Master the fundamentals first for a smoother ride through math Basic Math & Pre-Algebra Workbook For Dummies is your ticket to finally getting a handle on math! Designed to help you strengthen your weak spots and pinpoint problem areas, this book provides hundreds of practice problems to help you get over the hump. Each section includes a brief review of key concepts and full explanations for every practice problem, so you'll always know exactly where you went wrong. The companion website gives you access to quizzes for each chapter, so you can test your understanding and identify your sticking points before moving on to the next topic. You'll brush up on the rules of basic operations, and then learn what to do when the numbers just won't behave--negative numbers, inequalities, algebraic expressions, scientific notation, and other tricky situations will become second nature as you refresh what you know and learn what you missed. Each math class you take builds on the ones that came before; if you got lost somewhere around fractions, you'll have a difficult time keeping up in Algebra, Geometry, Trigonometry, and Calculus--so don't fall behind! This book provides plenty of practice and patient guidance to help you slay the math monster once and for all. * Make sense of fractions, decimals, and percentages * Learn how to handle inequalities, exponents, square roots, and absolute values * Simplify expressions and solve simple algebraic equations * Find your way around a triangle, circle, trapezoid, and more Once you get comfortable with the rules and operations, math takes on a whole new dimension. Curiosity replaces anxiety, and problems start feeling like puzzles rather than hurdles. All it takes is practice. Basic Math & Pre-Algebra Workbook For Dummies is your ultimate math coach, with hundreds of guided practice practice problems to help you break through the math barrier.
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Basic Math & Pre-Algebra Workbook For Dummies®, 3rd Edition with Online Practice
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Table of Contents
About This Book
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: Getting Started with Basic Math and Pre-Algebra
Chapter 1: We’ve Got Your Numbers
Getting in Place with Numbers and Digits
Rollover: Rounding Numbers Up and Down
Using the Number Line with the Big Four
The Column Lineup: Adding and Subtracting
Multiplying Multiple Digits
Cycling through Long Division
Solutions to We’ve Got Your Numbers
Chapter 2: Smooth Operators: Working with the Big Four Operations
Switching Things Up with Inverse Operations and the Commutative Property
Getting with the In-Group: Parentheses and the Associative Property
Becoming Unbalanced: Inequalities
Special Times: Powers and Square Roots
Answers to Problems in Smooth Operators
Chapter 3: Getting Down with Negative Numbers
Understanding Where Negative Numbers Come From
Sign-Switching: Understanding Negation and Absolute Value
Adding with Negative Numbers
Subtracting with Negative Numbers
Knowing Signs of the Times (And Division) for Negative Numbers
Answers to Problems in Getting Down with Negative Numbers
Chapter 4: It’s Just an Expression
Evaluating Expressions with Addition and Subtraction
Evaluating Expressions with Multiplication and Division
Making Sense of Mixed-Operator Expressions
Handling Powers Responsibly
Pulling Apart Parentheses and Powers
Figuring Out Nested Parentheses
Bringing It All Together: The Order of Operations
Solutions to It’s Just an Expression
Chapter 5: Dividing Attention: Divisibility, Factors, and Multiples
Checking for Leftovers: Divisibility Tests
Understanding Factors and Multiples
One Number, Indivisible: Identifying Prime (And Composite) Numbers
Generating a Number’s Factors
Decomposing a Number into Its Prime Factors
Finding the Greatest Common Factor
Generating the Multiples of a Number
Finding the Least Common Multiple
Solutions to Divisibility, Factors, and Multiples
Part 2: Slicing Things Up: Fractions, Decimals, and Percents
Chapter 6: Fractions Are a Piece of Cake
Getting Down the Basic Fraction Stuff
In Mixed Company: Converting between Mixed Numbers and Improper Fractions
Increasing and Reducing the Terms of Fractions
Comparing Fractions with Cross-Multiplication
Working with Ratios and Proportions
Solutions to Fractions Are a Piece of Cake
Chapter 7: Fractions and the Big Four
Multiplying Fractions: A Straight Shot
