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An easy, straightforward, and fun guide for learning fractions and its counterparts
Fractions For Dummies is the perfect strategy guide for both understanding and using one of math's most common (and most challenging) topic areas. You'll explore current solving strategies for fraction-related problems, and you'll also discover how to solve problems involving fractions' closely related cousins, decimals and percentages. Dive deep into the basics of these topics before moving on to more advanced uses, such as word problems, with the help of author and experienced math teacher Mark Zegarelli.
This straightforward and intuitive book also includes:
It's time you showed these math areas who's boss. Fractions For Dummies shows you just how easy fractions and the like can be!
Mark Zegarelli is a math teacher and tutor. He holds degrees in Mathematics and English from Rutgers University and is the author of several math and puzzle books, including Basic Math & Pre-Algebra For Dummies, Basic Math & Pre-Algebra Workbook For Dummies, and Logic For Dummies.
Introduction 1
About This Book 1
Foolish Assumptions 2
Icons Used in This Book 3
Beyond the Book 3
Where to Go from Here 4
Part 1: An Overview of Fractions, Decimals, and Percents 5
Chapter 1: What Are Fractions, Decimals, and Percents? 7
Taking a First Look at Fractions 8
Understanding how fractions work 8
Making sense of equivalent fractions 9
Understanding improper fractions 10
Seeing how improper fractions and mixed numbers relate 10
Seeing Why Decimals Are So User-Friendly 11
Understanding decimals that have one decimal place 12
Understanding decimals that have two decimal places 13
Understanding fraction-decimal equivalence 14
Putting Percents into Perspective 15
Making sense of percents 16
Converting percents to decimals and fractions 17
Chapter 2: Boosting Basic Skills 19
Knowing Basic Math Words 20
Knowing a number from a numeral 20
Getting your digits sorted 20
Finding values in numbers 21
Counting on integers 21
Examining mathematical operations 22
Talking the talk for the basic four operations 23
Knowing the Basics of Number Sets 24
Describing sets of numbers 24
Integers 25
Rational numbers 25
Irrational numbers 26
Real numbers 26
Reviewing Place Value in Whole Numbers 27
Putting values in their place 28
Looking at leading and placeholding zeros 28
Making Sense of Integer Division 29
Getting a remainder with division 30
Following the cycle of remainders 30
Chapter 3: Understanding Factors and Multiples 33
Understanding Divisibility 34
Making sense of divisibility 34
Knowing some divisibility tricks 35
Finding Factors and Multiples 40
Identifying factors 40
Making sense of multiples 47
Relating factors and multiples 47
Going Big with the Greatest Common Factor 48
Finding common factors 48
Identifying the GCF 49
Tricks to help you find the GCF 50
Playing Small with the Least Common Multiple 51
Finding the LCM 51
Tricks for calculating the LCM 51
Part 2: Fractions 55
Chapter 4: Figuring Fractions 57
Visualizing Fractions 58
Using divided circles to represent parts of the whole 58
Working with fractions on the number line 60
Understanding Basic Fraction Terminology 62
Numerators and denominators 62
Proper and improper fractions 62
Representing whole numbers as fractions 63
Understanding why a zero denominator is undefined 63
Fractions and their reciprocals 64
Understanding Equivalent Fractions 65
Using models to understand equivalent fractions 65
Changing a fraction to an equivalent form 68
Working with Improper Fractions and Mixed Numbers 73
Figuring out improper fractions 73
Mixing it up with mixed numbers 75
Changing mixed numbers to improper fractions by using Popcorn 78
Changing improper fractions to mixed numbers 79
Comparing Fractions' Values 82
Comparing fractions that have the same denominator 82
Comparing fractions by finding a common denominator 83
Chapter 5: Adding and Subtracting Fractions 89
Adding and Subtracting Fractions that Have a Common Denominator 90
Adding fractions that have a common denominator 90
Subtracting fractions that have a common denominator 94
Adding and subtracting three or more fractions 96
Adding and Subtracting Fractions that Have Different Denominators 97
Adding and subtracting two fractions by increasing terms 98
Adding and subtracting two fractions by cross-multiplying 101
Solving Bigger Problems That Have Different Denominators 104
