15
U
BY MARK ZEGARELLI
BASIC MATH &
PRE‐ALGEBRA
U Can: Basic Math & Pre‐Algebra For Dummies®Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030‐5774, www.wiley.com
Copyright © 2015 by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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Library of Congress Control Number: 2014956254
ISBN 978‐1‐119‐06796‐2 (pbk); ISBN 978‐1‐119‐06795‐5 (ebk); ISBN 978‐1‐119‐06794‐8 (ebk)
Manufactured in the United States of America
10 9 8 7 6 5 4 3 2 1
CONTENTS AT A GLANCEINTRODUCTION ...............................................................................................................1
PART I GETTING STARTED WITH BASIC MATH AND PRE-ALGEBRA .......................................................................................5
chapter 1: Playing the Numbers Game ....................................................................................................... 7
chapter 2: It’s All in the Fingers: Numbers and Digits ............................................................................ 15
PART II GETTING A HANDLE ON WHOLE NUMBERS ..................21 chapter 3: Putting the Big Four Operations to Work .............................................................................. 23
chapter 4: A Question of Values: Evaluating Arithmetic Expressions .................................................. 59
chapter 5: Say What? Turning Words into Numbers .............................................................................. 81
chapter 6: Divisibility and Prime Numbers............................................................................................ 103
chapter 7: Fabulous Factors and Marvelous Multiples ........................................................................ 121
PART III PARTS OF THE WHOLE: FRACTIONS, DECIMALS, AND PERCENTS .......................................................135
chapter 8: Fooling with Fractions ........................................................................................................... 137
chapter 9: Parting Ways: Fractions and the Big Four Operations ....................................................... 161
chapter 10: Dallying with Decimals .......................................................................................................... 213
chapter 11: Playing with Percents ............................................................................................................ 253
chapter 12: Word Problems with Fractions, Decimals, and Percents .................................................. 273
PART IV PICTURING AND MEASURING: MEASURES, STATS, AND SETS ..................................................289
chapter 13: A Perfect Ten: Condensing Numbers with Scientific Notation ......................................... 291
chapter 14: How Much Have You Got? Weights and Measures ............................................................. 305
chapter 15: Picture This: Basic Geometry ............................................................................................... 329
chapter 16: Figuring Your Chances: Statistics and Probability ............................................................. 361
chapter 17: Setting Things Up with Basic Set Theory ............................................................................ 377
PART V THE X-FILES: INTRODUCTION TO ALGEBRA .............. 387chapter 18: Enter Mr. X: Algebra and Algebraic Expressions ................................................................ 389
chapter 19: Unmasking Mr. X: Algebraic Equations................................................................................ 421
chapter 20: Putting Mr. X to Work: Algebra Word Problems ................................................................. 447
PART VI THE PART OF TENS ......................................................................... 465chapter 21: Ten Little Math Demons That Trip People Up .................................................................... 467
chapter 22: Ten Important Number Sets to Know .................................................................................. 473
chapter 23: Ten Curious Types of Numbers ............................................................................................ 479
chapter 24: Ten Alternative Numeral and Number Systems ................................................................. 485
INDEX .........................................................................................................................................493
TABLE OF CONTENTSINTRODUCTION .............................................................................................. 1
About This Book........................................................................................ 1Foolish Assumptions ................................................................................ 1Beyond the Book ....................................................................................... 2Where to Go from Here ............................................................................. 3
PART I GETTING STARTED WITH BASIC MATH AND PRE-ALGEBRA ................................................................ 5
chapter 1: Playing the Numbers Game . . . . . . . . . . . . . . . . . . . . 7Inventing Numbers.................................................................................... 7Understanding Number Sequences ........................................................ 8
Evening the odds ............................................................................. 8Counting by threes, fours, fives, and so on .................................. 9Getting square with square numbers ............................................ 9Composing yourself with composite numbers .......................... 10Stepping out of the box with prime numbers ............................ 11Multiplying quickly with exponents ............................................ 11
Four Important Sets of Numbers ........................................................... 12Counting on the counting numbers ............................................. 13Introducing integers ...................................................................... 13Staying rational .............................................................................. 14Getting real ..................................................................................... 14
chapter 2: It’s All in the Fingers: Numbers and Digits . . . . . . 15Knowing Your Place Value ...................................................................... 16