Flipping for Fraction Division
Reaching the Common Denominator: Adding Fractions
The Other Common Denominator: Subtracting Fractions
Multiplying and Dividing Mixed Numbers
Carried Away: Adding Mixed Numbers
Borrowing from the Whole: Subtracting Mixed Numbers
Solutions to Fractions and the Big Four
Chapter 8: Getting to the Point with Decimals
Getting in Place: Basic Decimal Stuff
Simple Decimal-Fraction Conversions
New Lineup: Adding and Subtracting Decimals
Counting Decimal Places: Multiplying Decimals
Points on the Move: Dividing Decimals
Decimals to Fractions
Fractions to Decimals
Solutions to Getting to the Point with Decimals
Chapter 9: Playing the Percentages
Converting Percents to Decimals
Changing Decimals to Percents
Switching from Percents to Fractions
Converting Fractions to Percents
Solving a Variety of Percent Problems Using Word Equations
Solutions to Playing the Percentages
Part 3: A Giant Step Forward: Intermediate Topics
Chapter 10: Seeking a Higher Power through Scientific Notation
On the Count of Zero: Understanding Powers of Ten
Exponential Arithmetic: Multiplying and Dividing Powers of Ten
Representing Numbers in Scientific Notation
Multiplying and Dividing with Scientific Notation
Answers to Problems in Seeking a Higher Power through Scientific Notation
Chapter 11: Weighty Questions on Weights and Measures
The Basics of the English System
Going International with the Metric System
Converting Between English and Metric Units
Answers to Problems in Weighty Questions on Weights and Measures
Chapter 12: Shaping Up with Geometry
Getting in Shape: Polygon (And Non-Polygon) Basics
Squaring Off with Quadrilaterals
Making a Triple Play with Triangles
Getting Around with Circle Measurements
Building Solid Measurement Skills
Answers to Problems in Shaping Up with Geometry
Chapter 13: Getting Graphic: Xy-Graphs
Getting the Point of the
Drawing the Line on the
Answers to Problems in Getting Graphic:
Part 4: The X Factor: Introducing Algebra
Chapter 14: Expressing Yourself with Algebraic Expressions
Plug It In: Evaluating Algebraic Expressions
Knowing the Terms of Separation
Adding and Subtracting Like Terms
Multiplying and Dividing Terms
Simplifying Expressions by Combining Like Terms
Simplifying Expressions with Parentheses
FOILing: Dealing with Two Sets of Parentheses
Answers to Problems in Expressing Yourself with Algebraic Expressions
Chapter 15: Finding the Right Balance: Solving Algebraic Equations
Solving Simple Algebraic Equations
Equality for All: Using the Balance Scale to Isolate x
Switching Sides: Rearranging Equations to Isolate x
Barring Fractions: Cross-Multiplying to Simplify Equations
Answers to Problems in Finding the Right Balance: Solving Algebraic Equations
Part 5: The Part of Tens
Chapter 16: Ten Alternative Numeral and Number Systems
Bundled Tally Marks
Ancient Greek Numerals
Base-2 (Binary) Numbers
Base-16 (Hexadecimal) Numbers
Chapter 17: Ten Curious Types of Numbers
Powers of Two
About the Author
Connect with Dummies
End User License Agreement
Table of Contents
When you approach math correctly, it’s almost always easier than you think. And a lot of the stuff that hangs you up when you first see it probably isn’t all that scary after all. Lots of students feel they got lost somewhere along the way on the road between learning to count to ten and their first day in an algebra class — and this may be true whether you’re 14 or 104. If this is you, don’t worry. You’re not alone, and help is right here!
Basic Math & Pre-Algebra Workbook For Dummies with Online Practice can give you the confidence and math skills you need to succeed in any math course you encounter on the way to algebra. One of the easiest ways to build confidence is to get experience working problems, allowing you to build those skills quickly. Everything in this book is designed to help clear the path on your math journey. Every section of every chapter contains a clear explanation of what you need to know, with plenty of practice problems and step-by-step solutions to every problem. Just grab a pencil, open this book to any page, and begin strengthening your math muscles!
This book is for anyone who wants to improve his or her math skills. You may already be enrolled in a math class or preparing to register for one or simply studying on your own. In any case, practice makes perfect, and in this book you get plenty of practice solving a wide variety of math problems.
