whole
number by moving the decimal point all the way to the right; at the
same time, move the decimal point in the dividend (the number
you’re dividing) the same number of places to the right
...
274 by 0
...
Write the problem as usual:
0
...
274
Turn 0
...
11 two
places to the right, giving you 11
...
274 two places to the right, giving you 1,027
...
g1027
...
Place a decimal point in the quotient (the answer) directly above
where the decimal point now appears in the dividend
...
11
...
4
3
...
To start out, notice that 11 is too large to go into either 1 or 10
...
So write the first digit of the quotient just
above the 2 and continue:
9
...
g1027
...
This time, 11 goes into
37 three times
...
11
...
4
99
37
33
44
Chapter 11: Dallying with Decimals
I left off after bringing down the next number, 4
...
Again, be careful to place the next digit in the quotient just
above the 4, and complete the division:
93
...
g1027
...
4
...
Dealing with more zeros in the dividend
Sometimes, you may have to add one or more trailing zeros to the dividend
...
For example, suppose you want
to divide 67
...
333:
0
...
8
1
...
333 into a whole number by moving the decimal point
three places to the right; at the same time, move the decimal point in
67
...
g 67800
...
8, you run out of
room, so you have to add a couple of zeros to the dividend
...
2
...
333
...
3
...
This time, 333 doesn’t go into 6 or 67, but it does go into
678 (two times)
...
333
...
666
120
169
170
Part III: Parts of the Whole: Fractions, Decimals, and Percents
I’ve jumped forward in the division to the place where I bring down the
first 0
...
Now, 333 does
go into 1,200, so place the next digit in the answer (3) over the second 0:
203
...
g 67800
...
If this were a problem with
whole numbers, you’d finish by writing down a remainder of 201
...
) But decimals are a different story
...
Completing decimal division
When you’re dividing whole numbers, you can complete the problem simply
by writing down the remainder
...
A common way to complete a problem in decimal division is to round off the
answer
...
)
To complete a decimal division problem by rounding it off, you need to add
at least one trailing zero to the dividend
...
333
...
0
666
1200
999
2010
Chapter 11: Dallying with Decimals
Attaching a trailing zero doesn’t change a decimal, but it does allow you to
bring down one more number, changing 201 into 2,010
...
6
333
...
0
666
1200
999
2010
1998
12
At this point, the you can round the answer to the nearest whole number, 204
...
Converting between Decimals
and Fractions
Fractions (see Chapters 9 and 10) and decimals are similar in that they both
allow you to represent parts of the whole — that is, these numbers fall on the
number line between whole numbers
...
For example, calculators love decimals, but aren’t so crazy about fractions
...
As another example, some units of measurement (such as inches) use fractions while others (such as meters) use decimals
...
In this section, I show you how to convert back and forth between fractions
and decimals
...
)
Making simple conversions
Some decimals are so common that you should memorize how to represent
them as fractions
...
1
...
3
...
5
...
7
...
9
3/10
2/5
1/2
3/5
7/10
4/5
9/10
And, here are few more common decimals that translate easily to fractions:
...
25
...
625
...
875
1/8
1/4
3/8
5/8
3/4
7/8
Changing decimals to fractions
Converting a decimal to a fraction is pretty simple
...
In this section, I first show you the easy case, when no further work is necessary
...
I also show you a great time-saving trick
...
Draw a line (fraction bar) under the decimal and place a 1 underneath it
...
3763 into a fraction
...
3763 and place a 1 underneath it:
0
...
2
...
3
...
Repeat Step 2 until the decimal point moves all the way to the right,
so you can drop the decimal point entirely
...
63 = 376
...
Chapter 11: Dallying with Decimals
Note: Moving a decimal point one place to the right is the same thing as
multiplying a number by 10
...
3763 and the 1 by
10,000
...
4
...
The fraction 3,763⁄10,000 may seem like a large number, but it’s smaller than 1
because the numerator is less than the denominator (bottom number)
...
This
fraction is also in the lowest possible terms (you can’t reduce it — see
Chapter 9), so this problem is complete
...
On better terms: Mixing numbers and reducing fractions
In some cases, you may have to reduce a fraction to lowest terms after you
convert it (see Chapter 9 for more on reducing fractions)
...
When you’re converting a decimal to a fraction, if the decimal ends in an even
number or in 5, you can reduce the fraction; otherwise, you can’t
...
Suppose you want to convert the decimal 12
...
Do the following:
1
...
16 and place a 1 underneath it:
12
...
Move the decimal point one place to the right and add a 0 after the 1:
121
...
Repeat Step 2:
1, 216
100
173
174
Part III: Parts of the Whole: Fractions, Decimals, and Percents
4
...
This time, the numerator is greater than the denominator, so
1, 216
100
is an improper fraction and must be changed to a mixed number
...
In this case, the original decimal was 12
...
Doing the following is perfectly okay:
1, 216
= 12 16
100
100
This trick works only when you’ve changed a decimal to a fraction — don’t
try it with other fractions, or you’ll get a wrong answer
...
If you need to get up to speed on this, check out
“Dividing decimals” earlier in this chapter
...
Set up the fraction as a decimal division, dividing the numerator (top
number) by the denominator (bottom number)
...
Attach enough trailing zeros to the numerator so that you can continue
dividing until you find that the answer is either a terminating decimal or
a repeating decimal
...
Chapter 11: Dallying with Decimals
The last stop: Terminating decimals
Sometimes, when you divide the numerator of a fraction by the denominator,
the division eventually works out evenly
...
For example, suppose you want to change the fraction 2⁄5 to a decimal
...
But watch what happens when I add a few trailing zeros
...
This step is important — you can read more
about it in “Dividing decimals”:
...
000
Now you can divide because although 5 doesn’t go into 2, 5 does go into 20
four times:
0
...
000
20
0
You’re done! As it turns out, you only needed one trailing zero, so you can
ignore the rest:
2 = 0
...
As another example, suppose you want to find out how to represent 7⁄16 as a
decimal
...
437
16 g 7
...
4375
16 g 7
...
Therefore, 7⁄16 = 0
...
The endless ride: Repeating decimals
Sometimes when you try to convert a fraction to a decimal, the division never
works out evenly
...
You may recognize these pesky little critters from your calculator, when an
apparently simple division problem produces a long string of numbers
...
As in the last
section, start out by adding three trailing zeros and see where it leads:
0
...
000
18
20
18
20
18
2
At this point, you still haven’t found an exact answer
...
No matter how many trailing zeros you attach to the number 2, the same pattern will continue forever
...
666
...
You can write 2⁄3 as
2/3 = 0
...
You can represent many simple fractions as repeating decimals
...
Now suppose you want to find the decimal representation of 5⁄11
...
4545
11 g 5
...
Attaching more trailing zeros to the original decimal
will only string out this pattern indefinitely
...
45
This time, the bar is over both the 4 and the 5, telling you that these two
numbers alternate forever
...
In fact,
as soon as you can show that a decimal division is repeating, you’ve found
your answer
...
Some decimals never end and never repeat
...
177
178
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Chapter 12
Playing with Percents
In This Chapter
ᮣ Understanding what percents are
ᮣ Converting percents back and forth between decimals and fractions
ᮣ Solving both simple and difficult percent problems
ᮣ Using the percent circle to solve three different types of percent problems
L
ike whole numbers and decimals, percents are a way to talk about parts
of a whole
...
” So if you have 50% of
something, you have 50 out of 100
...
Of course, if you have 100% of anything, you have all of it
...
Because percents
resemble decimals, I first show you how to convert numbers back and forth
between percents and decimals
...
Next, I show you how to convert back and forth between percents and
fractions — also not too bad
...
Making Sense of Percents
The word percent literally means “for 100,” but in practice, it means closer to
“out of 100
...
You can say that “50 out of 100” children are girls — or you
can shorten it to simply “50 percent
...
Saying that 50% of the students are girls is the same as saying that 1⁄2 of them
are girls
...
5 of all
the students are girls
...
In this case,
the whole is the total number of children in the school
...
You probably won’t ever really cut a cake into 100 pieces, but that doesn’t matter
...
Whether you’re talking about cake, a dollar, or a group
of children, 50% is still half, 25% is still one-quarter, and 75% is still threequarters, and so on
...
You probably know this fact well from the
school grading system
...
And 90% is
usually A work, 80% is a B, 70% is a C, and, well, you know the rest
...
Dealing with Percents
Greater than 100%
100% means “100 out of 100” — in other words, everything
...
What about percentages more than 100%? Well, sometimes percentages like
these don’t make sense
...
But lots of times, percentages larger than 100% are perfectly reasonable
...
That’s three times as many
...
Spend a little time thinking about this example until it makes sense
...
Chapter 12: Playing with Percents
Converting to and from Percents,
Decimals, and Fractions
To solve many percent problems, you need to change the percent to either a
decimal or a fraction
...
That’s why I show you how to convert to and
from percents before I show you how to solve percent problems
...
This similarity makes converting percents to decimals and vice versa mostly
a matter of moving the decimal point
...
Percents and fractions both express the same idea — parts of a whole — in
different ways
...
In this
section, I cover the ways to convert to and from percents, decimals, and fractions, starting with percents to decimals
...
That’s all there is to it
...
For example,
2
...
025
4% = 0
...
36
111% = 1
...
07 = 7%
0
...
375 = 37
...
Remember that the
word percent means “out of 100
...
To convert a percent to a fraction, use the number in the percent as your
numerator (top number) and the number 100 as your denominator (bottom
number):
39% = 39
100
86% = 86
100
217% = 217
100
As always with fractions, you may need to reduce to lowest terms or convert
an improper fraction to a mixed number (flip to Chapter 9 for more on these
topics)
...
However, 86⁄100 can be reduced because the numerator and denominator are
both even numbers:
86 = 43
100 50
And 217⁄100 can be converted to a mixed number because the numerator (217) is
greater than the denominator (100):
217 = 2 17
100
100
Once in a while, you may start out with a percentage that’s a decimal such as
99
...
The rule is still the same, but now you have a decimal in the numerator (top number), which most people don’t like to see
...
9% = 99
...
9% converts to the fraction 999⁄1,000
...
Here’s how to
convert a fraction to a percent:
Chapter 12: Playing with Percents
1
...
For example, suppose you want to convert the fraction 4⁄5 to a percent
...
8
4
2
...
Convert 0
...
0
...
Follow these
steps:
1
...
625
8 g 5
...
625
...
Convert 0
...
625 = 62
...
Others,
however, require a bit more work
...
183
184
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Figuring out simple percent problems
A lot of percent problems turn out to be easy when you give them a little
thought
...
5)
2
2
ߜ Finding 25% of a number: Remember that 25% equals 1⁄4, so to find 25%
of a number, divide it by 4:
25% of 40 is 10
25% of 88 is 22
25% of 15 is 15 (or 3 3 or 3
...
Because 20% equals 1⁄5, you can find 20% of a number by dividing it by 5
...
1 ⋅ 2 = 8
...
To do this, just move the decimal point one
place to the left:
10% of 30 is 3
10% of 41 is 4
...
7
Chapter 12: Playing with Percents
ߜ Finding 200%, 300%, and so on of a number: Working with percents
that are multiples of 100 is easy
...
)
Turning the problem around
Here’s a trick that makes certain tough-looking percent problems so easy
that you can do them in your head
...
Suppose someone wants you to figure out the following:
88% of 50
Finding 88% of anything isn’t an activity that anybody looks forward to
...
As I discuss in the preceding section, “Figuring out simple percent problems,” 50% of
88 is simply half of 88:
88% of 50 = 50% of 88 = 44
As another example, suppose you want to find
7% of 200
Again, finding 7% is tricky, but finding 200% is simple, so switch the problem
around:
7% of 200 = 200% of 7
In the preceding section, I tell you that to find 200% of any number, you just
multiply that number by 2:
7% of 200 = 200% of 7 = 2 ⋅ 7 = 14
185
186
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Deciphering more-difficult
percent problems
You can solve a lot of percent problems using the tricks I show you earlier in
this chapter
...
When
the numbers in a percent problem become a little more difficult, the tricks no
longer work, so you want to know how to solve all percent problems
...
Change the word of to a multiplication sign and the percent to a decimal (as I show you earlier in this chapter)
...
This change
turns something unfamiliar into a form that you know how to work with
...
Here’s how you start:
35% of 80 = 0
...
Solve the problem using decimal multiplication (see Chapter 11)
...
35
# 80
28
...
As another example, suppose you want to find 12% of 31
...
12 ⋅ 31
Now you can solve the problem with decimal multiplication:
0
...
72
So 12% of 31 is 3
...
Chapter 12: Playing with Percents
Putting All the Percent
Problems Together
In the preceding section, “Solving Percent Problems,” I give you a few ways to
find any percent of any number
...
But percents are commonly used in a wide range of business applications
such as banking, real estate, payroll, and taxes
...
) And depending
on the situation, two other common types of percent problems may present
themselves
...
I also give you a
simple tool to make quick work of all three types
...
(See “Solving Percent Problems” for details on
how to get this answer
...
Now suppose instead that I leave out the percent but give you the starting
and ending numbers:
? % of 2 is 1
You can still fill in the blank without too much trouble
...
187
188
Part III: Parts of the Whole: Fractions, Decimals, and Percents
If you get this basic idea, you’re ready to solve percent problems
...
Table 12-1
The Three Main Types of Percent Problems
Problem Type
What to Find
Example
Type #1
The ending number
50% of 2 is what?
Type #2
The percentage
What percent of 2 is 1?
Type #3
The starting number
50% of what is 1?
In each case, the problem gives you two of the three pieces of information, and
your job is to figure out the remaining piece
...
Introducing the percent circle
The percent circle is a simple visual aid that helps you make sense of percent
problems so that you can solve them easily
...
For example, Figure 12-1 shows how to
record the information that 50% of 2 is 1
...
1
0
...
5 (for more on changing percents to decimals, see
“Going from percents to decimals” earlier in this chapter)
...
5 ⋅ 2 = 1
ߜ If you make a fraction out of the top number and either bottom number,
that fraction equals the other bottom number:
1 =2
1 = 0
...
5
These features are the heart and soul of the percent circle
...
Most percent problems give you enough information to fill in two of the three
sections of the percent circle
...
Finding the ending number
Suppose you want to find out the answer to this problem:
What is 75% of 20?
You’re given the percent and the starting number and asked to find the ending
number
...
Figure 12-2:
Putting 75%
of 20 into a
percent
circle
?
0
...
75 and 20 are both bottom numbers in the circle, multiply them to
get the answer:
0
...
00
So 75% of 20 is 15
...
You still use multiplication to get your
answer, but with the percent circle, you’re less likely to get confused
...
Here’s an example:
What percent of 50 is 35?
In this case, the starting number is 50 and the ending number is 35
...
Figure 12-3:
Determining
what
percent of
50 is 35
...
” First, convert 35⁄50 to a decimal:
0
...
0
350
0
Now convert 0
...
7 = 70%
Tracking down the starting number
In the third type of problem, you get the percentage and the ending number,
and you have to find the starting number
...
Figure 12-4:
Working out
the answer
to 15%
of what
number
is 18?
18
0
...
15 in the circle, make a fraction out of these two
numbers:
18
0
...
^
...
g1800
...
191
192
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Chapter 13
Word Problems with Fractions,
Decimals, and Percents
In This Chapter
ᮣ Adding and subtracting fractions, decimals, and percents in word equations
ᮣ Translating the word of as multiplication
ᮣ Changing percents to decimals in word problems
ᮣ Tackling business problems involving percent increase and decrease
I
n Chapter 6, I show you how to solve word problems (also known as story
problems) by setting up word equations that use the Big Four operations
(adding, subtracting, multiplying, and dividing)
...
First, I show you how to solve relatively easy problems, in which all you need
to do is add or subtract fractions, decimals, or percents
...
Such problems
are easy to spot because they almost always contain the word of
...
Finally, I show you how to handle
problems of percent increase and decrease
...
Adding and Subtracting Parts of
the Whole in Word Problems
Certain word problems involving fractions, decimals, and percents are really
just problems in adding and subtracting
...
(In Chapter 15, I discuss these
applications in depth
...
Sharing a pizza: Fractions
You may have to add or subtract fractions in problems that involve splitting
up part of a whole
...