Breaking a big problem into steps 104
Finding a common denominator for more than two fractions 105
Chapter 6: Multiplying and Dividing Fractions 107
Multiplying Fractions 107
Multiplying a pair of fractions 108
Multiplying more than two fractions 108
Canceling factors to make multiplication easier 109
Dividing Fractions 112
Using Keep-Change-Flip to turn division into multiplication 112
Canceling factors after Keep-Change-Flip 113
Converting results of division from improper fractions to mixed numbers 114
Chapter 7: Working with Mixed Numbers 117
Adding Mixed Numbers 118
Adding mixed numbers that have a common denominator 118
Adding a pair of mixed numbers that have different denominators 120
Adding a pair of mixed numbers when carrying is required 123
Subtracting Mixed Numbers 126
Subtracting mixed numbers that have a common denominator 126
Subtracting a pair of mixed numbers that have different denominators 128
Subtracting mixed numbers when borrowing is required 130
Multiplying Mixed Numbers 134
Multiplying a pair of mixed numbers 135
Canceling factors to make multiplying mixed numbers easier 135
Dividing Mixed Numbers 136
Dividing a pair of mixed numbers 137
Canceling factors to make dividing mixed numbers easier 138
Chapter 8: Word Problems with Fractions 141
Solving Word Problems by Adding and Subtracting Fractions 142
Putting the pieces together: Adding fractions 142
Finding the difference: Subtracting fractions 144
Figuring out what's left over 146
Multiplying Fractions to Solve Word Problems 147
Multiplying a fraction by a whole number 147
Multiplying a pair of fractions 148
Using multiplication to translate the word of 149
Handling problems that have multiple steps, including multiplication 150
Solving Word Problems by Dividing Fractions 151
Dividing a fraction by a whole number 151
Dividing a fraction by a fraction 152
Solving Word Problems by Using Ratios 153
Ratios that have two values 154
Ratios that have more than two values 155
Ratios that include the total value 156
Using Proportional Equations to Solve Word Problems 157
Building proportional equations 157
Using the total value in a proportional equation 158
Part 3: Decimals 165
Chapter 9: Deciphering Decimals 167
Making Cents of Decimals by Using Money 168
Understanding Decimals by Using Visual Aids 169
Seeing decimals as divided circles 169
Working with decimals on the number line 170
Seeing the Value in Decimal Place Value 170
Multiplying and Dividing by Moving the Decimal Point 172
Multiplying decimals by 10, 100, 1,000, and beyond 173
Dividing decimals by 10, 100, 1,000, and more 173
Converting Between Decimals and Fractions 174
Working with common decimal-fraction conversions 174
Changing decimals less than 1 to fractions 175
Changing decimals greater than 1 to fractions 177
Changing a fraction to a terminating decimal 178
Changing a fraction to a repeating decimal 179
Chapter 10: Applying the Basic Four Operations to Decimals 183
Adding Decimals 183
Subtracting Decimals 185
Multiplying Decimals 188
Dividing Decimals 190
Adding trailing zeros when needed 191
Dealing with repeating decimal quotients 193
Chapter 11: Word Problems with Decimals 197
Solving Word Problems by Adding and Subtracting Decimals 198
Adding things up with decimals 198
Using decimals to find the difference 198
Calculating what's left over with decimals 199
Combining addition and subtraction to solve decimal word problems 199
Multiplying Decimals to Solve Word Problems 201
Multiplying by 10 and powers of 10 201
Multiplying by other numbers 202
Multiplying decimals to find area 202
Solving multi-step word problems that involve decimal multiplication 203
Solving Word Problems that Involve Decimal Division 204
Dividing by 10 and powers of 10 205
Dividing by other numbers 205
Dividing decimals by other decimals 206
Dividing decimals to solve area problems 206
Solving multi-step word problems that involve decimal division 207
Part 4: Percents 211
Chapter 12: Probing Percents 213
Understanding Percentages 214
Calculating Common Percents in Your Head 216
Calculating 100%, 200%, and 50% 216
Calculating 10%, 20%, and 5% 216
Calculating 25% and 75% 217
Switching Around Numbers to Calculate Percents 218
Changing Percents to Decimals, and Vice Versa 219
Converting decimals to percents 219
Converting percents to decimals 220
Converting Between Percents and Fractions 221
Changing fractions to percents 221
Changing percents to fractions 222