Each chapter covers a different topic in math: negative numbers, fractions, decimals, geometry, graphing, basic algebra — it’s all here. In every section within a chapter, you find problems that allow you to practice a different skill. Each section features the following:
A brief introduction to that section’s topic
An explanation of how to solve the problems in that section
Sample questions with answers that show you all the steps to solving the problem
Practice problems with space to work out your answer
Go ahead and write in this book — that’s what it’s for! When you’ve completed a problem or group of problems, flip to the end of the chapter. You’ll find the correct answer followed by a detailed, step-by-step explanation of how to get there.
Although you can certainly work all the exercises in this book from beginning to end, you don’t have to. Feel free to jump directly to whatever chapter has the type of problems you want to practice. When you’ve worked through enough problems in a section to your satisfaction, feel free to jump to a different section. If you find the problems in a section too difficult, flip back to an earlier section or chapter to practice the skills you need — just follow the cross-references.
You probably realize that the best way to figure out math is by doing it. You only want enough explanation to get down to business so you can put your math skills to work right away. If so, you’ve come to the right place. If you’re looking for a more in-depth discussion, including tips on how all these math concepts fit into word problems, you may want to pick up the companion book, Basic Math & Pre-Algebra For Dummies.
I’m willing to bet my last dollar on earth that you’re ready for this book. I assume only that you have some familiarity with the basics of the number system and the Big Four operations (adding, subtracting, multiplying, and dividing). To make sure that you’re ready, take a look at these four arithmetic problems and see whether you can answer them:
If you can do these problems, you’re good to go!
Throughout this book, I highlight some of the most important information with a variety of icons. Here’s what they all mean:
This icon points out some of the most important pieces of information. Pay special attention to these details — you need to know them!
Tips show you a quick and easy way to do a problem. Try these tricks as you’re solving the problems in that section.
Warnings are math booby traps that unwary students often fall into. Reading these bits carefully can help you avoid unnecessary heartache.
This icon highlights the example problems that show you techniques before you dive into the exercises.
In addition to the book you’re reading right now, be sure to check out the free Cheat Sheet for a set of quick reference notes including the order of operations, mathematical inequalities, basic algebra conventions, and more. To get this Cheat Sheet, simply go to www.dummies.com and search for “Basic Math & Pre-Algebra Workbook” in the Search box.
The online practice that comes free with this book contains extra practice questions that correspond with each chapter in the book. To gain access to the online practice, all you have to do is register. Just follow these simple steps:
Find your PIN access code located on the inside front cover of this book.
Go to Dummies.com and click
Find your product (
Basic Math & Pre-Algebra Workbook For Dummies with Online Practice
) and then follow the on-screen prompts to activate your PIN.
Now you’re ready to go! You can go back to the program at http://testbanks.wiley.com as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.
Tip: If you have trouble with your PIN or can’t find it, contact Wiley Product Technical Support at 877-762-2974 or go to http://support.wiley.com.
You can turn to virtually any page in this book and begin improving your math skills. Chapters 3 through 6 cover topics that tend to hang up math students: negative numbers, order of operations, factors and multiples, and fractions. A lot of what follows later in the book builds on these important early topics, so check them out. When you feel comfortable doing these types of problems, you have a real advantage in any math class.
Of course, if you already have a good handle on these topics, you can go anywhere you want (though you may still want to skim these chapters for some tips and tricks). My only advice is that you do the problems before reading the answer key!
And by all means, while you’re at it, pick up Basic Math & Pre-Algebra For Dummies, which contains more-detailed explanations and a few extra topics not covered in this workbook. Used in conjunction, these two books provide a powerful one-two punch to take just about any math problem to the mat.
IN THIS PART …
Understand place value.
Use the Big Four operations: addition, subtraction, multiplication, and division.
Calculate with negative numbers.
Simplify expressions using the order of operations (PEMDAS).
Work with factors and multiples.
IN THIS CHAPTER
Understanding how place value turns digits into numbers
Rounding numbers to the nearest ten, hundred, or thousand
Calculating with the Big Four operations: Adding, subtracting, multiplying, and dividing
Getting comfortable with long division
In this chapter, I give you a review of basic math, and I do mean basic. I bet you know a lot of this stuff already. So, consider this a trip down memory lane, a mini-vacation from whatever math you may be working on right now. With a really strong foundation in these areas, you’ll find the chapters that follow a lot easier.
First, I discuss how the number system you’re familiar with — called the Hindu-Arabic number system (or decimal numbers) — uses digits and place value to express numbers. Next, I show you how to round numbers to the nearest ten, hundred, or thousand.