What fraction of the
pizza was left when they were finished?
In this problem, just jot down the information that’s given as word equations:
Joan = 1
6
Tony = 1
4
Sylvia = 1
3
These fractions are part of one total pizza
...
Here’s one way:
All three = 2 + 3 + 4 = 9 = 3
12 12 12 12 4
However, the question asks what fraction of the pizza was left after they finished, so you have to subtract that amount from the whole:
1- 3 = 1
4 4
Thus, the three people left 1⁄4 of a pizza
...
The following problem
Chapter 13: Word Problems with Fractions, Decimals, and Percents
requires you to add and subtract decimals, which I discuss in Chapter 11
...
53 pounds of beef and 3
...
Lance
bought 5
...
7 pounds of pork
...
53 + 3
...
63
Lance = 5
...
7 = 5
...
To find how much
more, subtract:
7
...
94 = 1
...
69 pounds more than Lance
...
In real life, you may see such
info organized as a pie chart (which I discuss in Chapter 17)
...
Here’s an example:
In a recent mayoral election, five candidates were on the ballot
...
What percentage of voters wrote in their selection?
The candidates were in a single election, so all the votes have to total 100%
...
Then subtract that
value from 100%:
39% + 31% + 18% + 7% + 3% = 98%
100% – 98% = 2%
Because 98% of voters voted for one of the five candidates, the remaining 2%
wrote in their selections
...
So whenever you see the word of following a fraction, decimal, or percent, you can
usually replace it with a times sign
...
For example, when you point to an item in a store and say, “I’ll
take three of those,” in a sense you’re saying, “I’ll take that one multiplied by
three
...
When you divide up a single thing — such as one pizza or one death-bychocolate cake — the word of still means to multiply; you’re technically
multiplying each fraction by 1
...
Because anything times 1 is
itself, you don’t have to write the 1 at all — you can just add the fractions, as
I do earlier in “Sharing a pizza: Fractions
...
For instance, you can figure out how much
you’ll spend if you don’t buy food in the quantities listed on the signs
...
However, you want to know the cost
...
Use the rules of multiplying
fractions from Chapter 10 and solve:
Chapter 13: Word Problems with Fractions, Decimals, and Percents
= 5 $ $4 = $ 20
8$1
8
This fraction reduces to $5⁄2
...
So convert this fraction to a
decimal using the rules I show you in Chapter 11:
$ 5 = $5 ÷ 2 = $2
...
5 is more commonly written as $2
...
Easy as pie: Working out
what’s left on your plate
Sometimes, when you’re sharing something such as a pie, not everyone gets
to it at the same time
...
When someone takes a part of the leftovers, you can do a bit of multiplication to see how much of the whole pie that portion represents
...
Then his wife Doreen ate 1⁄6 of what was
left
...
He started with a whole pie,
so subtract his portion from 1:
Pie left after Jerry = 1 – 1 = 4
5 5
Next, Doreen ate 1⁄6 of this amount
...
This answer tells you how much of the whole pie Doreen ate:
Doreen = 1 $ 4 = 4
6 5 30
To make the numbers a little smaller before you go on, notice that you can
reduce the fraction:
Doreen = 2
15
197
198
Part III: Parts of the Whole: Fractions, Decimals, and Percents
Now you know how much Jerry and Doreen both ate, so you can add these
amounts together:
Jerry + Doreen = 1 + 2
5 15
Solve this problem as I show you in Chapter 10:
=3 + 2 = 5
15 15 15
This fraction reduces to 1⁄3
...
So finish up with some subtraction and write the answer:
1- 1 = 2
3 3
The amount of pie left over was 2⁄3
...
This idea is also true in word problems involving decimals and percents
...
You can easily solve word problems involving percents by changing the percents into decimals (see Chapter 12 for details)
...
25
50% = 0
...
75
99% = 0
...
Here’s an example:
Chapter 13: Word Problems with Fractions, Decimals, and Percents
Maria’s grandparents gave her $125 for her birthday
...
How much did the dress cost?
Start at the beginning, forming a word equation to find out how much money
Maria put in the bank:
Money in bank = 40% of $125
To solve this word equation, change the percent to a decimal and the word of
to a multiplication sign; then multiply:
Money in bank = 0
...
If you need to work with
the portion that remains, you may have to subtract the amount used from the
amount you started with
...
Again, change the percent to a decimal and the word of to a multiplication
sign:
Shoes = 35% of $75 = 0
...
25
She spent the rest of the money on a dress, so
Dress = $75 – $26
...
75
Therefore, Maria spent $48
...
Finding out how much you started with
Some problems give you the amount that you end up with and ask you to find
out how much you started with
...
Here’s an example, and it’s kind of a
tough one, so fasten your seat belt:
Maria received some birthday money from her aunt
...
How much did her aunt give her?
199
200
Part III: Parts of the Whole: Fractions, Decimals, and Percents
This problem is similar to the one in the preceding section, but you need to
start at the end and work backwards
...
The problem tells you that
she ends up with $12 after two transactions — putting money in the bank and
buying a purse — and asks you to find out how much you started with
...
The first tells you that
Maria took the money from her aunt, subtracted some money to put in the
bank, and left the bank with a new amount of money, which I’m calling money
after bank
...
It tells
you that Maria took the money left over from the bank, subtracted some
money for a purse, and ended up with $12
...
To solve this problem, realize that Maria spent 75% of her money at that time
on the purse — that is, 75% of the money she still had after the bank:
Money after bank – 75% of money after bank = $12
I’m going to make one small change to this equation so you can see what it’s
really saying:
100% of money after bank – 75% of money after bank = $12
Adding 100% of doesn’t change the equation because it really just means
you’re multiplying by 1
...
”
In this particular case, however, these words help you to make a connection
because 100% – 75% = 25%; here’s an even better way to write this equation:
25% ⋅ money after bank = $12
Before moving on, make sure you understand the steps that have brought
you here
...
25 gives
you the remaining amount of $48
...
First, Maria placed 40% of the money from her
aunt in the bank:
Money from aunt – 40% of money from aunt = $48
Again, rewrite this equation to make what it’s saying clearer:
100% of money from aunt – 40% of money from aunt = $48
Now, because 100% – 40% = 60%, rewrite it again:
60% ⋅ money from aunt = $48
At this point, you can use the percent circle, which I show you in Chapter 12,
to solve the equation (see Figure 13-1)
...
6 gives you the remaining amount of $80
...
$48
Figure 13-1:
Money
from aunt =
$48 ÷ 0
...
0
...
Typical percent-increase problems involve calculating the amount of a salary plus a raise, the cost of merchandise plus tax,
or an amount of money plus interest or dividend
...
201
202
Part III: Parts of the Whole: Fractions, Decimals, and Percents
To tell you the truth, you may have already solved problems of this kind earlier
in “Multiplying Decimals and Percents in Word Problems
...
Raking in the dough: Finding
salary increases
A little street smarts should tell you that the words salary increase or raise
means more money, so get ready to do some addition
...
What will she earn this year?
To solve this problem, first realize that Alison got a raise
...
The key to setting up
this type of problem is to think of percent increase as “100% of last year’s
salary plus 5% of last year’s salary
...
In other words,
3(1 + 5) = 3(1) + 3(5)
If you evaluate both sides of the equation, here’s
what you get
...
You may be able to do the
addition or subtraction in your head, and as
soon as you do the multiplication, you have
your answer
...
Chapter 13: Word Problems with Fractions, Decimals, and Percents
Now you can just add the percentages (see the nearby sidebar for why this
works):
This year’s salary = (100% + 5%) of last year’s salary
= 105% of last year’s salary
Change the percent to a decimal and the word of to a multiplication sign;
then fill in the amount of last year’s salary:
This year’s salary = 1
...
Earning interest on top of interest
The word interest means more money
...
And when you pay interest on a loan, you pay
more money
...
Here’s an example:
Bethany placed $9,500 in a one-year CD that paid 4% interest
...
How much
did Bethany earn on her investment in those two years?
This problem involves interest, so it’s another problem in percent increase,
only this time, you have to deal with two transactions
...
The first transaction is a percent increase of 4% on $9,500
...
04 ⋅ $9,500
Multiplication gives you this result:
Money after first year = $9,880
203
204
Part III: Parts of the Whole: Fractions, Decimals, and Percents
At this point, you’re ready for the second transaction
...
06 ⋅ $9,880 = $10,472
...
80 – $9,500 = $972
...
80 on her investment
...
Here’s
an example:
Greg has his eye on a television with a listed price of $2,100
...
What will the television
cost with the discount?
In this problem, you need to realize that the discount lowers the price of the
television, so you have to subtract:
Sale price = 100% of regular price – 30% of regular price
Now subtract percentages:
Sale price = (100% – 30%) of regular price = 70% of regular price
At this point, you can fill in the details, as I show you throughout this chapter:
Sale price = 0
...
Part IV
Picturing and
Measuring —
Graphs, Measures,
Stats, and Sets
I
In this part
...
I discuss two important systems of measurement — the
English system and the metric system — and show you
how to convert from one to the other
...
I provide some quick information on how to read a
variety of graphs and present the Cartesian graph
...
Finally, I show you how to apply these skills to solve
word problems involving measurement and geometry
...
To save on time and space — and to make calculations easier — people
developed a sort of shorthand called scientific notation
...
Each number in the sequence is 10 times more than the preceding number
...
They’re also easy to represent in exponential form (as I show you in
Chapter 4):
100
101
102
103
104
105
106
107
...
It uses both decimals and exponents
(so if you need a little brushing up on decimals, flip to Chapter 11)
...
I also
explain the order of magnitude of a number
...
First Things First: Powers
of Ten as Exponents
Scientific notation uses powers of ten expressed as exponents, so you need
a little background before you can jump in
...
Counting up zeros and writing exponents
Numbers starting with a 1 and followed by only 0s (such 10, 100, 1,000,
10,000, and so forth) are called powers of ten, and they’re easy to represent
as exponents
...
To represent a number that’s a power of 10 as an exponential number, count
the zeros and raise 10 to that exponent
...
Table 14-1 shows a list of some powers of ten
...
For example, a really big number is
a googol, which is 1 followed by a hundred 0s
...
You can
save yourself some trouble and write 10100
...
You can also represent decimals using negative exponents
...
1
10–2 = 0
...
001
10–4 = 0
...
For example, to find the value of 107, start with 1 and make it larger by moving
the decimal point 7 spaces to the right:
107 = 10,000,000
Similarly, to find the value of 10–7, start with 1 and make it smaller by moving
the decimal point 7 spaces to the left:
10–7 = 0
...
In this example, notice that 10–7 has six 0s
between them
...
For example,
10–23 = 0
...
209
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Adding exponents to multiply
An advantage of using the exponential form to represent powers of ten is that
this form is a cinch to multiply
...
Here are a few examples:
ߜ 101 ⋅ 102 = 101+ 2 = 103
Here, I simply multiply these numbers:
10 ⋅ 100 = 1,000
14
ߜ 10 ⋅ 1015 = 1014 + 15 = 1029
Here’s what I’m multiplying:
100,000,000,000,000 ⋅ 1,000,000,000,000,000
= 100,000,000,000,000,000,000,000,000,000
You can verify that this multiplication is correct by counting the 0s
...
In each of these cases, you can think of multiplying powers of ten as adding
extra 0s to the number
...
For example,
103 ⋅ 10–5 = 10(3 – 5) = 10–2 = 0
...
Every number can be written in scientific notation as the product of two numbers (two numbers multiplied
together):
ߜ A decimal greater than or equal to 1 and less than 10 (see Chapter 11 for
more on decimals)
ߜ A power of ten written as an exponent (see the preceding section)
Chapter 14: A Perfect Ten: Condensing Numbers with Scientific Notation
Writing in scientific notation
Here’s how to write any number in scientific notation:
1
...
Suppose you want to change the number 360,000,000 to scientific notation
...
0
2
...
Move the decimal point to the right or left so that only one nonzero digit
comes before the decimal point
...
Using 360,000,000
...
So
move the decimal point eight places to the left, drop the trailing zeros,
and get 3
...
0 becomes 3
...
Multiply the new number by 10 raised to the number of places you
moved the decimal point in Step 2
...
6 ⋅ 108
4
...
You moved the decimal point to the left, so you don’t have to take any
action here
...
6 ⋅ 108
...
For example, suppose you want to change the number 0
...
Write 0
...
00006113
2
...
00006113 to a new number between 1 and 10, move the
decimal point five places to the right and drop the leading zeros:
6
...
Because you moved the decimal point five places, multiply the new
number by 105:
6
...
Because you moved the decimal point to the right, put a minus sign on
the exponent:
6
...
00006113 in scientific notation is 6
...
After you get used to writing numbers in scientific notation, you can do it all
in one step
...
7400 $ 10 4
212
...
1204 $ 10 2
0
...
002 $ 10 - 3
Why scientific notation works
After you understand how scientific notation works, you’re in a better position to understand why it works
...
First of all, you can multiply any number by 1 without changing it, so
here’s a valid equation:
4,500 = 4,500 ⋅ 1
Because 4,500 ends in a 0, it’s divisible by 10 (see Chapter 7 for info on divisibility)
...
At this
point, you have no more 0s to drop, but you can continue the pattern by
moving the decimal point one place to the left:
4,500 = 4
...
45 ⋅ 10,000
4,500 = 0
...
But you can just as easily move the
decimal point one place to the right and multiply by 0
...
01, and three places right by multiplying by 0
...
1
4,500 = 450,000 ⋅ 0
...
001
As you can see, you have total flexibility to express 4,500 as a decimal multiplied by a power of ten
...
5 ⋅ 1,000
The final step is to change 1,000 to exponential form
...
5 ⋅ 103
The net effect is that you moved the decimal point three places to the left
and raised 10 to an exponent of 3
...
Understanding order of magnitude
A good question to ask is why scientific notation always uses a decimal
between 1 and 10
...
Order of
magnitude is a simple way to keep track of roughly how large a number is so
you can compare numbers more easily
...
For example,
703 = 7
...
00095 = 9
...
Every number starting with 100 but less than 1,000 has an order of
magnitude of 2
...
” Here’s how to multiply two numbers that are in scientific notation:
1
...
Suppose you want to multiply the following:
(4
...
And because of the
associative property, you can also change how you group the numbers
...
3 ⋅ 2)(105 ⋅ 107)
Multiply what’s in the first set of parentheses — 4
...
3 ⋅ 2 = 8
...
Multiply the two exponential parts by adding their exponents
...
Write the answer as the product of the numbers you found in Steps 1
and 2
...
6 ⋅ 1012
4
...
Because 8
...
6 ⋅ 1012
...
Because scientific notation uses positive decimals less than 10, when you
multiply two of these decimals, the result is always a positive number
less than 100
...
This method even works when one or both of the exponents are negative
numbers
...
02 ⋅ 1023)(9 ⋅ 10–28)
Chapter 14: A Perfect Ten: Condensing Numbers with Scientific Notation
1
...
02 by 9 to find the decimal part of the answer:
6
...
18
2
...
Write the answer as the product of the two numbers:
54
...
Because 54
...
418 ⋅ 10–4
Note: In decimal form, this number equals 0
...
Scientific notation really pays off when you’re multiplying very large and very
small numbers
...
0000000000000000000000000009
As you can see, scientific notation makes the job a lot easier
...
Apples, coins, and hats are easy to count
because they’re discrete — that is, you can easily see where one ends and
the next one begins
...
For example, how do you
count water — by the drop? Even if you tried, exactly how big is a drop?
That’s where units of measurement come in
...
In this chapter, I discuss two important systems of measurement: English and
metric
...
Each of these measurement systems provides a different way to measure distance, volume, weight
(or mass), time, and speed
...
Finally, I show how to convert from English units to metric
and vice versa
...
218
Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Most Americans learn the units of the English system — for example, pounds
and ounces, feet and inches, and so forth — and use them every day
...
English units
such as inches and fluid ounces are often measured in fractions, which (as
you know from Chapters 9 and 10) can be difficult to work with
...
Metric units are based on the number 10, which makes them much easier
to work with
...
Yet despite these advantages, the metric system has been slow to catch on in
the U
...