Chapter 13: Solving Percentage Problems 227
Calculating Percents 228
Meeting Three Types of Percentage Problems 228
Solving Percentage Problems by Building Equations 230
Using an equation to find the starting number 230
Using an equation to find the percentage value 232
Chapter 14: Word Problems with Percents 235
Solving Straightforward Percentage Word Problems 236
Solving More Difficult Percentage Word Problems 237
Finding the starting value 237
Calculating the percentage value 238
Working Simple Problems That Involve Percent Increase and Decrease 239
Calculating percent increase 240
Calculating percent decrease 240
Solving Tricky Percent Increase and Decrease Word Problems 242
Finding the starting value 242
Finding the percentage value 244
Part 5: the Part of Tens 249
Chapter 15: Ten Important Points for Working with Fractions 251
Simplifying Fractions 251
Increasing the Terms of Fractions 252
Changing Mixed Numbers to Improper Fractions 253
Knowing When to Find a Common Denominator 254
Knowing When You Don't Need a Common Denominator 255
Canceling Factors Before Multiplying Fractions 255
Using Keep-Change-Flip Only When Dividing Fractions 256
Using Keep-Change-Flip Before Canceling Factors 257
When to Change Mixed Numbers to Improper Fractions 257
When Not to Change Mixed Numbers to Improper Fractions 258
Chapter 16: Ten Real-World Uses for Fractions 261
Cooking and Baking 261
Construction and Building 262
Money and Finance 263
Sports 264
Telling Time 264
Music and Visual Art 265
Gambling 265
Nursing and Medicine 266
Health and Fitness 266
Travel and Fuel Efficiency 267
Index 269
Chapter 1
IN THIS CHAPTER
Expressing fractions as a numerator divided by a denominator
Working with place value in decimals
Understanding percents as a number out of 100
Representing part of a whole by using fractions, decimals, or percents
Welcome to your journey into the world of fractions! If you pick up this book, you're probably feeling a bit unsure about fractions, and that's completely okay. You're not alone - many people find fractions tricky. The good news is that understanding fractions is like honing any new skill. In the same way that you may have figured out how to balance so that you could ride a bike or caught onto the basic rules of play for a new game, you can grasp the concepts and calculations associated with fractions if you have patience, practice, and the right guidance.
This book is designed to be your friendly companion while you explore fractions and their close cousins, decimals and percents. I give you the basics, breaking down what fractions are and why they matter. I also help you tackle problems one step at a time, with plenty of examples, tips, and encouragement along the way. My goal is to keep you from ever feeling lost or overwhelmed when working with fractions.
Remember, everyone figures things out at their own pace. So take your time, ask questions, and even make mistakes - you have to do all of those things to get good at anything. And with the help offered in this book, not only can you understand fractions, but you might also discover that they're not so scary, after all.
At a basic level, fractions give you a way to represent parts of a whole. Imagine that you have a pizza, and you cut it into eight equal slices. If you eat one slice out of eight, you've eaten one part of the eight slices, or of the pizza. That's a fraction! Fractions help us understand and work with pieces of something, whether it's food, time, money, or anything else that can be divided into equal sections.
In the following sections, I explain how fractions work to represent parts of the whole, how to read and write fractions for use in calculations, and why fractions are so useful.
Fractions use two numbers to show how much of an item you're working with after you divide the whole into equal parts. For example, if you have a chocolate bar that's divided into 4 equal pieces and you eat 3 pieces, the fraction that represents what you ate is . In this fraction, you use two numbers, 3 and 4, which you write one over the other with a vertical line in between.
Here are the names for these two numbers:
As another example, imagine that you have a rope that's 10 meters long, and you cut off a 7-meter piece. The fraction representing the part of the rope that you cut off is . The numerator is 7 (because you cut 7 meters), and the denominator is 10 (because the whole rope was 10 meters long).