After that, I discuss the Big Four operations: adding, subtracting, multiplying, and dividing. You see how to use the number line to make sense of all four operations. Then I give you practice doing calculations with larger numbers. To finish up, I make sure you know how to do long division both with and without a remainder.
Some math books use a dot (·) to indicate multiplication. In this book, I use the more familiar times sign .
The number system used most commonly throughout the world is the Hindu-Arabic number system. This system contains ten digits (also called numerals), which are symbols like the letters A through Z. I’m sure you’re quite familiar with them:
Like letters of the alphabet, individual digits aren’t very useful. When used in combination, however, these ten symbols can build numbers as large as you like using place value. Place value assigns each digit a greater or lesser value depending upon where it appears in a number. Each place in a number is ten times greater than the place to its immediate right.
Although the digit 0 adds no value to a number, it can act as a placeholder. When a 0 appears to the right of at least one nonzero digit, it’s a placeholder. Placeholders are important for giving digits their proper place value. In contrast, when a 0 isn’t to the right of any nonzero digit, it’s a leading zero. Leading zeros are unnecessary and can be removed from a number.
Q. In the number 284, identify the ones digit, the tens digit, and the hundreds digit.
A. The ones digit is 4, the tens digit is 8, and the hundreds digit is 2.
Q. Place the number 5,672 in a table that shows the value of each digit. Then use this table and an addition problem to show how this number breaks down digit by digit.
The numeral 5 is in the thousands place, 6 is in the hundreds place, 7 is in the tens place, and 2 is in the ones place, so here’s how the number breaks down:
Q. Place the number 040,120 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit. Which 0s are placeholders, and which are leading zeros?
The first 0 is in the hundred-thousands place, 4 is in the ten-thousands place, the next 0 is in the thousands place, 1 is in the hundreds place, 2 is in the tens place, and the last 0 is in the ones place, so
The first 0 is a leading zero, and the remaining 0s are placeholders.
1 In the number 7,359, identify the following digits:
The ones digit
The tens digit
The hundreds digit
The thousands digit
2 Place the number 2,136 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit.
3 Place the number 03,809 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit. Which 0 is a placeholder and which is a leading zero?
4 Place the number 0,450,900 in a table that shows the value of each digit. Then use this table to show how this number breaks down digit by digit. Which 0s are placeholders and which are leading zeros?
Rounding numbers makes long numbers easier to work with. To round a two-digit number to the nearest ten, simply increase it or decrease it to the nearest number that ends in 0:
When a number ends in 1, 2, 3, or 4, bring it down; in other words, keep the tens digit the same and turn the ones digit into a 0.
When a number ends in 5, 6, 7, 8, or 9, bring it up; add 1 to the tens digit and turn the ones digit into a 0.
To round a number with more than two digits to the nearest ten, use the same method, focusing only on the ones and tens digits.
After you understand how to round a number to the nearest ten, rounding a number to the nearest hundred, thousand, or beyond is easy. Focus only on two digits: The digit in the place you’re rounding to and the digit to its immediate right, which tells you whether to round up or down. All the digits to the right of the number you’re rounding to change to 0s.
Occasionally when you’re rounding a number up, a small change to the ones and tens digits affects the other digits. This is a lot like when the odometer in your car rolls a bunch of 9s over to 0s, such as when you go from 11,999 miles to 12,000 miles.
Q. Round the numbers 31, 58, and 95 to the nearest ten.
A. 30, 60, and 100.
The number 31 ends in 1, so round it down:
The number 58 ends in 8, so round it up:
The number 95 ends in 5, so round it up:
Q. Round the numbers 742, 3,820, and 61,225 to the nearest ten.
A. 740, 3,820, and 61,230.
The number 742 ends in 2, so round it down:
The number 3,820 already ends in 0, so no rounding is needed:
The number 61,225 ends in 5, so round it up:
5 Round these two-digit numbers to the nearest ten:
6 Round these numbers to the nearest ten:
7 Round these numbers to the nearest hundred:
8 Round these numbers to the nearest thousand:
The number line is just a line with numbers marked off at regular intervals. You probably saw your first number line when you were learning how to count to ten. In this section, I show you how to use this trusty tool to perform the Big Four operations (adding, subtracting, multiplying, and dividing) on relatively small numbers.