Counting to ten and beyond ......................................................... 16Telling placeholders from leading zeros ..................................... 16Reading long numbers .................................................................. 17
Close Enough for Rock ‘n’ Roll: Rounding and Estimating ................ 18Rounding numbers ........................................................................ 18Estimating value to make problems easier ................................. 19
v
U Can: Basic Math & Pre-Algebra For Dummies vi
PART II GETTING A HANDLE ON WHOLE NUMBERS .....................................................................................21
chapter 3: Putting the Big Four Operations to Work . . . . . . . . 23The Big Four Operations ........................................................................ 23
Adding things up: Addition ........................................................... 23Take it away: Subtraction.............................................................. 24A sign of the times: Multiplying ................................................... 25Doing division lickety‐split ........................................................... 26
Switching Things Up with Inverse Operations and the Commutative Property ................................................................. 26
Getting with the In‐Group: Parentheses and the Associative Property ........................................................................... 31
Distribution to lighten the load.................................................... 34Understanding Where Negative Numbers Come From ...................... 36Sign‐Switching: Understanding Negation and Absolute Value .......... 38Addition and Subtraction with Negative Numbers ............................. 41Knowing Signs of the Times (And Division) for
Negative Numbers ............................................................................... 46Understanding Inequalities .................................................................... 50
Doesn’t equal (≠) ............................................................................ 50Less than (<) and greater than (>) ............................................... 50Less than or equal to (≤) and greater than or equal to (≥) ....... 51Approximately equals (≈) ............................................................. 51
Moving Beyond the Big Four: Exponents and Square Roots ............. 53Understanding exponents............................................................. 54Discovering your roots ................................................................. 55
chapter 4: A Question of Values: Evaluating Arithmetic Expressions . . . . . . . . . . . . . . . . . . . . . . 59
Seeking Equality for All: Equations ....................................................... 59Hey, it’s just an expression ........................................................... 60Evaluating the situation ................................................................ 60Putting the Three E’s together ..................................................... 61
Introducing Order of Operations .......................................................... 61Expressions with only addition and subtraction ....................... 62Expressions with only multiplication and division ................... 64Mixed‐operator expressions ......................................................... 66
Handling Powers Responsibly ............................................................... 68Prioritizing parentheses ................................................................ 70Pulling apart parentheses and powers ....................................... 72Figuring out nested parentheses ................................................. 75
Bringing It All Together: The Order of Operations ............................. 77
Table of Contents vii
chapter 5: Say What? Turning Words into Numbers . . . . . . . . 81Dispelling Two Myths about Word Problems ...................................... 81
Word problems aren’t always hard.............................................. 82Word problems are useful ............................................................. 82
Solving Basic Word Problems ................................................................ 82Turning word problems into word equations ............................ 83Plugging in numbers for words .................................................... 88
Solving More‐Complex Word Problems ................................................ 92When numbers get serious ........................................................... 93Too much information ................................................................... 96
chapter 6: Divisibility and Prime Numbers . . . . . . . . . . . . . . 103Knowing the Divisibility Tricks ........................................................... 103
Counting everyone in: Numbers you can divide everything by ............................................................................ 103
In the end: Looking at the final digits ........................................ 104Count it up: Checking divisibility by adding and
subtracting digits ..................................................................... 107Less is more: Checking divisibility by subtracting .................. 112Cross‐checking: Using multiple tests ........................................ 114
Identifying Prime and Composite Numbers....................................... 115
chapter 7: Fabulous Factors and Marvelous Multiples . . . . . 121Knowing Six Ways to Say the Same Thing .......................................... 121Understanding Factors and Multiples ................................................ 122Finding Fabulous Factors ..................................................................... 124
Deciding when one number is a factor of another .................. 124Understanding factor pairs......................................................... 125
Generating a Number’s Factors ........................................................... 125Decomposing a Number into Its Prime Factors ................................ 127Finding the Greatest Common Factor ................................................ 129Generating the Multiples of a Number ............................................... 132Finding the Least Common Multiple ................................................... 133