Unlike whole numbers, where different numbers always represent different values (for example, 3 is always less than 5), fractions are a bit more flexible. A pair of fractions can look different, each containing a different numerator (top number) and denominator (bottom number), and yet their value can be exactly the same.
Different number combinations can represent the same value because fractions rely on the relationship between the numerator and the denominator. For example, and might seem different at first glance, but they actually represent the same amount. In both cases, you're talking about half of something - whether it's 1 out of 2 parts, or 2 out of 4 parts.
Pairs of fractions like this (such as the and example) are called equivalent fractions. Even though the numerators and denominators are different numbers, the value that they represent is the same. You really need to keep this idea in mind when you add, subtract, and compare fractions! (I go in depth about equivalent fractions in Chapter 4, so flip there if you want all the details.)
When you think about fractions, you might think of them as numbers that are less than 1 - that is, numbers that fall between 0 and 1 on the number line. These types of fractions, where the numerator is smaller than the denominator, are called proper fractions. For example, is a proper fraction because 4 is less than 5, so the value it represents is less than 1.
However, fractions can also be equal to or greater than 1, and these types of fractions are called improper fractions. In an improper fraction, the numerator is equal to or larger than the denominator, which means that the fraction represents one whole or more.
Improper fractions are common in the real world. For example, is an improper fraction. It tells us that we have 5 parts, but each whole is made up of only 4 parts. So, if you have of a cake, you have 1 whole cake and an extra of another identical cake.
Similarly, if you have 7 meters of fabric divided into 2-meter sections, you have , which is 3 and a half sections.
In the preceding section, you can explore improper fractions - fractions where the numerator is equal to or greater than the denominator. But in this section, you can find out about another way to express these fractions as mixed numbers. A mixed number is simply a combination of a whole number and a proper fraction. It gives you another way of showing the same value that an improper fraction does, but in a form that you can often more easily visualize or understand.
For example, I revisit the improper fraction that I talk about in the preceding section. This fraction tells you that you have more than one whole, but how much more? To express as a mixed number, you break it down into one whole () and the remaining . So, is the same as the mixed number . Both represent the same amount - 1 whole and of another whole.
In real life, mixed numbers come up often. Imagine that you're baking (something really sweet) and using 2 full cups of sugar, plus another half cup. Instead of saying that you have cups, you probably say you have cups of sugar. Both expressions are valid, but the mixed number is a little easier to understand in a real-world situation.
Consider another example when measuring length: If you have a board that's 3 full meters, plus an additional of a meter, you could express that as of a meter. However, this amount is probably easier to understand if you use the equivalent mixed number meters.
Every mixed number is just another way of writing an improper fraction, and it offers you a clearer picture of how many whole items and how many parts you have than an improper fraction does.
In this section, I want to help you explore a way to express values that fall between whole numbers, allowing you to sidestep using fractions. (See the section "Taking a First Look at Fractions," earlier in this chapter, for an overview of fractions.) Decimals are a more user-friendly alternative to fractions because they align neatly with our base-10 number system, which makes them easier to understand and use in everyday situations.
Although fractions such as or represent parts of a whole, decimals express the same ideas, but in a different format. For example, instead of saying , you can use the decimal 0.5. Similarly, you can express as 0.75.
The digits that fall to the right of the period in a decimal represent the decimal places, which I describe in the following two sections.
Decimals are especially handy in contexts such as money, measurements (especially the metric system), and technology, where precision is a key aspect. Whether you calculate the total cost of items at the store, measure ingredients for a recipe, or read a digital scale, decimals offer a straightforward and precise way to represent values that aren't whole numbers. And decimals also bypass many of the difficulties that calculating with fractions can carry (for example, finding a common denominator and simplifying your answer - see Chapter 4 for more on these subtleties of fractions).
In the sections that follow, you can find out how decimals work, how to convert between fractions and decimals, and how to use decimals in various real-world scenarios. And you can discover just how versatile and helpful decimals can be.
Decimal place value, meaning how far each decimal digit is from the decimal point, helps you understand the value of the digits in a decimal number. For decimals that have only one decimal place (that is, one digit to the right of the period), each digit represents tenths of the whole.
For example, in the...
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