Many Americans feel comfortable with English units and reluctant to
part with them
...
For example, if I ask, you to carry a
20 lb
...
However, if I ask you to
carry a bag weighing 10 kilograms half a kilometer, you may not be sure
...
If you want an example of the importance of converting carefully, you may
want to look to NASA — they kind of lost a Mars orbiter in the late 1990s
because an engineering team used English units and NASA used metric to
navigate!
Looking at the English system
The English system of measurement is most commonly used in the United
States (but, ironically, not in England)
...
I also show you some equivalent values
that can help you do conversions from one type of unit to another
...
), feet (ft
...
), and miles (mi
...
I discuss
volume when I talk about geometry in Chapter 16
...
oz
...
), pints (pt
...
), and gallons (gal
...
The volume of solid objects is more commonly measured in cubic units of distance such as cubic inches, cubic
feet, and so forth
...
Weight is measured in ounces (oz
...
), and tons
...
These units are two completely different types of
measurements!
ߜ Units of time: Time is hard to define, but everybody knows what it is
...
A year is closer to 365
...
I left months out of the picture because the definition of a month is
imprecise — it can vary from 28 to 31 days
...
The most common unit of speed is miles
per hour (mph)
...
This object can be a glass of water, a turkey in the oven, or
the air surrounding your house
...
Looking at the metric system
Like the English system, the metric system provides units of measurement for
distance, volume, and so on
...
Table 15-1 shows five important basic units in the metric system
...
Each basic SI unit correlates
directly to a measurable scientific process that defines it
...
For technical reasons, scientists tend to use the more rigidly defined SI, but most other people use the
looser metric system
...
Table 15-2 shows ten metric prefixes, with the three most commonly used in
bold (see Chapter 14 for more information on powers of ten)
...
1
10–1
Centi-
One hundredth
0
...
001
10 –3
Micro-
One millionth
0
...
000000001
10–9
Large and small metric units are formed by linking a basic unit with a prefix
...
Similarly, linking the prefix milli- to the
basic unit liter gives you the milliliter, which means 0
...
Here’s a list giving you the basics:
ߜ Units of distance: The basic metric unit of distance is the meter (m)
...
Another common unit is the milliliter (mL):
1 liter = 1,000 milliliters
Note: One milliliter is equal to 1 cubic centimeter (cc)
...
Weight is the measurement of how strongly gravity
pulls an object toward Earth
...
If you traveled to the moon, your weight
would change, so you would feel lighter
...
Unless you’re planning a trip
into outer space or performing a scientific experiment, you probably
don’t need to know the difference between weight and mass
...
The basic unit of weight in the metric system is the gram (g)
...
ߜ Units of time: As in the English system, the basic metric unit of time is a
second (s)
...
For many scientific purposes, the second is the only unit used to measure time
...
ߜ Units of speed: For most purposes, the most common metric unit of
speed (also called velocity) is kilometers per hour (km/hr)
...
ߜ Units of temperature (degrees Celsius or Centigrade): The basic metric
unit of temperature is the Celsius degree (°C), also called the Centigrade
degree
...
Scientists often use another unit — the Kelvin (K) — to talk about temperature
...
Absolute zero is
approximately equal to –273
...
Estimating and Converting between
the English and Metric Systems
Most Americans use the English system of measurement all the time but have
only a passing acquaintance with the metric system
...
Also, if you travel abroad, you need to know how far 100
kilometers is or how long you can drive on 10 liters of gasoline
...
I also show you how to convert between English and metric units,
which is a common type of math problem
...
In contrast, when I talk
about converting, I mean using an equation to change from one system of
units to the other
...
Estimating in the metric system
One reason people sometimes feel uncomfortable using the metric system
is that when you’re not familiar with it, estimating amounts in practical
terms is hard
...
And if I tell you that it’s
10 miles away, you head for the car
...
And if I tell you it’s 40 degrees
Fahrenheit, you’ll probably wear a coat
...
In
each case, I show you how a common metric unit compares with an English
unit that you already feel comfortable with
...
26 feet
...
By this estimate, a 6-foot man stands about 2 meters tall
...
And a football field that’s 100 yards long is about 100 meters
long
...
A mountain that’s 3,000 meters tall is about 9,000 feet
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Estimating longer distances and speed
Here’s how to convert kilometers to miles: 1 kilometer ≈ 0
...
For a ballpark estimate, you can remember that 1 kilometer is about 1⁄2 a mile
...
This guideline tells you that if you live 2 miles from the nearest supermarket,
then you live about 4 kilometers from there
...
And if you run on a treadmill at 6 miles per hour, then you can
run at about 12 kilometers per hour
...
If the Tour de France is about 4,000 kilometers, then it’s about
2,000 miles
...
Approximating volume: 1 liter is about 1 quart (1⁄4 gallon)
Here’s how to convert liters to gallons: 1 liter ≈ 0
...
A good estimate
here is that 1 liter is about 1 quart (that is, there are about 4 liters to the
gallon)
...
If you
put 10 gallons of gasoline in your tank, this is about 40 liters
...
If you
buy an aquarium with a 100-liter capacity, it holds about 25 gallons of water
...
Estimating weight: 1 kilogram is about 2 pounds
Here’s how to convert kilograms to pounds: 1 kilogram ≈ 2
...
For
estimating, figure that 1 kilogram is equal to about 2 pounds
...
If you
can bench press 70 kilograms, then you can bench press about 140 pounds
...
Similarly, if a baby weighs 8 pounds at birth,
he or she weighs about 4 kilograms
...
And if your New Year’s resolution is to lose 20 pounds,
then you want to lose about 10 kilograms
...
The formula for converting from Celsius to Fahrenheit
is kind of messy:
Fahrenheit = Celsius ⋅ 9⁄5 + 32
Instead, use the handy chart in Table 15-3
...
Most of the time, the temperature falls in this middling range
...
When it’s
14°C, you may want a sweater or at least long sleeves
...
People often find this conversion method confusing because they have trouble remembering which formula
to use in which direction
...
Understanding conversion factors
When you multiply any number by 1, that number stays the same
...
And when a fraction has the same numerator (top number)
and denominator (bottom number), that fraction equals 1 (see Chapter 10 for
details)
...
For example:
36 ⋅ 5 = 36
5
If you multiply a measurement by a special fraction that equals 1, you can
switch from one unit of measurement to another without changing the value
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Take a look at some equations that show how metric and English units are
related (all conversions between English and metric units are approximate):
ߜ 1 meter ≈ 3
...
62 miles
ߜ 1 liter ≈ 0
...
20 pounds
Because the values on each side of the equations are equal, you can create
fractions that are equal to 1, such as
ߜ 1 meter
3
...
26 feet
1 meter
ߜ 1 kilometer
0
...
62 miles
1 kilometer
ߜ
1 liter
0
...
26 gallons
1 liter
ߜ
1 kilogram
2
...
2 pounds
1 kilogram
After you understand how units of measurement cancel (which I discuss in
the next section), you can easily choose which fractions to use to switch
between units of measurement
...
Just as with
numbers, you can also cancel out units of measurement in fractions
...
But you can also cancel out the unit gallons in both the
numerator and denominator:
3
=
6 gallons
2 gallons
So this fraction simplifies as follows:
=3
Chapter 15: How Much Have You Got? Weights and Measures
Converting units
After you understand how to cancel out units in fractions and how to set up
fractions equal to 1 (see the preceding sections), you have a foolproof system
for converting units of measurement
...
Using the equation 1 meter =
3
...
26 feet
or
3
...
So you can multiply the quantity you’re trying to convert (7 meters) by
one of these fractions without changing it
...
You already have the word meters in the numerator
(to make this clear, place 1 in the denominator), so use the fraction that puts
1 meter in the denominator:
7 meters 3
...
26 feet
$ 1 meter
1
At this point, the only value in the denominator is 1, so you can ignore it
...
26 feet
Now do the multiplication (Chapter 11 shows how to multiply decimals):
= 22
...
You can get more practice converting units of measurement in Chapter 18,
where I show you how to set up conversion chains and tackle word problems
involving measurement
...
Because geometry is the math of physical space, it’s
one of the most useful areas of math
...
Although geometry is usually a year-long course in high school, you may be
surprised at how quickly you can pick up what you need to know about basic
geometry
...
In this chapter, I give you a quick and practical overview of geometry
...
Then I give you the basics on geometric shapes, from flat circles
to solid cubes
...
Of course, if you want to know more about geometry, the ideal place to look
beyond this chapter is Geometry For Dummies (published by Wiley)!
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Getting on the Plane: Points,
Lines, Angles, and Shapes
Plane geometry is the study of figures on a two-dimensional surface — that is,
on a plane
...
Technically, a plane doesn’t end at the edge of the paper — it continues
forever
...
Making some points
A point is a location on a plane
...
Although in reality a
point is too small to be seen, you can represent it visually in a drawing by
using a dot
...
Additionally, each corner of a polygon is a point
...
)
Knowing your lines
A line — also called a straight line — is pretty much what it sounds like; it
marks the shortest distance between two points, but it extends infinitely in
both directions
...
Chapter 16: Picture This: Basic Geometry
Given any two points, you can draw exactly one line that passes through both
of them
...
When two lines intersect, they share a single point
...
A good visual aid for parallel lines is a set of
railroad tracks
...
Arrows on either end of a line mean that the line goes on forever (as you can
see in Chapter 1, where I discuss the number line)
...
A ray is a piece of a line that starts at a point and extends infinitely in one
direction, kind of like a laser
...
Figuring the angles
An angle is formed when two rays extend from the same point
...
They’re also used in navigation to indicate a sudden change in direction
...
”
The sharpness of an angle is usually measured in degrees
...
Shaping things up
A shape is any closed geometrical figure that has an inside and an outside
...
Chapter 16: Picture This: Basic Geometry
Much of plane geometry focuses on different types of shapes
...
Later in this chapter,
I show you how to measure these shapes
...
The area of a shape is the measurement of the size inside that shape
...
However, many shapes don’t have names, as you can see in
Figure 16-1
...
Measuring the perimeter and area of shapes is useful for a variety of applications, from land surveying (to get information about a parcel of land that you’re
measuring) to sewing (to figure out how much material you need for a project)
...
Later in the
chapter, I show you how to find the perimeter and area of each, but for now I
just acquaint you with them
...
The distance from any point on the circle to its center is called the
radius of the circle
...
Unlike the polygons I discuss later, a circle has no straight edges
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Polygons
A polygon is any shape whose sides are all straight
...
Following are a few of the most common polygons
...
You find out all about triangles when you study trigonometry (and what
better place to begin than Trigonometry For Dummies?)
...
Take a look at the differences (and
see Figure 16-2):
ߜ Equilateral: An equilateral triangle has three sides that are all the same
length and three angles that all measure 60 degrees
...
ߜ Scalene: Scalene triangles have three sides that are all different lengths
and three angles that are all unequal
...
It may be isosceles or scalene
...
Equilateral
Isosceles
Scalene
Right
Quadrilaterals
A quadrilateral is any shape that has four straight sides
...
If you doubt this, look around
and notice that most rooms, doors, windows, and tabletops are quadrilaterals
...
ߜ Rectangle: Like a square, a rectangle has four right angles and two pairs
of opposite sides that are parallel
...
ߜ Rhombus: Imagine taking a square and collapsing it as if its corners
were hinges
...
All four sides are equal in
length, and both pairs of opposite sides are parallel
...
This shape is a parallelogram — both pairs of opposite
sides are equal in length, and both pairs of opposite sides are parallel
...
ߜ Kite: A kite is a quadrilateral with two pairs of adjacent sides that are
the same length
...
Parallelogram
Kite
Trapezoid
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
A quadrilateral can fit into more than one of these categories
...
Every rectangle and rhombus is also both a
parallelogram and a trapezoid
...
In practice, however, it’s common to identify a quadrilateral as
descriptively as possible — that is, use the first word in the list above that
accurately describes it
...
Polygons with more than four sides
are not as common as triangles and quadrilaterals, but are still worth knowing
about
...
A regular polygon has equal sides and equal angles
...
See Figure 16-4
...
Every other polygon is an irregular polygon (see Figure 16-5)
...
Chapter 16: Picture This: Basic Geometry
Taking a Trip to Another Dimension:
Solid Geometry
Solid geometry is the study of shapes in space — that is, the study of shapes
in three dimensions
...
Every solid has an inside and an outside separated by the
surface of the solid
...
The many faces of polyhedrons
A polyhedron is the three-dimensional equivalent of a polygon
...
Similarly, a polyhedron is a solid that has only straight edges and flat
faces (that is, faces that are polygons)
...
As you can see,
a cube has 6 flat faces that are polygons — in this case, all of the faces are
square — and 12 straight edges
...
Later in this chapter, I show you how to measure the surface area and volume
of a cube
...
Figure 16-7 shows a few common polyhedrons
...
Cube
Pyramid
Later in this chapter, I show you how measure each of these polyhedrons
to determine its volume — that is, the amount of space contained inside its
surface
...
Each regular solid has identical faces that are regular polygons
...
Similarly, the tetrahedron is a pyramid with
four faces that are equilateral triangles
...
Icosahedron
Dodecahedron
3-D shapes with curves
Many solids aren’t polyhedrons because they contain at least one curved surface
...
A ball is a perfect visual aid for a sphere
...
A good visual aid for a cylinder is a can of soup
...
A good visual aid for a cone is an ice cream cone
...
Sphere
Cylinder
Cone
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
In the next section, I show you how to measure each of these solids to determine its volume — that is, the amount of space contained inside its surface
...
These formulas use letters to stand
for numbers that you can plug in to make specific measurements
...
2-D: Measuring on the flat
Two important skills in geometry — and real life — are finding the perimeter
and the area of shapes
...
You use perimeter for measuring the distance around the edges of a
room, building, or circular pathway
...
You use area when measuring the size of a wall, a table, or a tire
...
Figure 16-10:
Measuring 2 in
...
2 in
...
2 in
...
2 in
...
When every side of a shape is straight, you can measure its perimeter by
adding up the lengths of all of its sides
...
Figure 16-11:
The areas of
figures
...
²
3 in
...
73 in
...
2 ),
square feet (ft
...
2 ), square kilometers (km2 ), and so on —
even if you’re talking about the area of a circle! (For more on measurements,
flip to Chapter 15
...
(For more information on
the names of shapes, refer back to “Closed Encounters: Shaping Up Your
Understanding of 2-D Shapes
...
This distance is called the radius of the circle, or r for short
...
See Figure 16-12
...
Center
Radius
Diameter
As you can see, the diameter of any circle is made up of one radius plus
another radius — that is, two radii (pronounced ray-dee-eye)
...
Early
mathematicians went to a lot of trouble figuring out how to measure the circumference of a circle
...
The symbol π is called pi (pronounced pie)
...
14
So given a circle with a radius of 5 mm, you can figure out the approximate
circumference:
C ≈ 2 ⋅ 3
...
4 mm
The formula for the area (A) of a circle also uses π:
A=π⋅r2
Here’s how to use this formula to find the approximate area of a circle with a
radius of 5 mm:
A ≈ 3
...
14 ⋅ 25 mm2 = 78
...
Then, I show you a special feature of right triangles that allows you to measure
them more easily
...
To find the area of a triangle, you need to know the length of one side — the
base (b for short) — and the height (h)
...
Figure 16-13 shows a triangle with a base of 5 cm and a
height of 2 cm:
Figure 16-13:
The base
and height
of a triangle
...
The most important right triangle formula is the Pythagorean theorem:
a2 + b2 = c2
Figure 16-14:
The Leg (b)
hypotenuse
and legs of
a right
triangle
...
For example, suppose the legs of a triangle are 3 and 4
units
...
Therefore,
c=5
The length of the hypotenuse is 5 units
...
For example, if the side
of a square is 3 inches, then you say s = 3 in
...
Here’s the formula
for the perimeter of a square:
P=4⋅s
For example, if the length of the side is 3 inches, substitute 3 inches for s in
the formula:
P = 4 ⋅ 3 in
...
Finding the area of a square is also easy: Just multiply the length of the side
by itself — that is, take the square of the side
...
)2 = 3 in
...
= 9 in
...
The short side
is called the width, or w for short
...
and w = 4 ft
...
+ 4 yd
...