PART III PARTS OF THE WHOLE: FRACTIONS, DECIMALS, AND PERCENTS ...................................135
chapter 8: Fooling with Fractions . . . . . . . . . . . . . . . . . . . . . . 137Slicing a Cake into Fractions ................................................................ 137Knowing the Fraction Facts of Life ..................................................... 139
Telling the numerator from the denominator .......................... 139Flipping for reciprocals ............................................................... 140Using ones and zeros ................................................................... 140Mixing things up........................................................................... 140Knowing proper from improper ................................................. 141
U Can: Basic Math & Pre-Algebra For Dummies viii
Increasing and Reducing Terms of Fractions .................................... 144Increasing the terms of fractions ............................................... 145Reducing fractions to lowest terms........................................... 146
Converting between Improper Fractions and Mixed Numbers ....... 151Knowing the parts of a mixed number ...................................... 151Converting a mixed number to an improper fraction ............. 152Converting an improper fraction to a mixed number ............. 152
Comparing Fractions with Cross‐Multiplication ............................... 155Working with Ratios and Proportions ................................................ 158
chapter 9: Parting Ways: Fractions and the Big Four Operations . . . . . . . . . . . . . . . . . . . . . . . . 161
Multiplying and Dividing Fractions ..................................................... 161Multiplying numerators and denominators
straight across .......................................................................... 161Doing a flip to divide fractions ................................................... 166
Adding and Subtracting Fractions with the Same Denominator ...... 169Adding fractions with different denominators ......................... 173Trying a quick trick ..................................................................... 176Relying on the traditional way ................................................... 179
Taking It Away: Subtracting Fractions ................................................ 183Subtracting fractions with different denominators ................. 183
Working Properly with Mixed Numbers ............................................. 190Multiplying and dividing mixed numbers ................................. 190Adding mixed numbers ............................................................... 195Subtracting mixed numbers ....................................................... 204
chapter 10: Dallying with Decimals . . . . . . . . . . . . . . . . . . . . 213Understanding Basic Decimal Stuff ..................................................... 213
Counting dollars and decimals .................................................. 214Identifying the place value of decimals..................................... 216Knowing the decimal facts of life ............................................... 217
Performing the Big Four with Decimals ............................................. 223Adding decimals........................................................................... 223Subtracting decimals ................................................................... 225Multiplying decimals ................................................................... 228Dividing decimals......................................................................... 231
Converting between Decimals and Fractions .................................... 238Simple Decimal‐Fraction Conversions ................................................ 239
Changing decimals to fractions.................................................. 242Changing fractions to decimals.................................................. 246
chapter 11: Playing with Percents . . . . . . . . . . . . . . . . . . . . . 253Making Sense of Percents .................................................................... 253Dealing with Percents Greater than 100% .......................................... 254
Table of Contents ix
Converting to and from Percents, Decimals, and Fractions ............ 254Converting Percents to Decimals ........................................................ 255Changing Decimals to Percents ........................................................... 256Switching from Percents to Fractions ................................................ 257Converting Fractions to Percents ....................................................... 259Solving Percent Problems .................................................................... 261
Figuring out simple percent problems ...................................... 262Turning the problem around ...................................................... 264Deciphering more‐difficult percent problems .......................... 265
Putting All the Percent Problems Together ....................................... 267Identifying the three types of percent problems ..................... 267Solving percent problems with equations ................................ 268
chapter 12: Word Problems with Fractions, Decimals, and Percents . . . . . . . . . . . . . . . . . . . . 273
Adding and Subtracting Parts of the Whole in Word Problems ...... 273Sharing a pizza: Fractions ........................................................... 274Buying by the pound: Decimals ................................................. 274Splitting the vote: Percents ........................................................ 275
Problems about Multiplying Fractions ............................................... 276Renegade grocery shopping: Buying less than
they tell you to .......................................................................... 277Easy as pie: Working out what’s left on your plate .................. 277
Multiplying Decimals and Percents in Word Problems .................... 280To the end: Figuring out how much money is left ................... 280Finding out how much you started with ................................... 281
Handling Percent Increases and Decreases in Word Problems ....... 284Raking in the dough: Finding salary increases ......................... 284Earning interest on top of interest ............................................. 285Getting a deal: Calculating discounts ........................................ 286