The number line can be a useful tool for adding and subtracting small numbers:
When you add, move
the number line, to the right.
When you subtract, move
the number line, to the left.
To multiply on the number line, start at 0 and count by the first number in the problem as many times as indicated by the second number.
To divide on the number line, first block off a segment of the number line from 0 to the first number in the problem. Then divide this segment evenly into the number of pieces indicated by the second number. The length of each piece is the answer to the division.
Q. Add on the number line.
A. 13. The expression means start at 6, up 7, which brings you to 13 (see Figure 1-1).
Q. Subtract on the number line.
A. 8. The expression means start at 12, down 4, which brings you to 8 (see Figure 1-2).
Q. Multiply on the number line.
A. 10. Starting at 0, count by twos a total of five times, which brings you to 10 (see Figure 1-3).
Q. Divide on the number line.
A. 4. Block off the segment of the number line from 0 to 12. Now divide this segment evenly into three smaller pieces, as shown in Figure 1-4. Each of these pieces has a length of 4, so this is the answer to the problem.
9 Add the following numbers on the number line:
10 Subtract the following numbers on the number line:
11 Multiply the following numbers on the number line:
12 Divide the following numbers on the number line:
FIGURE 1-1 Adding on the number line.
FIGURE 1-2 Subtracting on the number line.
FIGURE 1-3 Multiplying on the number line.
FIGURE 1-4 Dividing on the number line.
To add or subtract large numbers, stack the numbers on top of each other so that all similar digits (ones, tens, hundreds, and so forth) form columns. Then work from right to left. Do the calculations vertically, starting with the ones column, then going to the tens column, and so forth:
When you’re adding and a column adds up to 10 or more, write down the ones digit of the result and carry the tens digit over to the column on the immediate left.
When you’re subtracting and the top digit in a column is less than the bottom digit, borrow from the column on the immediate left.
Q. Add .
A. 203. Stack the numbers and add the columns from right to left:
Notice that when I add the ones column (), I write the 3 below this column and carry the 1 over to the tens column. Then, when I add the tens column (), I write the 0 below this column and carry the 1 over to the hundreds column.
Q. Subtract .
A. 752. Stack the numbers and subtract the columns from right to left:
When I try to subtract the tens column, 4 is less than 9, so I borrow 1 from the hundreds column, changing the 8 to 7. Then I place this 1 above the 4, changing it to 14. Now I can subtract .
13 Add .
14 Find the following sum:
16 Subtract .
To multiply large numbers, stack the first number on top of the second. Then multiply each digit of the bottom number, from right to left, by the top number. In other words, first multiply the top number by the ones digit of the bottom number. Then write down a 0 as a placeholder and multiply the top number by the tens digit of the bottom number. Continue the process, adding placeholders and multiplying the top number by the next digit in the bottom number.
When the result is a two-digit number, write down the ones digit and carry the tens digit to the next column. After multiplying the next two digits, add the number you carried over.
Add the results to obtain the final answer.
Q. Multiply .
A. 100,912. Stack the first number on top of the second:
Now multiply 6 by every number in 742, starting from the right. Because , a two-digit number, you write down the 2 and carry the 1 to the tens column. In the next column, you multiply , and add the 1 you carried over, giving you a total of 25. Write down the 5, and carry the 2 to the hundreds column. Multiply , and add the 2 you carried over, giving you 44:
Next, write down a 0 all the way to the right in the row below the one that you just wrote. Multiply 3 by every number in 742, starting from the right and carrying when necessary:
Write down two 0s all the way to the right of the row below the one that you just wrote. Repeat the process with 1:
To finish, add up the results:
17 Multiply .
19 Solve .
20 Multiply .
To divide larger numbers, use long division. Unlike the other Big Four operations, long division moves from left to right. For each digit in the divisor (the number you’re dividing), you complete a cycle of division, multiplication, and subtraction.
In some problems, the number at the very bottom of the problem isn’t a 0. In these cases, the answer has a remainder, which is a leftover piece that needs to be accounted for. In those cases, you write r followed by whatever number is left over.
A. 239. Start off by writing the problem like this:
To begin, ask how many times 4 goes into 9 — that is, what’s The answer is 2 (with a little left over), so write 2 directly above the 9. Now multiply
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