= 18 yd
...
⋅ 4 yd
...
2
Chapter 16: Picture This: Basic Geometry
Calculating with rhombuses
As with a square, use s to represent the length of a rhombus’s side
...
The height of a rhombus (h for short) is the shortest distance from one side to the opposite side
...
4 cm
2 cm
Figure 16-15:
Measuring
a rhombus
...
Here’s the formula:
A=s⋅h
So here’s how you determine the area of a rhombus with a side of 4 cm and a
height of 2 cm:
A = 4 cm ⋅ 2 cm = 8 cm2
You can read 8 cm2 as 8 square centimeters or, less commonly, as 8 centimeters squared
...
And as with rhombuses, another
important measurement of a parallelogram is its height (h), the shortest
distance between the bases
...
, s = 3 in
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Figure 16-16:
Measuring
a parallelogram
...
2 in
...
Each parallelogram has two equal bases and two equal sides
...
+ 3 in
...
= 18 in
...
⋅ 2 in
...
2
Measuring trapezoids
The parallel sides of a trapezoid are called its bases
...
The height (h) of a trapezoid is
the shortest distance between the bases
...
, b2 = 3 in
...
2 in
...
Figure 16-17:
Measuring
a trapezoid
...
Because a trapezoid can have sides of four different lengths, you really don’t
have a special formula for finding the perimeter of a trapezoid
...
Chapter 16: Picture This: Basic Geometry
Here’s the formula for the area of a trapezoid:
A = 1 $ _b 1+ b 2i $ h
2
So, here’s how to find the area of the pictured trapezoid:
A = 1 $ ^ 2 in
...
h $ 2 in
...
2 in
...
2 = 5 in
...
⋅ 2 in
...
Spacing out: Measuring in three dimensions
In three dimensions, the concepts of perimeter and area have to be tweaked
a little
...
In
3-D, the boundary of a solid is called its surface area and what’s inside a solid
is called its volume
...
2 ), square feet (ft
...
The volume (V) of a solid is a measurement of the
space it occupies, as measured in cubic units such as cubic inches (in
...
3 ), cubic meters (m3 ), and so forth
...
)
You can find the surface area of a polyhedron (solid whose faces are all
polygons — see the earlier “The many faces of polyhedrons” section) by
adding together the areas of all its faces
...
In most other cases, you don’t need to know the formulas for finding the surface area of a solid
...
”)
Finding the volume of solids, however, is something mathematicians love for
you to know
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Spheres
The center of a sphere is a point that’s the same distance from any point on
the sphere itself
...
If you
know the radius of a sphere, you can find out its volume using the following
formula:
V= 4 $π$r3
3
Because this formula includes π, using 3
...
For example, here’s how to figure
out the approximate volume of a ball whose radius is 4 inches:
3
V
...
14 $ ^ 4 in
...
14 $ 64 in
...
4
...
3
= 268
...
3
Cubes
The main measurement of a cube is the length of its side (s)
...
Boxes (Rectangular solids)
The three measurements of a box (or rectangular solid) are its length (l),
width (w), and height (h)
...
2m
Figure 16-18:
Measuring
a box
...
One measurement is the height (h) of the prism
...
The base is the polygon that extends vertically from the plane
...
)
Here’s the formula for finding the volume of a prism:
V = Ab ⋅ h
For example, suppose a prism has a base with an area of 5 square centimeters and a height of 3 centimeters
...
Cylinders
You find the volume of cylinders the same way you find the area of prisms —
by multiplying the area of the base (Ab) by the cylinder’s height (h):
V = Ab ⋅ h
Suppose you want to find the volume of a cylindrical can whose height is 4
inches and whose base is a circle with a radius of 2 inches
...
14 ⋅ (2 in
...
14 ⋅ 4 in
...
56 in
...
14 as an approximate value for π
...
)
Now use this area to find the volume of the cylinder:
V ≈ 12
...
2 ⋅ 4 in
...
24 in
...
2 ) by inches gives a result in cubic
inches (in
...
Pyramids and cones
The two key measurements for pyramids and cones are the same as those for
prisms and cylinders (see the previous sections): the height (h) and the area
of the base (Ab )
...
Here’s how
you do it:
V = 1 _ 3 in
...
i
3
1 _12 in
...
3
Similarly, suppose you want to find the volume of a pyramid in Egypt whose
height is 60 meters with a square base whose sides are each 50 meters
...
Most students find graphs relatively easy because they provide a picture to work with rather than just a bunch of numbers
...
In this chapter, I introduce you to three common styles of graphs: the bar
graph, the pie chart, and the line graph
...
I also show you how to answer the
types of questions people may ask to check your understanding
...
This system is so common
that when math folks talk about a graph, they’re usually talking about this
type
...
At the end of the chapter, you see how you can solve
math problems using a graph
...
They aren’t the only types of
graphs, but they’re very common, and understanding them can give you a leg
up on reading other types of graphs when you see them
...
ߜ The pie chart allows you to show how a whole is cut up into parts
...
Bar graph
A bar graph gives you an easy way to compare numbers or values
...
25
20
15
Figure 17-1:
The number
of new
clients
recorded
this quarter
...
The advantage of such a graph is that you can
see at a glance, for example, that Edna has the most new clients and Iris has
the fewest
...
For example, if Iris gets another new client, it doesn’t
necessarily affect any other trainer’s performance
...
Here are a few types of
questions someone could ask about the bar graph in Figure 17-1:
ߜ Individual values: How many new clients does Jay have? Find the bar
representing Jay’s clients and notice that he has 23 new clients
...
ߜ Totals: Together, how many clients do the three women have? Notice that
the three women — Edna, Iris, and Rita — have 25, 16, and 20 new
clients, respectively, so they have 61 new clients altogether
...
Pie charts are most often used to represent percentages
...
Car 20%
Rent 25%
Miscellaneous 5%
Figure 17-2:
Eileen’s
monthly
expenses
...
Unlike the bar graph, the pie chart shows numbers
that are dependent upon each other
...
Here are a few typical questions you may be asked about a pie chart:
ߜ Individual percentages: What percentage of her monthly expenses does
Eileen spend on food? Find the slice that represents what Eileen spends
on food, and notice that she spends 10% of her income there
...
ߜ How much a percent represents in terms of dollars: If Eileen brings
home $2,000 per month, how much does she put away in savings each
month? First notice that Eileen puts 15% every month into savings
...
Using your skills from Chapter 12,
solve this problem by turning 15% into a decimal and multiplying:
0
...
Line graph
The most common use of a line graph is to plot how numbers change over
time
...
$40,000
$35,000
$30,000
$25,000
$20,000
$15,000
Figure 17-3:
Gross
receipts
for Tami’s
Interiors
...
At a glance, you can tell that
Tami’s business tended to rise strongly at the beginning of the year, drop off
during the summer, rise again in the fall, and then drop off again in December
...
ߜ Total over a period of time: How much did she bring in altogether the
last quarter of the year? A quarter of a year is three months, so the last
quarter is the last three months of the year
...
ߜ Greatest change: In what month did the business show the greatest gain in
revenue as compared with the previous month? You want to find the line
segment on the graph that has the steepest upward slope
...
Cartesian Coordinates
When math folks talk about using a graph, they’re usually referring to a
Cartesian graph (also called the Cartesian coordinate system), as shown in Figure 17-4
...
You see a lot of this graph
when you study algebra, so getting familiar with it now is a good idea
...
0
1 2
x-axis
3
4
5
6
7
8
A Cartesian graph is really just two number lines that cross at 0
...
The place where these two axes (plural of axis) cross is
called the origin
...
(Flip to Chapter 1 for more on using the number line
...
To plot any point,
start at the origin, where the two axes cross
...
The second number tells you how far to go up (if positive) or down (if negative) along the vertical axis
...
Start at the origin,
(0, 0)
...
To plot
point B, count 4 spaces to the left (the negative direction) and then 1 space
up
...
And to plot point D, count 6 spaces to the right and
then 0 spaces up or down
...
A (2, 3)
2
D (6, 0)
1
−2
−3
−4
−5
−6
C (0, −5)
2
3
4
5
6
Chapter 17: Seeing Is Believing: Graphing as a Visual Tool
Drawing lines on a Cartesian graph
After you understand how to plot points on a graph (see the preceding
section), you can begin to plot lines and use them to show mathematical
relationships
...
The horizontal axis represents Xenia’s money,
and the vertical axis represents Yanni’s
...
To do this, make a chart as follows:
Xenia
1
2
3
4
5
Yanni
Now fill in each column of the chart assuming that Xenia has that number of
dollars
...
And if Xenia has $2,
then Yanni has $1
...
Next, draw a straight line through
these points, as shown in Figure 17-6
...
For example, notice how the point (6, 5) is on the line
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Yanni
6
5
4
3
2
1
Figure 17-6:
All possible
values of
Xenia’s and
Yanni’s
money if
Xenia has $1
more than
Yanni
...
Again, start by making the usual chart:
Xenia
1
2
3
4
5
Yanni
You can fill in this chart by supposing that Xenia has a certain amount of
money and then figuring out how much money Yanni would have in that case
...
And if Xenia has $2, then twice that amount is $4, so
Yanni has $3 more, or $7
...
As in the other examples, this graph represents all possible values that Xenia
and Yanni could have
...
Chapter 17: Seeing Is Believing: Graphing as a Visual Tool
Yanni
9
8
7
6
5
4
3
2
Figure 17-7:
All possible
values of
Xenia’s and
Yanni’s
money if
Yanni has $3
more than
twice the
amount
Xenia has
...
When you draw two lines that represent
different parts of a word problem, then the point at which the lines intersect
(where they cross) is your answer
...
How old are they?
To solve this problem, first make a chart to show that Jacob is 5 years
younger than Marnie:
Jacob
1
2
3
4
5
Marnie
6
7
8
9
10
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Then make another chart to show that together, the two children’s ages add
up to 15:
Jacob
1
2
3
4
5
Marnie
14
13
12
11
10
Finally, plot both lines on a graph (see Figure 17-8) where the horizontal axis
represents Jacob’s age and the vertical axis represents Marnie’s age
...
Marnie
15
14
13
12
11
(5, 10)
10
9
8
7
6
5
4
3
2
Figure 17-8:
Both lines
plotted on a
graph
...
In a word problem involving measurement, you’re often asked to perform a conversion from one type of unit
to another
...
Another common type of word problem requires the geometric formulas that
I provide in Chapter 16
...
In other cases, you have to draw the picture yourself by
reading the problem carefully
...
The Chain Gang: Solving Measurement
Problems with Conversion Chains
In Chapter 15, I give you a set of basic conversion equations for converting
units of measurement
...
This information is useful as far as it goes, but you may not always have an equation
for the exact conversion that you want to perform
...
A conversion chain links together a sequence of unit conversions
...
How many 1-ounce servings of strawberries
is that?
You don’t have an equation to convert tons directly to ounces
...
You
can use these equations to build a bridge from one unit to another:
tons → pounds → ounces
So here are the two equations that you’ll want to use:
1 ton = 2,000 lbs
...
= 16 oz
...
or
2,000 lbs
...
16 oz
...
1 lb
...
But when you know the basic
idea, you set up a conversion chain instead
...
Because you already have tons on top, you want the tonsand-pounds fraction that puts ton on the bottom
...
16 oz
...
Chapter 18: Solving Geometry and Measurement Word Problems
The net effect here is to take the expression 7 tons and multiply it twice by 1,
which doesn’t change the value of the expression
...
16 oz
...
If any units don’t cancel out properly, you probably made a mistake when you
set up the chain
...
Now you can simplify the expression:
= 7 ⋅ 2,000 ⋅ 16 oz
...
Working with more links
After you understand the basic idea of a conversion chain, you can make a
chain as long as you like to solve longer problems easily
...
You forgot to get her a present, but
you decide that offering her your mathematical skills is the greatest gift
of all — you’ll recalculate how old she is
...
60 min
...
$ 1 year $ 1 day $ 1 hr
...
1
Cancel out all units that appear in both a numerator and a denominator:
=
12 years 365 days 24 hrs
...
60 sec
...
$ 1 min
...
When the smoke clears, here’s what’s left:
= 12 ⋅ 365 ⋅ 24 ⋅ 60 ⋅ 60 sec
...
The conversion chain from 12 years to 378,432,000 seconds doesn’t change
the value of the expression, just the unit of measurement
...
Take this problem for example:
A furlong is 1⁄8 of a mile, and a fathom is 2 yards
...
Every unit you want to cancel has
to appear once in the numerator and once in the denominator:
1 yard
24 furlongs
$ 8 1 mile $ 5,280 feet $ 3 feet $ 12fathom
1
1 mile
yards
furlongs
Next, you can cancel out all the units except for fathoms:
=
24 furlongs
1 mile
5,280 feet 1 yard
$ 8 furlongs $ 1 mile $ 3 feet $ 12fathom
1
yards
Another way you can make this problem a little easier is to notice that the
number 24 is in the numerator and 3 and 8 are in the denominator
...
Rounding off: Going for the short answer
Sometimes real-life measurements just aren’t that accurate
...
When you perform calculations with such
measurements, finding the answer to a bunch of decimal places doesn’t make
sense because the answer’s already approximate
...
Here’s a problem
that asks you to do just that:
Heather weighed her new pet hamster, Binky, and found that he weighs
4 ounces
...
20 pounds
Notice that this conversion equation includes only kilograms and pounds,
but the problem includes ounces and grams
...
1 lb
...
$ 2
...
$ 1 kg
As always, after you set up the expression, you can cancel out every unit
except for the one you’re converting to:
=
4 oz
...
1 kg 1,000 g
1 $ 16 oz
...
2 lb
...
The numbers that were originally in the numerators of fractions remain in the numerator
...
Then just put a multiplication sign between
each pair of numbers
...
2
Chapter 18: Solving Geometry and Measurement Word Problems
At this point, you can begin calculating
...
In this case, you cancel out a 4 in the numerator and denominator, changing the 16 in the denominator to a 4:
=
4 $ 1, 000
g
416 $ 2
...
2
At this point, here’s what’s left:
= 250 g
2
...
2 to get your answer:
≈ 113
...
Because the number
after the decimal point is 6, I need to round my answer up to the next highest
gram
...
)
So to the nearest gram, Binky weighs 114 grams
...
Solving Geometry Word Problems
Some geometry word problems present you with a picture
...
Sketching figures is always a good idea
because it can usually give you an idea of how to proceed
...
(To solve these word problems, you need some of the geometry formulas I discuss in Chapter 16
...
Read the
problem carefully, recognize shapes in the drawing, pay attention to labels,
and use whatever formulas you have to help you answer the question
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Mr
...
He gave them a rectangular
piece of land with a creek running through it diagonally, as shown in
Figure 18-1
...
What is the area of each boy’s land in square feet?
350 ft
...
250 ft
...
200 ft
...
To find the area of the smaller, triangular plot, use the formula for the area of
a triangle, where A is the area, b is the base, and h is the height:
A = 1 ^ b $ hh
2
The whole piece of land is a rectangle, so you know that the corner the triangle shares with the rectangle is a right angle
...
Find the area of
this plot by plugging the base and height into the formula:
A = 200 feet $ 250 feet
2
To make this calculation a little easier, notice that you can cancel a factor of 2
from the numerator and denominator:
A=
100
200 feet $ 250 feet
= 25, 000 square feet
2
Chapter 18: Solving Geometry and Measurement Word Problems
The shape of the remaining area is a trapezoid
...
Because you know the
area of the triangular plot, you can use this word equation to find the area of
the trapezoid:
Area of trapezoid = area of whole plot – area of triangle
To find the area of the whole plot, remember the formula for the area of a rectangle
...
⋅ 250 ft
...
2
Now just substitute the numbers that you know into the word equation you
set up:
Area of trapezoid = 87,500 square feet – 25,000 square feet
= 62,500 square feet
So the area of the elder boy’s land is 62,500 square feet, and the area of the
younger boy’s land is 25,000 square feet
...
Here’s an example of a geometry problem without a picture provided:
In Elmwood Park, the flagpole is due south of the swing set and exactly
20 meters due west of the treehouse
...
Start with the first sentence, depicted in Figure 18-2
...