PART IV PICTURING AND MEASURING: MEASURES, STATS, AND SETS..............................289
chapter 13: A Perfect Ten: Condensing Numbers with Scientific Notation . . . . . . . . . . . . . . . . . . . . 291
First Things First: Using Powers of Ten as Exponents ..................... 291Counting zeros and writing exponents ..................................... 292
Exponential Arithmetic: Multiplying and Dividing Powers of Ten ..................................................................................... 295
Working with Scientific Notation ......................................................... 297Writing in scientific notation ...................................................... 297Understanding order of magnitude ........................................... 300Multiplying and dividing with scientific notation .................... 302
Multiplying and Dividing with Scientific Notation ............................ 303
U Can: Basic Math & Pre-Algebra For Dummies x
chapter 14: How Much Have You Got? Weights and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Understanding Units ............................................................................. 305Adding and subtracting units ..................................................... 305Multiplying and dividing units ................................................... 306
Examining Differences between the English and Metric Systems ........................................................................... 307
Looking at the English system ................................................... 307Looking at the metric system ..................................................... 313
Estimating and Converting between the English and Metric Systems ........................................................................... 318
Estimating in the metric system ................................................ 318Converting units of measurement ............................................. 321
Converting Between English and Metric Units .................................. 323
chapter 15: Picture This: Basic Geometry . . . . . . . . . . . . . . . 329Getting on the Plane: Points, Lines, Angles, and Shapes ................. 329
Making some points..................................................................... 330Knowing your lines ...................................................................... 330Figuring the angles....................................................................... 331Shaping things up ........................................................................ 332
Getting in Shape: Polygon (And Non‐Polygon) Basics ..................... 332Closed Encounters: Shaping Up Your Understanding
of 2‐D Shapes ...................................................................................... 333Polygons ........................................................................................ 333Circles ............................................................................................ 335
Squaring Off with Quadrilaterals ........................................................ 336Making a Triple Play with Triangles ................................................... 340Getting Around with Circle Measurements ........................................ 344Taking a Trip to Another Dimension: Solid Geometry ..................... 347
The many faces of polyhedrons ................................................. 3473‐D shapes with curves ............................................................... 348
Building Solid Measurement Skills ...................................................... 349Solving Geometry Word Problems ...................................................... 354
Working from words and images ............................................... 354Breaking out those sketching skills ........................................... 355
chapter 16: Figuring Your Chances: Statistics and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Gathering Data Mathematically: Basic Statistics .............................. 361Understanding differences between qualitative
and quantitative data ............................................................... 362
Table of Contents xi
Working with qualitative data .................................................... 362Working with quantitative data .................................................. 367
Looking at Likelihoods: Basic Probability ......................................... 370Figuring the probability .............................................................. 371Oh, the possibilities! Counting outcomes with
multiple coins ........................................................................... 372
chapter 17: Setting Things Up with Basic Set Theory . . . . . 377Understanding Sets ............................................................................... 377
Elementary, my dear: Considering what’s inside sets ............. 378Sets of numbers ........................................................................... 380
Performing Operations on Sets ........................................................... 380Union: Combined elements ........................................................ 381Intersection: Elements in common ............................................ 381Relative complement: Subtraction (sorta) ............................... 383Complement: Feeling left out ..................................................... 384
PART V THE X-FILES: INTRODUCTION TO ALGEBRA ..................................................................................387
chapter 18: Enter Mr . X: Algebra and Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Seeing How X Marks the Spot .............................................................. 389Expressing Yourself with Algebraic Expressions .............................. 390Evaluating Algebraic Expressions ....................................................... 391Knowing the Terms of Separation ....................................................... 396
Making the commute: Rearranging your terms ....................... 397Identifying the coefficient and variable .................................... 399
Adding and Subtracting Like Terms ................................................... 400Identifying like terms ................................................................... 400Adding and subtracting terms ................................................... 401