As you can see, I’ve drawn a right triangle whose
corners are the swing set (S), the flagpole (F), and the treehouse (T)
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
North
S
Figure 18-2:
A labeled
sketch
shows the
important
information
in a word
problem
...
Because you know the area of the triangle, you may find the
formula for the area of a triangle helpful:
A = 1 ^ b $ hh
2
Here, b is the base and h is the height
...
So you already know the area of the triangle, and you also know the
length of the base
...
Start by simplifying:
150 = 10 ⋅ h
You can turn this into a division problem using inverse operations, as I
show you in Chapter 4:
150 ÷ 10 = h
15 = h
Now you know that the height of the triangle is 15 m, so you can add this
information to your picture, as shown in Figure 18-3
...
T
20 m
To solve the problem, though, you still need to find out the distance from
S to T
...
(See Chapter 16 for more on the
Pythagorean theorem
...
Start with the two exponents
and then move on to addition:
225 + 400 = c2
625 = c2
At this point, remember that c2 means c ⋅ c:
625 = c ⋅ c
So c is a number that, when multiplied by itself, results in the number 625
...
A bit of trial and error gives you the following:
625 = 25 ⋅ 25
So the distance from the swing set to the treehouse is 25 meters
...
This is especially popular (among teachers, not students!) on a final exam, when you’re being
tested on everything that a class covered for a whole semester
...
What is the width of
the path in inches to the nearest inch? (Use 3
...
)
First, draw a picture of what the problem is asking, like the one shown in
Figure 18-4
...
Path
Circumference = 120 feet
Chapter 18: Solving Geometry and Measurement Word Problems
As you can see, I drew a circle for the fountain and labeled its diameter as
32 feet
...
The problem is asking for the width of the path
...
The radius of a circle is the distance from its center to the circle itself (see
Chapter 16)
...
Look at the diagram and make
sure you understand it before continuing
...
= 2 ⋅ radius of inner circle
You can probably solve this problem in your head:
Radius of inner circle = 16 ft
...
1 for π gives you
120 ft
...
1 ⋅ radius of outer circle
This equation can be simplified a little as follows:
120 ft
...
2 ⋅ radius of outer circle
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Now, you can use inverses to rearrange this multiplication problem into a
division problem (see Chapter 4 for more on how to do this):
120 ft
...
2 = radius of outer circle
Rounding off this division to the nearest tenth gives you this answer:
Radius of outer circle ≈ 19
...
Now that you have the radius of both circles, you can substitute into the
word equation that you started with:
Width of path ≈ 19
...
– 16 ft
...
4 ft
...
Here’s a conversion
equation:
1 ft
...
You can set up an expression as follows:
3
...
12 in
...
Now, you can cancel out feet, because this unit appears in both the numerator and denominator:
=
3
...
12 in
...
This expression now simplifies to the following:
= 3
...
= 40
...
So the width of the path is about 41 inches
...
Copy it down and see whether you can work through it from
beginning to end
...
They’re applicable to virtually every aspect of the
real world — business, biology, city planning, politics, meteorology, and
many more areas of study
...
In this chapter, I give you a basic understanding of these two mathematical
ideas
...
I show you how to work with both types of
data to find meaningful answers
...
I
show you how the probability that an event will occur is always a fraction from
0 to 1
...
Finally, I put these ideas to work by
showing you how to calculate the probability of tossing coins and rolling dice
...
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An individual statistic is a conclusion drawn from this data
...
7 cups of coffee every day
...
ߜ The cat is the most popular pet in the United States
...
Statisticians do their work by identifying a population that they’d like to
study: working people, law students, pet owners, buyers of electronics, whoever
...
Much of statistics concerns itself with gathering data that’s reliable and
accurate
...
In this section, I give you a short introduction to the more mathematical
aspects of statistics
...
Qualitative data divides a data set (the pool of data that you’ve
gathered) into discrete chunks based on a specific attribute
...
For example, four attributes of
Emma are that she’s female, her favorite color is green, she owns a dog, and
she walks to school
...
For example, quantitative data
on this same classroom of students can include the following:
Chapter 19: Figuring Your Chances: Statistics and Probability
ߜ Each child’s height in inches
ߜ His or her weight in pounds
ߜ The number of siblings he or she has
ߜ The number of words he or she spelled correctly on the most recent
spelling test
You can identify quantitative data by noticing that it links a number to
each member of the data set
...
Working with qualitative data
Qualitative data usually divides a sample into discrete chunks
...
For example, suppose all 25 children in Sister Elena’s class
answer the three yes-or-no questions in Table 19-1
...
Table 19-2
Favorite Colors in Sister Elena’s Class
Color
Number of Students
Color
Number of Students
Blue
8
Orange
1
Red
6
Yellow
1
Green
5
Gold
1
Purple
3
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Even though the information that each child provided is non-numerical, you
can handle it numerically by counting how many students made each
response and working with these numbers
...
For instance,
ߜ Exactly 20 children have at least one brother or sister
...
ߜ Only one child’s favorite color is yellow
...
Here’s how you do so:
1
...
Suppose you want to know what percentage of students in Sister Elena’s
class are only children
...
So you can begin to
answer this question as follows:
Five out of 25 children are only children
...
Rewrite this statement, turning the numbers into a fraction:
Number who share attribute
Number in sample
In the example, 5⁄25 of the children are only children
...
Turn the fraction into a percent, using the method I show you in
Chapter 12
...
2, so 20% of the children are only children
...
This time, the chart tells you that 16 children take the bus, so
you can write this statement:
Sixteen out of 25 children take the bus to school
...
25
Chapter 19: Figuring Your Chances: Statistics and Probability
Finally, turn this fraction into a percent
...
64,
or 64%:
64% of the children take the bus to school
...
For
example, in the poll of Sister Elena’s class (see Tables 19-1 and 19-2), the
mode groups are children who
ߜ Have at least one brother or sister (20 students)
ߜ Own at least one pet (14 students)
ߜ Take the bus to school (16 students)
ߜ Chose blue as their favorite color (8 students)
When a question divides a data set into two parts (as with all yes-or-no questions), the mode group represents more than half of the data set
...
For example, 14 children own at least one pet, and the other 11 children don’t
own one
...
But 8 of the 25 children chose blue as their favorite color
...
With a small sample, you may have more than one mode — for example, perhaps the number of students who like red is equal to the number who like
blue
...
Working with quantitative data
Quantitative data assigns a numerical value to each member of the sample
...
Suppose that the information in Table 19-3 has been gathered
about each team member’s height and most recent spelling test
...
Both terms refer to ways to calculate the average value in a quantitative data set
...
For example, the average height of Sister Elena’s fifth grade
class is probably less than the average height of the Los Angeles Lakers
...
Finding the mean
The mean is the most commonly used average
...
Here’s how you find the mean
of a set of data:
1
...
For example, to find the average height of the five team members, first
add up all their heights:
55 + 60 + 59 + 58 + 63 = 295
2
...
Divide 295 by 5 (that is, by the total number of boys on the team):
295 ÷ 5 = 59
So the mean height of the boys on Sister Elena’s team is 59 inches
...
4
As you can see, when you divide, you end up with a decimal in your answer
...
(For more information about rounding, see Chapter 2
...
For example, suppose that the president of a company tells you, “The average
salary in my company is $200,000 a year!” But on your first day at work, you
find out that the president’s salary is $19,010,000 and each of his 99 employees
earns $10,000
...
However, the skew in salaries resulted in a misleading mean
...
Here’s how to find the median of a set
of data:
1
...
To find the median height of the boys in Table 19-3, arrange their five
heights in order from lowest to highest
...
Choose the middle number
...
To find the median number of words that the boys spelled correctly (see
Table 19-3), arrange their scores in order from lowest to highest:
14
17
18
18
20
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
This time, the middle value is 18, so 18 is the median score
...
For instance, consider the following:
2
3
5
7
9
11
The two center numbers are 5 and 7
...
The median in this list is 6
...
Here’s how this data looks:
10,000
10,000
10,000
...
The median salary is $10,000, and this result
is much more reflective of what you’d probably earn if you were to work at
this company
...
For
example,
ߜ What’s the likelihood that the lottery ticket I bought will win?
ߜ What’s the likelihood that my new car will need repairs before the
warranty runs out?
ߜ What’s the likelihood that more than 100 inches of snow will fall in
Manchester, New Hampshire, this winter?
Probability has a wide variety of applications in insurance, weather prediction,
biological sciences, and even physics
...
You can read all about the details of probability in Probability For Dummies
(Wiley)
...
Chapter 19: Figuring Your Chances: Statistics and Probability
Figuring the probability
The probability that an event will occur is a fraction whose numerator (top
number) and denominator (bottom number) are as follows (for more on fractions, flip to Chapter 9):
Number of favorable outcomes
Total number of possible outcomes
In this case, a favorable outcome is simply an outcome in which the event
you’re examining does happen
...
For example, suppose you want to know the probability that a tossed coin will
land heads up
...
To find the probability of this event, make a fraction as follows:
Number of favorable outcomes = 1
Total number of outcomes
2
So the probability that the coin will land heads up is 1⁄2
...
To find the probability
of this outcome, make a fraction as follows:
Number of favorable outcomes = 1
Total number of outcomes
6
So the probability that the number 3 will land face up is 1⁄6
...
So
Number of favorable outcomes = 4
Total number of outcomes
52
So the probability that you’ll pick an ace is 4⁄52, which reduces to 1⁄13 (see
Chapter 9 for more on reducing fractions)
...
When the probability of
an outcome is 0, the outcome is impossible
...
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Part IV: Picturing and Measuring — Graphs, Measures, Stats, and Sets
Oh, the possibilities! Counting outcomes
with multiple coins and dice
Although the basic probability formula isn’t difficult, sometimes finding the
numbers to plug into it can be tricky
...
In this section, I focus
on tossing coins and rolling dice
...
When you flip two coins at the same time — say, a penny and a nickel —
you can get four possible outcomes:
Outcome
Penny
Nickel
#1
Heads
Heads
#2
Heads
Tails
#3
Tails
Heads
#4
Tails
Tails
When you flip three coins at the same time — say, a penny, a nickel, and a
dime — eight outcomes are possible:
Outcome
Penny
Nickel
Dime
#1
Heads
Heads
Heads
#2
Heads
Heads
Tails
#3
Heads
Tails
Heads
#4
Heads
Tails
Tails
#5
Tails
Heads
Heads
#6
Tails
Heads
Tails
#7
Tails
Tails
Heads
#8
Tails
Tails
Tails
Notice the pattern: Every time you add an additional coin, the number of possible outcomes doubles
...
Here’s a handy formula for calculating the number of outcomes when you’re
flipping, shaking, or rolling multiple coins, dice, or other objects at the same
time:
Number of outcomes per objectNumber of objects
Suppose you want to find the probability that six tossed coins will all fall
heads up
...
Only one outcome is favorable, so the numerator is 1:
Number of favorable outcomes = 1
Total number of outcomes
64
So the probability that six tossed coins will all fall heads up is 1⁄64
...
To find the numerator (favorable outcomes), think about it this way: If the first coin falls tails up, then all the rest
must fall heads up
...
This is true of all six coins, so you have six favorable outcomes:
Number of favorable outcomes = 6
Total number of outcomes
64
Therefore, the probability that exactly five out of six coins will fall heads up
is 6⁄64, which reduces to 3⁄32 (see Chapter 9 for more on reducing fractions)
...
However, when you roll two dice, this number jumps to 36
...
Every time you add an additional die, the number of possible outcomes is multiplied by 6
...
The probability is a fraction, and you already know that the denominator of this fraction is
1,296
...
So the probability that you’ll roll four 6s is 1⁄1,296 — a very small probability,
indeed
...
To find the numerator, think about it this way: For the first die, there are
three favorable outcomes (4, 5, or 6)
...
So for all four dice,
there are 3 ⋅ 3 ⋅ 3 ⋅ 3 = 81 favorable outcomes
...
This fraction reduces to 1⁄16 (see Chapter 9 for more on reducing fractions)
...
But in their simplicity, sets are profound
...
Set theory provides a way to talk about collections of numbers such as even
numbers, prime numbers, or counting numbers with ease and clarity
...
For these reasons, set theory becomes more important the higher you
go up the math food chain — especially when you begin writing mathematical
proofs
...
In this chapter, I show you the basics of set theory
...
I also show you the simple idea of a set’s cardinality
...
After that, I discuss four operations on
sets: union, intersection, relative complement, and complement
...
These things can be buildings,
earmuffs, lightning bugs, numbers, qualities of historical figures, names you
call your little brother, whatever
...
When the set is too large, simply
use an ellipsis (
...
For
example, to list the set of numbers from 1 to 100, you can write {1, 2,
3,
...
To list the set of all the counting numbers, you can write
{1, 2, 3,
...
For instance, the set
of the four seasons is pretty clear-cut, but you may run into some debate
on the set of words that describe my cooking skills because different
people have different opinions
...
Check out Algebra II For Dummies, by Mary Jane
Sterling (Wiley), for details
...
(Chapter 21 talks about
using variables
...
For example,
here I define three sets:
A = {Empire State Building, Eiffel Tower, Roman Colosseum}
B = {Albert Einstein’s intelligence, Marilyn Monroe’s talent, Joe
DiMaggio’s athletic ability, Senator Joseph McCarthy’s ruthlessness}
C = the four seasons of the year
Set A contains three tangible objects: famous works of architecture
...
And set C also
contains intangible objects: the four seasons
...
In the following sections, I show you the basics of set theory
...
Consider the first two sets I define in the section intro:
Chapter 20: Setting Things Up with Basic Set Theory
A = {Empire State Building, Eiffel Tower, Roman Colosseum}
B = {Albert Einstein’s intelligence, Marilyn Monroe’s talent, Joe
DiMaggio’s athletic ability, Senator Joseph McCarthy’s ruthlessness}
The Eiffel Tower is an element of A, and Marilyn Monroe’s talent is an element
of B
...
You can write this statement
using the symbol , which means is not an element of:
Eiffel Tower
B
These two symbols become more common as you move higher in your study
of math
...
Cardinality of sets
The cardinality of a set is just a fancy word for the number of elements in
that set
...
Set B, which is {Albert Einstein’s
intelligence, Marilyn Monroe’s talent, Joe DiMaggio’s athletic ability, Senator
Joseph McCarthy’s ruthlessness}, has four elements, so the cardinality of B
is four
...
The order of elements in the sets
doesn’t matter
...
Suppose I define some sets as follows:
C = the four seasons of the year
D = {spring, summer, fall, winter}
E = {fall, spring, summer, winter}
F = {summer, summer, summer, spring, fall, winter, winter, summer}
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Set C gives a clear rule describing a set
...
Set E lists the four seasons in a different order
...
Thus, all four sets are equal
...
For example, consider these sets:
C = {spring, summer, fall, winter}
G = {spring, summer, fall}
As you can see, every element of G is also an element of C, so G is a subset of C
...
This idea may seem odd until you realize that
all the elements of any set are obviously contained in that set
...
The symbol 8 is used to represent the empty set
...
You can also define an empty set using a rule
...
You can think of 8 as nothing
...
All empty sets are equal to each other, so in this case, H = I
...
Remember that 8 has no
elements, so technically speaking, every element in 8 is in every other set
...
As with all other sets,
you can do so either by listing the elements or by verbally describing a rule
that clearly tells you what’s included in the set and what isn’t
...
}
L = the set of counting numbers
My definitions of J and K list their elements explicitly
...
) to show that this set goes on forever
...
I discuss some especially significant sets of numbers in Chapter 25
...
Set theory also has four important operations:
union, intersection, relative complement, and complement
...
Here are definitions for three sets of numbers:
P = {1, 7}
Q = {4, 5, 6}
R = {2, 4, 6, 8, 10}
In this section, I use these three sets and a few others to discuss the four set
operations and show you how they work
...
Therefore, you shouldn’t have to flip back and forth to look up what each set
contains
...
For example, the
union of {1, 2} and {3, 4} is {1, 2, 3, 4}
...
For example, consider the union of Q and R
...
For example, the intersection of {1, 2, 3} and {2, 3,
4} is {2, 3}
...
You can write the following:
{1, 2, 3} + {2, 3, 4} = {2, 3}
Similarly, here’s how to write the intersection of Q and R:
Q + R = {4, 5, 6} + {2, 4, 6, 8, 10} = {4, 6}
When two sets have no elements in common, their intersection is the empty
set (8):
P + Q = {1, 7} + {4, 5, 6} = 8
Chapter 20: Setting Things Up with Basic Set Theory
The intersection of any set with itself is itself:
P+P=P
But the intersection of any set with 8 is 8:
P+8=8
Relative complement: Subtraction (sorta)
The relative complement of two sets is an operation similar to subtraction
...
Starting with the first set, you
remove every element that appears in the second set to arrive at their relative
complement
...
Both sets
share a 4 and a 6, so you have to remove those elements from R:
R – Q = {2, 4, 6, 8, 10} – {4, 5, 6} = {2, 8, 10}
Note that the reversal of this operation gives you a different result
...
In other words, order is important
...
)
Complement: Feeling left out
The complement of a set is everything that isn’t in that set
...
For example, suppose you define
the universal set like this:
U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
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Now, here are a couple of sets to work with:
M = {1, 3, 5, 7, 9}
N = {6}
The complement of each set is the set of every element in U that isn’t in the
original set:
U – M = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 3, 5, 7, 9} = {0, 2, 4, 6, 8}
U – N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} – {6} = {0, 1, 2, 3, 4, 5, 7, 8, 9}
The complement is closely related to the relative complement (see the preceding section)
...
The main difference
is that the complement is always subtraction of a set from U, but the relative
complement is subtraction of a set from any other set
...
introduce you to algebra, an amazing tool for solving a
wide variety of math problems with speed and accuracy
...
Then, I give you a bunch
of important tools for working with algebraic expressions
...
Finally, everything comes together as you
solve algebra word problems
...
X: Algebra and
Algebraic Expressions
In This Chapter
ᮣ Meeting Mr
...
Unfortunately
for some folks, remembering their first x in algebra is similar to remembering their first love who stood them up at the prom or their first car that
broke down someplace in Mexico
...
So, if you have a traumatic x-related tale, all I can say is
that the future will be brighter than the past
...
Algebra is used for solving problems that are just too difficult for
ordinary arithmetic
...
Anywhere that numbers are useful, algebra is there
...
298
Part V: The X Files: Introduction to Algebra
In this chapter, I introduce (or re-introduce) you to that elusive little fellow,
Mr
...
Then I show
you how algebraic expressions are similar to and different from the arithmetic
expressions that you’re used to working with
...
)
X Marks the Spot
In math, x stands for a number — any number
...
In contrast, a number in algebra is often called a constant,
because its value is fixed
...
For
example, consider the following:
2+2=x
Obviously, in this equation, x stands for the number 4
...
For example:
x>5
In this inequality, x stands for some number greater than 5 — maybe 6,
maybe 71⁄2, maybe 542
...
Expressing Yourself with
Algebraic Expressions
In Chapter 5, I introduce you to arithmetic expressions: strings of numbers
and operators that can be evaluated or placed on one side of an equation
...
5 – 2
2^4 – |–4| – 100
Chapter 21: Enter Mr
...
An algebraic expression is any string of mathematical symbols that can be placed on one side of an equation and that includes
at least one variable
...
In this section, I show you how to work with algebraic expressions
...
Then, I show you how to separate an algebraic expression into one or more terms and how to identify the coefficient and the variable part of each term
...
For each variable in the expression, substitute the number that
it stands for, then evaluate the expression
...
Briefly,
this means finding the value of that expression as a single number (flip to
Chapter 5 for more on evaluating)
...
For example, suppose you want to evaluate the following expression:
4x – 7
Note that this expression contains the variable x, which is unknown, so the
value of the whole expression is also unknown
...
You can use any
letter as a variable, but x, y, and z tend to get a lot of mileage
...
To evaluate the expression, substitute
2 for x everywhere it appears in the expression:
4(2) – 7
After you make the substitution, you’re left with an arithmetic expression, so
you can finish your calculations to evaluate the expression:
=8–7=1
So given x = 2, the algebraic expression 4x – 7 = 1
...
You
do powers first, so begin by evaluating the exponent 42, which equals 4 ⋅ 4:
= 2(16) – 5(4) – 15
Now proceed on to the multiplication, moving from left to right:
= 32 – 5(4) – 15
= 32 – 20 – 15
Then evaluate the subtraction, again from left to right:
= 12 – 15
= –3
So given x = 4, the algebraic expression 2x 2 – 5x – 15 = –3
...
As long as you know the value of every variable in the expression, you can
evaluate algebraic expressions with any number of variables
...
X: Algebra and Algebraic Expressions
To evaluate it, you need values of all three variables:
x=3
y = –2
z=5
The first step is to substitute the equivalent value for each of the three variables wherever you find them:
3(32 ) + 2(3)(–2) – 3(–2)(5)
Now use the rules for order of operations from Chapter 5
...
Evaluate from left to right,
remembering the rules for adding and subtracting negative numbers in
Chapter 4:
= 15 – (–30) = 15 + 30 = 45
So given the three values for x, y, and z, the algebraic expression
3x 2 + 2xy – xyz = 45
...
Coming to algebraic terms
A term in an algebraic expression is any chunk of symbols set off from the
rest of the expression by either addition or subtraction
...
Here are some examples:
301
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Part V: The X Files: Introduction to Algebra
Expression
Number of Terms
Terms
5x
One
5x
–5x + 2
x 2y + z – xyz + 8
3
Two
–5x and 2
x 2y, z , –xyz, and 8
3
Four
No matter how complicated an algebraic expression gets, you can always
separate it out into one or more terms
...
When a term has a variable, it’s called an algebraic term
...
For example, look at the following expression:
x 2y + z – xyz + 8
3
The first three terms are algebraic terms, and the last term is a constant
...
Terms are really useful because you can follow rules to move them, combine
them, and perform the Big Four operations on them
...
But for
now, this section explains a bit about terms and some of their traits
...
Each term moves as a unit, kind of like a group of people carpooling to work
together — everyone in the car stays together for the whole ride
...
You can
rearrange the two terms of this expression without changing its value
...
X: Algebra and Algebraic Expressions
Rearranging terms in this way doesn’t affect the value of the expression
because addition is commutative — that is, you can rearrange things that
you’re adding and without changing the answer
...
)
For example, suppose x = 3
...
As another example, suppose you
have this expression:
4x – y + 6
You can rearrange it in a variety of ways:
= 6 + 4x – y
= –y + 4x + 6
Because the term 4x has no sign, it’s positive, so you can write in a plus sign
as needed when rearranging terms
...
For example, suppose that x = 2 and y = 3
...
The coefficient is the
signed numerical part of a term in an algebraic expression — that is, the
number and the sign (+ or -) that goes with that term
...
So the
coefficient of –4x 3 is –4
...
So
the coefficient of x2 is 1, and the coefficient of –x is –1
...
So the coefficient of the term –7 is simply –7
...
The table above shows the four terms of the same expression, with
each term’s variable part
...
Here are some examples:
Chapter 21: Enter Mr
...
9x
8 x2
3
πy
22 xy
7
3
...
Only the coefficient changes, and it can be any real number:
positive or negative, whole number, fraction, decimal, or even an irrational
number such as π
...
)
Considering algebraic terms
and the Big Four
In this section, I get you up to speed on how to apply the Big Four to algebraic expressions
...
You
find how useful these tools are in Chapter 22, when you begin solving algebraic equations
...
For example, suppose you have the expression 2x + 3x
...
So when you add
them up, you get the following:
= x + x + x + x + x = 5x
As you can see, when the variable parts of two terms are the same, you add
these terms by adding their coefficients: 2x + 3x = (2 + 3)x
...
You cannot add non-similar terms
...
You’re faced with a situation
that’s similar to 2 apples + 3 oranges
...
(See Chapter 4 for more on how to
work with units
...
Subtract similar terms by finding the difference between their coefficients and keeping the same variable
part
...
Recall that 3x is simply shorthand for
x + x + x
...
You simply find (3 – 1)x
...
Here’s another example:
2x – 5x
Again, no problem, as long as you know how to work with negative numbers
(see Chapter 4 if you need details)
...
You cannot subtract non-similar terms
...
Think of
this as trying to figure out $7 – 4 pesos
...
(See Chapter 4 for more on working
with units
...
X: Algebra and Algebraic Expressions
Multiplying terms
Unlike adding and subtracting, you can multiply non-similar terms
...
For example, suppose you want to multiply 5x(3y)
...
To get the algebraic part, combine the variables x and y:
= 5(3)xy = 15xy
Now suppose you want to multiply 2x(7x)
...
Multiply all three coefficients together and gather up
the variables:
2x 2(3y)(4xy)
= 2(3)(4)x 2xyy
= 24x 3y 2
As you can see, the exponent 3 that’s associated with x is just the count of
how many x’s appear in the problem
...
A fast way to multiply variables with exponents is to add the exponents
together
...
Similarly, I added the exponents of the y’s
(3 + 5 + 1 = 9 — don’t forget that y = y 1!) to get the exponent of y in the solution
...
So division of algebraic terms really
looks like reducing a fraction to lowest terms (see Chapter 9 for more on
reducing)
...
Make a fraction of the two terms
...
Begin by turning the problem
into a fraction:
3xy
12x 2
2
...
In this case, you can cancel out a 3
...
Cancel out any variable that’s in both the numerator and denominator
...
As another example, suppose you want to divide –6x 2yz 3 by –8x 2y 2z
...
Notice that because both coefficients were originally negative, you can cancel out both minus signs as well:
=
3x 2 yz 3
4x 2 y 2 z
Chapter 21: Enter Mr
...
I do this in two steps as before:
=
3xxyzzz
4xxyyz
At this point, just cross out any occurrence of a variable that appears in both
the numerator and denominator:
= 3zz
4y
2
= 3z
4y
You can’t cancel out variables or coefficients if either the numerator or
denominator has more than one term in it
...
Simplifying an expression means (quite simply!)
making it smaller and easier to manage
...
For now, think of this section as a kind of algebra toolkit
...
In Chapter 22, I show you when to use them
...
This feature comes in handy when you’re trying to simplify an
expression
...
But three terms have the variable
x and the other three have the variable y
...
I do this in two steps, first for
the x terms and then for the y terms:
= 5x – 3y + y + 2y
= 5x + 0y = 5x
Notice that the x terms simplify to 5x and the y terms simplify to 0y, which is 0,
so the y terms drop out of the expression altogether!
Here’s a somewhat more complicated example that has variables with
exponents:
12x – xy – 3x 2 + 8y + 10xy + 3x 2 – 7x
This time, you have four different types of terms
...
In Chapter
5, I show you how to handle parentheses in an arithmetic expression
...
As you find when you begin
solving algebraic equations in Chapter 22, getting rid of parentheses is often
the first step toward solving a problem
...
Drop everything: Parentheses with a plus sign
When an expression contains parentheses that come right after a plus sign (+),
you can just remove the parentheses
...
X: Algebra and Algebraic Expressions
Now you can simplify the expression by combining similar terms:
= 5x + 4y
When the first term inside the parentheses is negative, when you drop the
parentheses, the minus sign replaces the plus sign
...
In this case, change the sign of every term inside the parentheses to
the opposite sign; then remove the parentheses
...
Notice that the
term 2xy appears to have no sign because it’s the first term inside the parentheses
...
For example,
2(3) = 6
4(4) = 16
10(15) = 150
This notation becomes much more common with algebraic expressions,
replacing even the dot multiplication sign (⋅):
3(4x) = 12x
4x(2x) = 8x 2
3x(7y) = 21xy
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Part V: The X Files: Introduction to Algebra
To remove parentheses without a sign, multiply the term outside the parentheses by every term inside the parentheses; then remove the parentheses
...
Here’s an example:
2(3x – 5y + 4)
In this case, multiply 2 by each of the three terms inside the parentheses:
= 2(3x) + 2(–5y) + 2(4)
For the moment, this expression looks more complex than the original one,
but now you can get rid of all three sets of parentheses by multiplying:
= 6x – 10y + 8
Multiplying by every term inside the parentheses is simply distribution of
multiplication over addition — also called the distributive property — which I
discuss in Chapter 4
...
I do this in two steps, first rearranging
and then combining:
= 6x 2 – 5x 2 – 2xy + 2xy – 12x
= x 2 – 12x
Parentheses by FOILing
Sometimes, expressions have two sets of parentheses next to each other
without a sign between them
...
Chapter 21: Enter Mr
...
The word FOIL is an acronym to help you make sure
you multiply the correct terms
...
Here’s how the process works:
1
...
Suppose you want to simplify the expression (2x – 2)(3x – 6)
...
Therefore, multiply 2x by 3x:
(2x – 2)(3x – 6)
2x(3x) = 6x 2
2
...
The two outside terms, 2x and –6, are on the ends:
(2x – 2)(3x – 6)
2x(–6) = –12x
3
...
The two terms in the middle are –2 and 3x:
(2x – 2)(3x – 6)
–2(3x) = –6x
4
...
The last term in the first set of parentheses is –2, and –6 is last term in
the second set:
(2x – 2)(3x – 6)
–2(–6) = 12
Add these four results together to get the simplified expression:
6x 2 – 12x – 6x + 12
In this case, you can simplify this expression still further by combining the
similar terms –12x and –6x:
= 6x 2 – 18x + 12
Notice that during this process, you multiply every term inside one set of
parentheses by every term inside the other set
...
FOILing is really just an application of the distributive property, which I discuss in the section before this one
...
Then, distributing
again gives you 6x 2 – 6x – 12x + 12
...
X:
Algebraic Equations
In This Chapter
ᮣ Using variables (such as x) in equations
ᮣ Knowing some quick ways to solve for x in simple equations
ᮣ Understanding the balance-scale method for solving equations
ᮣ Rearranging terms in an algebraic equation
ᮣ Isolating algebraic terms on one side of an equation
ᮣ Removing parentheses from an equation
ᮣ Cross multiplying to remove fractions
W
hen it comes to algebra, solving equations is the main event
...
Not surprisingly, this process is called solving for x, and
when you know how to do it, your confidence — not to mention your grades —
in your algebra class will soar through the roof
...
First, I show you a few informal methods
to solve for x when an equation isn’t too difficult
...
The balance-scale method is really the heart of algebra (yes, algebra has a
heart after all!)
...
You find out how
to extend these skills to algebraic equations
...
By the end of this chapter, you should have a solid grasp of a bunch of ways
to solve equations for the elusive and mysterious x
...
Solving an algebraic
equation means finding out what number x stands for
...
Then I show you a few quick ways to solve
for x when an equation isn’t too difficult
...
For example, here’s a perfectly good equation:
7 ⋅ 9 = 63
At its heart, a variable (such as x) is nothing more than a placeholder for a
number
...
Usually, this number comes after the
equal sign
...
X: Algebraic Equations
Four ways to solve algebraic equations
You don’t need to call an exterminator just to kill a bug
...
Generally speaking, you have four ways to solve algebraic equations such as
those I introduce earlier in this chapter:
ߜ Eyeballing them (also called inspection, or just looking at the problem to
get the answer)
ߜ Rearranging them, using inverse operations when necessary
ߜ Guessing and checking
ߜ Applying algebra
Eyeballing easy equations
You can solve easy problems just by looking at them
...
When a problem is this
easy and you can see the answer, you don’t need to take any particular trouble to solve it
...
This is the method that I use throughout most of this book,
especially for solving equations with formulas, like geometry problems and
problems involving weights and measures
...
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Part V: The X Files: Introduction to Algebra
Guessing and checking equations
You can solve some equations by guessing an answer and then checking to
see whether you’re right
...
Now, check to see
whether you’re right by substituting 2 for x in the equation:
3(2) + 7 = 6 + 7 = 13 < 19
WRONG!
When x = 2, the left side of the equation equals 13, not 19
...
Applying algebra to more difficult equations
When an algebraic equation gets hard enough, you find that looking at it and
rearranging it just isn’t enough to solve it
...
You also
can’t solve it just by rearranging it using an inverse operation
...
That’s where algebra comes into play
...
Throughout the rest of this chapter, I show you how to use
the rules of algebra to turn tough problems like this one into problems that
you can solve
...