Multiplying and Dividing Terms .......................................................... 403Simplifying Expressions by Combining Like Terms .......................... 408Removing Parentheses from an Algebraic Expression ..................... 411FOILing: Dealing with Two Sets of Parentheses ................................ 416
chapter 19: Unmasking Mr . X: Algebraic Equations . . . . . . . 421Understanding Algebraic Equations ................................................... 421
Using x in equations .................................................................... 422Choosing among four ways to solve algebraic equations ...... 422
The Balancing Act: Solving for x ......................................................... 428Striking a balance ......................................................................... 428Using the balance scale to isolate x .......................................... 429
Rearranging Equations and Isolating x ............................................... 432Rearranging terms on one side of an equation ........................ 433
U Can: Basic Math & Pre-Algebra For Dummies xii
Moving terms to the other side of the equals sign .................. 433Removing parentheses from equations .................................... 438Cross‐multiplying ......................................................................... 441
chapter 20: Putting Mr . X to Work: Algebra Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Solving Algebra Word Problems in Five Steps ................................... 447Declaring a variable ..................................................................... 448Setting up the equation ............................................................... 449Solving the equation .................................................................... 449Answering the question .............................................................. 450Checking your work ..................................................................... 450
Choosing Your Variable Wisely............................................................ 453Solving More‐Complex Algebraic Problems ...................................... 458
Charting four people ................................................................... 458Crossing the finish line with five people ................................... 459
PART VI THE PART OF TENS .........................................................465
chapter 21: Ten Little Math Demons That Trip People Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
Knowing the Multiplication Table ....................................................... 467Adding and Subtracting Negative Numbers ...................................... 467Multiplying and Dividing Negative Numbers ..................................... 468Knowing the Difference between Factors and Multiples ................. 469Reducing Fractions to Lowest Terms ................................................. 469Adding and Subtracting Fractions ...................................................... 469Multiplying and Dividing Fractions ..................................................... 470Identifying Algebra’s Main Goal: Find x .............................................. 470Knowing Algebra’s Main Rule: Keep the Equation in Balance ......... 470Seeing Algebra’s Main Strategy: Isolate x ........................................... 471
chapter 22: Ten Important Number Sets to Know . . . . . . . . 473Counting on Counting (or Natural) Numbers .................................... 473Identifying Integers ............................................................................... 474Knowing the Rationale behind Rational Numbers ............................ 474Making Sense of Irrational Numbers .................................................. 475Absorbing Algebraic Numbers ............................................................ 475Moving through Transcendental Numbers ........................................ 476Getting Grounded in Real Numbers .................................................... 476Trying to Imagine Imaginary Numbers ............................................... 476Grasping the Complexity of Complex Numbers ................................ 477Going beyond the Infinite with Transfinite Numbers ....................... 478
Table of Contents xiii
chapter 23: Ten Curious Types of Numbers . . . . . . . . . . . . . . 479Square Numbers .................................................................................... 479Triangular Numbers .............................................................................. 480Cubic Numbers ...................................................................................... 480Factorial Numbers ................................................................................ 481Powers of Two ....................................................................................... 482Perfect Numbers ................................................................................... 482Amicable Numbers ............................................................................... 482Prime Numbers...................................................................................... 483Mersenne Primes .................................................................................. 483Fermat Primes ....................................................................................... 484
chapter 24: Ten Alternative Numeral and Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 485
Tally Marks ............................................................................................. 485Bundled Tally Marks ............................................................................. 485Egyptian Numerals ................................................................................ 486Babylonian Numerals ........................................................................... 486Ancient Greek Numerals ...................................................................... 487Roman Numerals ................................................................................... 487Mayan Numerals ................................................................................... 488Base‐2 (Binary) Numbers ..................................................................... 489Base‐16 (Hexadecimal) Numbers ........................................................ 490Prime‐Based Numbers .......................................................................... 490
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .493