X: Algebraic Equations
it
...
I call this method the balancing scale
...
In this section, I
show you how to use the balancing scale method to solve algebraic equations
...
To keep that
equal sign, you have to maintain that balance
...
For example, here’s a balanced equation:
1+2=3
O
If you add 1 to one side of the equation, the scale would go out of balance
...
And in math, any number means x:
1+2+x=3+x
Remember that x is the same wherever it appears in a single equation or
problem
...
You can just as easily subtract an x, or even multiply or divide by x, as
long as you do the same thing to both sides of the equation:
Subtract:
1+2-x=3-x
Multiply:
^1 + 2 h x = 3x
1+2 = 3
x
x
Divide:
319
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Part V: The X Files: Introduction to Algebra
Using the balance scale to isolate x
The simple idea of balance is at the heart of algebra, and it enables you to
find out what x is in many equations
...
In algebraic equations of middling difficulty, this is a three-step process:
1
...
2
...
3
...
For example, take a look at the following problem:
11x – 13 = 9x + 3
As you follow the steps, notice how I keep the equation balanced at each step:
1
...
And now, the only non-x term (16) is
on the right side of the equation
...
Get all of the x-terms on the other side by subtracting 9x from both sides
of the equation:
11x = 9x + 16
- 9x -9x
2x =
16
Again, the balance is preserved, so the new equation is correct
...
Divide by 2 to isolate x:
2x = 16
2
2
x=8
Chapter 22: Unmasking Mr
...
Rearranging Equations and Isolating X
When you understand how algebra works like a balance scale, as I show you
in the previous section, you can begin to solve more difficult algebraic equations
...
In this section, I show you how to you put your skills from Chapter 21 to work
solving equations
...
Next, I show you
how removing parentheses from an equation can help you solve it
...
Rearranging terms on one
side of an equation
Rearranging terms becomes all-important when working with equations
...
And of course, that’s true of every equation
...
For example, you can rearrange the terms on one side of an equation
...
Moving terms to the other
side of the equal sign
Earlier in this chapter, I show you how an equation is similar to a balance
scale
...
Figure 22-1:
Showing
how an
equation is
similar to a
balance
scale
...
For example:
2x - 3 = 11
-2x
- 2x
- 3 = 11 - 2x
Now take a look at these two versions of this equation side by side:
2x – 3 = 11
–3 = 11 – 2x
In the first version, the term 2x is on the left side of the equal sign
...
This example illustrates an important rule
...
Chapter 22: Unmasking Mr
...
When you move the term 3x from the right side to the left side, you have to
change its sign from plus to minus (technically, you’re subtracting 3x from
both sides of the equation):
4x – 2 – 3x = 1
After that, you can simplify the expression on the left side of the equation by
combining similar terms:
x–2=1
At this point, you can probably see that x = 3 because 3 – 2 = 1
...
Removing parentheses from equations
Chapter 21 gives you a treasure trove of tricks for simplifying expressions,
and they come in very handy when you’re solving equations
...
This is also
indispensable when you’re solving equations
...
As the equation stands, however, x-terms and constants are “locked together” inside parentheses
...
So before you can isolate terms, you need to
remove the parentheses from the equation
...
So you can start working with the expression on the left side
...
This time, the parentheses
come right after a minus sign (–)
...
Move the –x from
the right side to the left, changing it to x:
5x + 6x – 15 + x = 30 + 7 + 8
Next, move –15 from the left side to the right, changing it to 15:
5x + 6x + x = 30 + 7 + 8 + 15
Now combine similar terms on both sides of the equation:
12x = 30 + 7 + 8 + 15
12x = 60
Finally, get rid of the coefficient 12 by dividing:
12x = 60
12
12
x = 5
As usual, you can check your answer by substituting 5 into the original equation wherever x appears:
5x + (6x – 15) = 30 – (x – 7) + 8
5(5) + [6(5) – 15] = 30 – (5 – 7) + 8
25 + (30 – 15) = 30 – (–2) + 8
25 + 15 = 30 + 2 + 8
40 = 40 ✓
Chapter 22: Unmasking Mr
...
This time, however, on the left side of the equation, you have no sign between
3 and (–3x + 1)
...
To
remove the parentheses, multiply 3 by both terms inside the parentheses:
11 – 9x + 3 = 25 – (7x – 3) – 12
On the right side, the parentheses begin with a minus sign, so remove the
parentheses by changing all the signs inside the parentheses:
11 – 9x + 3 = 25 – 7x + 3 – 12
Now, you’re ready to isolate the x terms
...
Begin with the multiplication inside the parentheses, which
I’ve underlined:
11 + 3(3 + 1) = 25 – (–7 – 3) – 12
Now you can simplify what’s inside each set of parentheses:
11 + 3(4) = 25 – (–10) – 12
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Part V: The X Files: Introduction to Algebra
And now you can remove the parentheses and complete the check:
11 + 12 = 25 + 10 – 12
23 = 23 ✓
Copy this example and work through it a few times with the book closed
...
As I discuss in Chapter 9, you can use cross-multiplication to find out whether two
fractions are equal
...
You can’t do the division or cancel anything out
because the fraction on the left has two terms in the denominator, and the fraction on the right has two terms in the numerator (see Chapter 21 for info on
dividing algebraic terms)
...
So if you
that you have is that the fraction
4x
2x - 2
cross-multiply these two fractions, you get two results that are also equal:
x(4x) = (2x + 3)(2x – 2)
At this point, you have something you know how to work with
...
X: Algebraic Equations
The right side requires a bit of FOILing (flip to Chapter 21 for details):
4x 2 = 4x 2 – 4x + 6x – 6
Now all the parentheses are gone, so you can isolate the x terms
...
You may be able to eyeball the correct
answer, but here’s how to finish:
6 = 2x
2
2
x= 3
To check your answer, substitute 3 back into the original equation:
2 ^ 3h + 3
3
=
2 ^ 3h - 2
4 ^ 3h
3 = 6+3
6-2
12
3= 9
4 12
3=3
4 4
This checks out, so the answer x = 3 is correct
...
X to Work:
Algebra Word Problems
In This Chapter
ᮣ Solving algebra word problems in simple steps
ᮣ Working through solving an algebra word problem
ᮣ Choosing variables
ᮣ Using charts
W
ord problems that require algebra are among the toughest problems
that students face — and the most common
...
And standardized tests virtually always
include these types of problems
...
Then, I give you a bunch of examples that take you through
all five steps
...
First, I show you how to choose a variable that makes your
equation as simple as possible
...
By the end of this chapter, you should
have a solid understanding of how to solve a wide variety of algebra word
problems
...
Throughout this section, I use the following word problem as an example:
In three days, Alexandra sold a total of 31 tickets to her school play
...
And on
Thursday, she sold exactly 7 tickets
...
Here’s what I came up with:
Tuesday:
twice as many as on Wednesday
Wednesday:
?
Thursday:
7
Total:
31
At this point, all the information is in the chart, but the answer still may not
be jumping out at you
...
Here are the five steps for solving most algebra word problems:
1
...
2
...
3
...
4
...
5
...
Declaring a variable
As you know from Chapter 21, a variable is a letter that stands for a number
...
That doesn’t mean you don’t need algebra to
Chapter 23: Putting Mr
...
It just means that you’re going to have to put x into the
problem yourself and decide what it stands for
...
Here are some examples of variable declarations:
Let m = the number of dead mice that the cat dragged into the house
...
Let c = the number of complaints Arnold received after he painted his
garage door purple
...
Notice that the earlier chart for the sample problem has a big question mark
next to Wednesday
...
Here’s how you do it:
Let w = the number of tickets that Alexandra sold on Wednesday
...
This practice makes remembering what the variable means a
lot easier, which will help you later in the problem
...
Setting up the equation
After you have a variable to work with, you can go through the problem again
and find other ways to use this variable
...
Now you have a lot more information to fill in the chart:
Tuesday:
Twice as many as on Wednesday
2w
Wednesday:
?
w
Thursday:
7
7
Total:
31
31
331
332
Part V: The X Files: Introduction to Algebra
You know that the total number of tickets, or the sum of the tickets she sold
on Tuesday, Wednesday, and Thursday, is 31
...
Here’s the equation one more time:
2w + w + 7 = 31
For starters, remember that 2w really means w + w
...
So on the left side of
the equation, you want to get rid of the 7
...
To do this, you
have to undo the multiplication by 3, so divide both sides by 3:
3w = 24
3
3
w=8
Answering the question
You may be tempted to think that after you’ve solved the equation, you’re
done
...
Look back at the problem, and
you see that it asks you this question:
How many tickets did Alexandra sell on each day, Tuesday through
Thursday?
Chapter 23: Putting Mr
...
The problem tells you that Alexandra sold 7 tickets on Thursday
...
And on
Tuesday, she sold twice as many on Wednesday, so she sold 16
...
Checking your work
To check your work, compare your answer to the problem, line by line, to
make sure every statement in the problem is true:
In three days, Alexandra sold a total of 31 tickets to her school play
...
On Tuesday, she sold twice as many tickets as on Wednesday
...
And on Thursday, she sold exactly 7 tickets
...
Choosing Your Variable Wisely
Declaring a variable is simple, as I show you earlier in this chapter, but you
can make the rest of your work a lot easier when you know how to choose
your variable wisely
...
For example, suppose you’re trying to solve this problem:
Irina has three times as many clients as Toby
...
” It’s significant because it indicates a relationship between Irina and
Toby that’s based on either multiplication or division
...
333
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Part V: The X Files: Introduction to Algebra
Whenever you see a sentence that indicates you should use either multiplication or division, choose your variable to represent the smaller number
...
Suppose you begin by declaring your variable as follows:
Let t = the number of clients that Toby has
...
To check this result — which I recommend highly earlier in this chapter! — note that 13 + 39 = 52
...
Given that variable, you’d have to represent Toby’s clients using the fraction i ,
3
which leads to the same answer but a lot more work
...
In this section, the complexity
increases from two or three people to four and then five
...
Chapter 23: Putting Mr
...
Here’s a problem that involves four people:
Alison, Jeremy, Liz, and Raymond participated in a canned goods drive at
work
...
Together,
the two women donated two more cans than the two men
...
Remember that to avoid fractions, you want to declare a variable based on the person who brought in the
fewest cans
...
Furthermore, Raymond donated more cans than Liz
...
The next sentence tells you that the women donated two more cans
than the men, so make a word problem, as I show you in Chapter 6:
Liz + Alison = Jeremy + Raymond + 2
You can now substitute into this equation as follows:
3j + 2j = j + 3j + 7 + 2
With your equation set up, you’re ready to solve
...
With this information, you can go back to the chart, plug in 9
for j, and find out how many cans the other children donated: Liz donated 27,
Alison donated 18, and Raymond donated 34
...
To check the numbers, read through the problem and make sure they work at
every point in the story
...
Crossing the finish line with five people
Here’s one final example, the most difficult in this chapter, in which you have
five people to work with
...
So far this
month, Mina has run 12 miles, Suzanne has run 3 more miles than Jake,
and Kyle has run twice as far as Victor
...
How far has each person
run so far?
The most important thing to notice in this problem is that there are two sets
of numbers: the miles that all five people have run up to today and their
mileage including tomorrow
...
Here’s how to set up a chart:
Today
Tomorrow (Today + 5)
Jake
Kyle
Mina
Suzanne
Victor
With this chart, you’re off to a good start to solve this problem
...
Here it is:
...
Because Victor has run fewer miles than Kyle, declare your variable as follows:
Let v = the number of miles that Victor has run up to today
...
X to Work: Algebra Word Problems
Notice that I added the word today to the declaration to be very clear that I’m
talking about Victor’s miles before the 5-mile run tomorrow
...
I’ve also begun to fill in the Tomorrow
column by adding 5 to my numbers in the Today column
...
Jake will have run as far as Mina and Victor combined
...
Now I can use the information that,
today, Suzanne has run 3 more miles than Jake:
Today
Tomorrow (Today + 5)
Jake
17 + v
17 + v + 5
Kyle
2v
2v + 5
Mina
12
17
Suzanne
17 + v + 3
17 + v + 8
Victor
v
v+5
337
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Part V: The X Files: Introduction to Algebra
With the chart filled in like this, you can begin to set up your equation
...
With this
information, you substitute 20 for v and fill in the chart as follows:
Today
Tomorrow (Today + 5)
Jake
37
42
Kyle
40
45
Mina
12
17
Suzanne
40
45
Victor
20
25
The Today column contains the answers to the question that the problem
asks
...
For example, tomorrow the five people will have run a total of 174 miles
because
42 + 45 + 17 + 45 + 25 = 174
Copy down this problem, close the book, and work through it for practice
...
ust for fun, this part of the book includes a couple of
top-ten lists on math-related topics
...
And I list some very important sets of numbers
...
But inside all this, certain concepts get so much airplay that in my humble
opinion, they make the Math Hall of Fame
...
Knowing them can
change your world, too — or at least give you a broader perspective of what
math is all about
...
Getting Set with Sets
A set is a collection of objects
...
Sets are such a simple and flexible way of organizing the world that you can
define all of math in terms of them
...
See Chapter 20 for more on set theory and Chapter 25 for some
significant number sets
...
Here are the first
ten prime numbers:
2
3
5
7
11
13
17
19
23
29
...
Beyond this, prime numbers are, in an important sense, the elements from
which all other numbers can be built
...
Far more than just a curious bit of trivia, the uniqueness of each number’s
prime factors goes by a much more important name: the Fundamental
Theorem of Arithmetic
...
Zero: Much Ado about Nothing
Zero may look like a big nothing, but it’s actually one of the greatest inventions of all time
...
The Greeks and Romans, who knew so much about math and logic, knew
nothing about zero
...
The concept of zero as a number arose independently in several different
places
...
And the Hindu-Arabic system used throughout most
of the world today developed from an earlier Arabic system that used
zero as a placeholder
...
)
In fact, zero isn’t really nothing — it’s simply a way to express nothing mathematically
...
Chapter 24: Ten Key Math Concepts You Shouldn’t Ignore
Going Greek: Pi (π)
The symbol π (pi — pronounced pie) is a Greek letter that stands for the ratio
of the circumference of a circle to its diameter (see Chapter 16 for the scoop
on circles)
...
1415926535
...
Circles are one of the
most basic shapes in geometry, and you need π to measure the area and
the circumference of a circle
...
ߜ Pi is an irrational number, which means that no fraction that equals it
exactly exists
...
Thus, even though π emerges from a very
simple operation (measuring a circle), it contains a deep complexity that
numbers such as 0, 1, –1, 1⁄2, and even 2 don’t share
...
)
ߜ Pi is everywhere in math
...
One example is trigonometry, the study of
triangles
...
On the Level: Equal Signs and Equations
Almost everyone takes the humble equal sign (=) for granted
...
But the fact that the equal sign
shows up practically everywhere only adds weight to the idea that the concept of equality — an understanding of when one thing is mathematically the
same as another — is one of the most important math concepts ever created
...
The equal sign
links two mathematical expressions that have the same value
...
That’s why nearly everything in math involves equations
...
The equal
sign provides a powerful way to connect expressions, which allows scientists
to connect ideas in new ways
...
Einstein’s famous equation — E = mc2 — links an expression that represents energy with one that represents matter
...
For details on how the concepts of equality and balance play out in algebra,
flip to Chapter 22
...
It
was invented by French philosopher and mathematician René Descartes
...
Algebra was exclusively the study of equations (see
Part V), and geometry was solely the study of figures on the plane or in space
(see Chapter 16)
...
The result was analytic geometry, a new mathematics that not only merged
the ancient sciences of algebra and geometry but also brought greater clarity
to both
...
In and Out: Relying on Functions
A function is a mathematical machine that takes in one number (called the
input) and gives back exactly one other number (called the output)
...
When you put in fruit, you
get a fruit smoothie
...
Suppose I invent a function called PlusOne that adds 1 to any number
...
And this will happen for every even number
...
This process may look rather simplistic, but as with sets, the simplicity of
functions gives them their power
...
Functions get a lot of play as you move forward in algebra
...
For a
deeper look at functions, see Algebra For Dummies by Mary Jane Sterling
(Wiley)
...
So does the symbol for infinity
(∞)
...
Yessiree, that’s big
...
Infinity, beyond any classification of size
or number, is the very quality of endlessness
...
In his invention of calculus, Sir Isaac Newton introduced the concept of a limit,
which allows you to calculate what happens to numbers as they get very large
and approach infinity
...
And here’s the kicker: This set of larger and larger infinities
itself is infinite
...
)
345
346
Part VI: The Part of Tens
The Real Number Line
The number line has been around for a very long time, and it’s one of the first
visual aids that teachers use to teach kids about numbers
...
Well, okay, that sounds pretty obvious, but
strange to say, this concept wasn’t fully understood for thousands of years
...
Then you have to go half the remaining distance
(1⁄4)
...
This
pattern continues forever:
⁄2
1
⁄4
1
⁄8
1
⁄16
1
⁄32
1
⁄64
1
⁄128
1
⁄256
...
Obviously, in the real world, you can and do walk across rooms all the time
...
The basic problem was this: All the fractions listed in the preceding sequence
are between 0 and 1 on the number line
...
But how can you have an infinite number of numbers in a finite space?
Mathematicians of the 19th century — Augustin Cauchy, Richard Dedekind,
Karl Weierstrass, and Georg Cantor foremost among them — solved this
paradox
...
The Imaginary Number i
The imaginary numbers are a set of numbers not found on the real number
line
...
But real-world applications in electronics, particle physics, and many other
areas of science have turned skeptics into believers
...
See Chapter 25 for info on imaginary and complex numbers
...
When
you’re working with just the counting numbers and a few simple operations, numbers seem to develop a landscape all their own
...
In this chapter, I take you
on a mind-expanding tour of ten sets of numbers
...
I continue with
the integers (positive and negative counting numbers and 0), the rational
numbers (integers and fractions), and real numbers (all numbers on the
number line)
...
The tour
ends with the bizarre and almost unbelievable transfinite numbers
...
Each of these sets of numbers serves a different purpose, some familiar (such
as accounting or carpentry), some scientific (such as electronics and
physics), and a few purely mathematical
...
They start with 1 and go up from there:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
...
The counting numbers are useful for keeping track of tangible objects:
stones, chickens, cars, cellphones — anything that you can touch and that
you don’t plan to cut into pieces
...
That is, if you add or multiply any two counting numbers, the result is
also a counting number
...
For example, if you subtract 2 – 3, you get –1, which is a negative
number rather than a counting number
...
If you place 0 in the set of counting numbers, you get the set of whole numbers
...
, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6,
...
Because the integers include the negative numbers, you can use them to keep
track of anything that can potentially involve debt
...
For example, if you have $100 in your checking account and
write a check for $120, you find that your new balance will drop to –$20 (not
counting any fees that the bank charges!)
...
In other words, if you add, subtract, or multiply any two integers, the result is
also an integer
...
For example, if you
divide the integer –2 by the integer 5, you get the fraction –2⁄5, which isn’t an
integer
...
Here, I list only the rational numbers from
–1 to 1 whose denominators (bottom numbers) are positive numbers less
than 5:
...
–3⁄4
...
–1⁄2
...
–1⁄4
...
1⁄4
...
1⁄2
...
3⁄4
...
The ellipses tell you that between any pair of rational numbers are an infinite
number of other rational numbers — a quality called the infinite density of
rational numbers
...
For example, a ruler wouldn’t be much good if it were to measure
length only to the nearest inch
...
Similarly, measuring
cups, scales, precision clocks, and thermometers that allow you to make
measurements to a fraction of a unit also use rational numbers
...
)
The set of rational numbers is closed under the Big Four operations
...
Making Sense of Irrational Numbers
In a sense, the irrational numbers are a sort of catchall; every number on the
number line that isn’t rational is irrational
...
349
350
Part VI: The Part of Tens
Instead, an irrational number can be approximated only as a non-terminating,
non-repeating decimal: the string of numbers after the decimal point goes on
forever without creating a pattern
...
Another common irrational number is 2, which represents the diagonal distance across a square
with a side of 1 unit
...
Irrational numbers fill out the spaces in the real number line
...
) These numbers are
used in many cases in which you need not just a high level of precision, as
with the rational numbers, but the exact value of a number that can’t be represented as a fraction
...
I discuss both of these types of numbers in the sections that
follow
...
A polynomial equation is an algebraic equation that meets
the following conditions:
ߜ Its operations are limited to addition, subtraction, and multiplication
...
ߜ Its variables are raised only to positive, whole-number exponents
...
Here are some polynomial equations:
2x + 14 = (x + 3)2
2x 2 – 9x – 5 = 0
Every algebraic number shows up as the solution of at least one polynomial
equation
...
Thus, 2 is an algebraic number
whose approximate value is 1
...
(see Chapter 4 for more information on square roots)
...
Like the irrational
numbers, transcendental numbers are also a sort of catchall: Every number
on the number line that isn’t algebraic is transcendental
...
1415926535
...
(See Chapters 16 and 24 for more on π
...
Sines, cosines, tangents, and other
trigonometric functions are often transcendental numbers
...
7182818285
...
People use e to
do problems on compound interest, population growth, radioactive decay,
and the like
...
The real numbers comprise every point on the number line
...
That is, if you take any two real numbers and add, subtract, multiply, or divide them, the result is always another real number
...
To understand what’s so strange about imaginary numbers, it helps to know
a bit about square roots
...
For example, the
square root of 9 is 3 because 3 ⋅ 3 = 9
...
(See Chapter 4 for more on square roots and multiplying
negative numbers
...
If it were on the real number line, it would
be a positive number, a negative number, or 0
...
And when you multiply
any negative number by itself, you also get a positive number
...
If -1 isn’t on the real number line, where is it? That’s a good question
...
They banished it to the mathematical
non-place called undefined, which is the same place where they kept fractions
with a denominator of 0
...
Mathematicians designated -1 with the symbol i
...
Figure 25-1 shows some numbers that form the imaginary
number line
...
−3i −2i −i
0
i
2i
3i
Even though these numbers are called imaginary, mathematicians today consider them no less real than the real numbers
...
Grasping the Complexity
of Complex Numbers
A complex number is any real number (see “Getting Grounded in Real
Numbers,” earlier in this chapter) plus or minus an imaginary number
(see the preceding section)
...
Mathematicians call these nesting sets
subsets
...
Because the set of counting
or natural numbers (represented as N) is completely contained within the set of integers, N
is a subset, or part, of Z
...
Because
the set of integers is completely contained
within the set of rational numbers, N and Z are
both subsets of Q
...
The set of complex numbers is called C
...
The symbol ʚ means is a subset of (see Chapter 20 for details on set notation)
...
Because the
set of rational numbers is completely contained
You can turn any real number into a complex number by just adding 0i
(which equals 0):
3 = 3 + 0i
–12 = –12 + 0i
3
...
14 + 0i
These examples show you that the real numbers are just a part of the larger
set of complex numbers
...
That is, if you take any two complex numbers and add, subtract, multiply, or divide them, the result is always another complex number
...
Consider this for a moment: The counting numbers (1, 2, 3,
...
But there are more real numbers than counting
numbers
...
Mathematician Georg Cantor proved this fact
...
He called
these numbers transfinite because they transcend, or go beyond, what you
think of as infinite
...
Because the counting numbers are
infinite, the familiar symbol for infinity (∞) and ℵ0 mean the same thing
...
This is a higher order of infinity than ∞
...
And the sets of irrational, transcendental, imaginary, and complex numbers
all have ℵ1 elements
...
Here’s the set of transfinite numbers:
ℵ0, ℵ1, ℵ2, ℵ3,
...
As you can see, on the surface, the
transfinite numbers look similar to the counting numbers (in the first section
of this chapter)
...
Index
• Symbols and Numerics •
≈ (approximately equals), 34–35, 67
{} (braces), 288
[=/] (doesn’t equal), 66
÷ (division sign), 51
· (dot), 44–45
...
See also formulas
circle formula, 241
defined, 239
parallelogram formula, 245
rectangle formula, 244
rhombus formula, 245
square formula, 243
trapezoid formula, 246, 268
triangle formula, 242, 268
absolute value, 70
acute angles, 232
addends, 38
adding
algebraic terms, 305–306
arithmetic equations, 76
associative operations, 60
carrying digits, 38–39
commutative operations, 59
decimals, 164–165
exponents, 210
expressions, 76
fractions, 136–143
inverse operations, 58–59
356
Basic Math and Pre-Algebra For Dummies
adding (continued)
large numbers in columns, 38–39
mixed numbers, 148–151
negative numbers, 62–63
on the number-line, 18–19
overview, 37–39
units, 65
word problems, 193–195
addition
...
See also algebraic
expressions; arithmetic equations
balancing scale, 318–321
cross-multiplication, 326–327
isolating x, 320–321
overview, 316
rearranging terms, 321–323
removing parentheses, 323–326
setting up, 331–332
solving, 317–318, 332
using x in, 316
algebraic expressions
...
See also formulas
circle formula, 241
defined, 239
parallelogram formula, 245
rectangle formula, 244
rhombus formula, 245
square formula, 243
trapezoid formula, 246, 268
triangle formula, 242, 268
arithmetic equations
...
See also formulas
balancing scale
defined, 318–319
using, 319–321
bar graph, 252–253
base (b), 242
...
See also formulas
C subset, 353
Cantor, Georg, 345, 354
capacity
...
See also formulas
closed set, 348
coefficient, 304
column addition, 38
commutative
operations, 59
property defined, 59
property of multiplication, 23
complement, of sets, 293–294
complex numbers, 352–353
composite numbers
defined, 15–16
identifying, 102–104
cones
defined, 238
finding the volume of, 249–250
constant, 298, 302
conventions, used in this book, 2
conversion
equations, 265–267
factors, 225–226
conversion chains
defined, 262
setting up, 262–264
word problems, 261–267
converting
decimals to fractions, 172–174
decimals to percents, 181
fractions to decimals, 174–175
fractions to percents, 182–183
measurement units, 225–227
percents to decimals, 181
percents to fractions, 182
units, 227
coordinates
Cartesian, 255–260
defined, 256
counting
by numbers, 13–14
outcomes, 284–286
counting numbers
defined, 18, 25
overview, 348
set, 26
cross-multiplication
fractions, 131–132
overview, 326–327
cubes, finding the volume of, 247–248
cylinders
defined, 238
finding the volume of, 249
•D•
d (diameter)
...
See also formulas
circle formula, 240–241
defined, 233
measuring, 240–241
difference, 40
digital root, finding, 99
digits
carrying, 38–39
numbers versus, 30
distance
estimating, 223–224
units of, 221
distribution
algebraic expressions, 311–312
overview, 61
distributive property
of multiplication over addition, 61
overview, 202
dividend, 51, 169
dividing
algebraic terms, 308–309
arithmetic equations, 77
decimals, 168–171
expressions, 77
fractions, 136
inverse operations, 58–59
long, 52–53
mixed numbers, 147–148
negative numbers, 64
noncommutative operations, 59
number-line, 23–24
overview, 51–54
remainders, 54
symbols, 51
units, 65–66
divisibility
checking, 99–102
composite numbers, 102–104
prime numbers, 102–104
tricks, 97–102
divisible, 97
division
...
(ellipsis), 288
empty sets
defined, 19
overview, 290–291
English system of measurement, 217–220
equal sets, 289–290
= (equal to), overview, 343–344
equality, properties, 72
equations
...
See also solving
algebraic expressions, 299–301
defined, 71
overview, 73–74
even numbers, 13
exponents
adding, 210
multiplying, 210
multiplying with, 17–18
order of operations in expressions with,
78–79
overview, 68–69
powers of ten, 208–210
expressions
addition, 76
algebraic, 298–301
applying order of operations to, 75–78
defined, 71
division, 77
mixed-operator, 77–78
multiplication, 77
order of operations and exponential,
78–79
order of precedence with parenthetical,
79–82
overview, 73–74
subtraction, 76
•F•
factorization, prime, 112–114, 115–116, 118
factorization tree, prime factors, 110–112
factors
conversion, 225–226
defined, 44
finding the GCF (greatest common
factor), 114–115
generating, 108–109
identifying, 107–108
multiples and, 106–107
prime, 109–114
Fahrenheit, Celsius versus, 225
favorable outcome, 283
finding the value of an expression
...
See elements
metric prefixes, basic, 221
metric system
estimating, 223–225
overview, 220–222
metric units, basic, 220
Micro-, ten metric prefix, 221
Mili-, ten metric prefix, 221
minuend, 40
– (minus sign), 39
mixed numbers
adding, 148–151
components, 130
defined, 125
dividing, 147–148
improper fractions and, 129–131
multiplying, 147–148
subtracting, 151–153
mixed-operator expressions, 77–78
mode, 279
multiples
defined, 22
factors and, 106–107
generating, 116
multiplicand, 44
multiplication
...
See zeros
number–line
adding, 18–19
dividing, 23–24
fractions, 24–25
multiplying, 22–23
Index
negative numbers, 21
overview, 18, 346
subtracting, 18–19
zero, 19–20
number sequences
composite numbers, 15–16
counting by numbers, 13–14
even, 13
exponents, 17–18
odd, 13
overview, 12
prime numbers, 16
square numbers, 14
number sets
algebraic numbers, 350
complex numbers, 352–353
counting numbers, 348
imaginary numbers, 351–352
integers, 348–349
irrational numbers, 349–350
natural numbers, 348
rational numbers, 349
real numbers, 351
subsets, 353
transcendental numbers, 351
transfinite numbers, 353–354
types, 347–354
numbers
...
See also order of
operations
expressions with parentheses, 79–82
organization, of this book, 3–6
origin, 255
ounces, fluid ounces versus, 219
outcomes, counting, 284–286
•P•
P (perimeter)
...
See also formulas
overview, 233, 239
parallelogram formula, 245
rectangle formula, 244
rhombus formula, 244
square formula, 243
periods, 32
π (pi)
measuring, 241
overview, 343
pie chart, 253–254
place value
chart, 31
decimals, 158–159
overview, 30–32
placeholders, leading zeros versus, 31–32
plane geometry, 230
plotting a point, 256
+ (plus sign), 38
points
geometry, 230
plotting, 256
polygons
defined, 234
types, 234–236
polyhedrons, 237
polynomial equation, 350
possible outcome, 283
powers
...
See word problems
product, 44, 106
proper fractions, 125–126
properties, of the Big Four Operations,
57–61
Index
pyramids, finding the volume of, 249–250
Pythagorean theorem, 242–243
•Q•
Q subset, 353
quadrilaterals
defined, 234
types, 234–235
qualitative data
defined, 276
working with, 277–279
quantitative data
defined, 276–277
working with, 279–282
quotient, 51
•R•
r (radius)
defined, 233, 247
measuring, 240–241
R subset, 353
radical
...
See number–line
real numbers
defined, 25
overview, 351
reciprocal fractions, 124
rectangles
area formula, 244
defined, 235
measuring, 243–244
perimeter formula, 244
reducing fractions, 127–129
reflexivity, 72
regular polygon, 236
relative complement, of sets, 293
remainder, 54
removing, parentheses from equations,
323–326
repeating decimals, 176
rhombus
area formula, 245
defined, 235
measuring, 244–245
perimeter formula, 244
right angle, 232
right triangle, 234
Roman numerals, 359–360
root
...
See also evaluation
algebraic expressions, 309–313
solid, defined, 236
solid geometry
defined, 236
overview, 236–239
solving
...
See subtracting
subtrahend, 40
sum, 38
...
See also word problems
defined, 85
turning word problems into, 85–88
•V•
variable
declaring, 330–331
defined, 298
identifying, 304
selecting, 333–334
vertical axis (y-axis), 255
volume
box formula, 248
cone formula, 249–250
cube formula, 247–248
cylinder formula, 249
defined, 247
estimating, 224
prism formula, 248–249
pyramid formula, 249–250
sphere formula, 247
units of, 221
367
368
Basic Math and Pre-Algebra For Dummies
word problems
Title: Pre-Algebra 365pg
Description: This will help you in pre-algebra and math in general. This PDF is aimed at people who want to learn or review pre-algebra/math.
Description: This will help you in pre-algebra and math in general. This PDF is aimed at people who want to learn or review pre-algebra/math.