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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2025
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:
- Techniques for working with mixed numbers (numbers that include whole amounts and fractions) and more
- Ways to add, subtract, multiply, and divide fractions with whole numbers and with each other
- Strategies for helping and supporting the young student in your life who's struggling with fractions, decimals, and/or percentages
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!
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Fractions For Dummies®
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Table of Contents
Cover
Title Page
Copyright
Introduction
About This Book
Foolish Assumptions
Icons Used in This Book
Beyond the Book
Where to Go from Here
Part 1: An Overview of Fractions, Decimals, and Percents
Chapter 1: What Are Fractions, Decimals, and Percents?
Taking a First Look at Fractions
Seeing Why Decimals Are So User-Friendly
Putting Percents into Perspective
Chapter 2: Boosting Basic Skills
Knowing Basic Math Words
Knowing the Basics of Number Sets
Reviewing Place Value in Whole Numbers
Making Sense of Integer Division
Chapter 3: Understanding Factors and Multiples
Understanding Divisibility
Finding Factors and Multiples
Going Big with the Greatest Common Factor
Playing Small with the Least Common Multiple
Part 2: Fractions
Chapter 4: Figuring Fractions
Visualizing Fractions
Understanding Basic Fraction Terminology
Understanding Equivalent Fractions
Working with Improper Fractions and Mixed Numbers
Comparing Fractions’ Values
Chapter 5: Adding and Subtracting Fractions
Adding and Subtracting Fractions that Have a Common Denominator
Adding and Subtracting Fractions that Have Different Denominators
Solving Bigger Problems That Have Different Denominators
Chapter 6: Multiplying and Dividing Fractions
Multiplying Fractions
Dividing Fractions
Chapter 7: Working with Mixed Numbers
Adding Mixed Numbers
Subtracting Mixed Numbers
Multiplying Mixed Numbers
Dividing Mixed Numbers
Chapter 8: Word Problems with Fractions
Solving Word Problems by Adding and Subtracting Fractions
Multiplying Fractions to Solve Word Problems
Solving Word Problems by Dividing Fractions
Solving Word Problems by Using Ratios
Using Proportional Equations to Solve Word Problems
Part 3: Decimals
Chapter 9: Deciphering Decimals
Making Cents of Decimals by Using Money
Understanding Decimals by Using Visual Aids
Seeing the Value in Decimal Place Value
Multiplying and Dividing by Moving the Decimal Point
Converting Between Decimals and Fractions
Chapter 10: Applying the Basic Four Operations to Decimals
Adding Decimals
Subtracting Decimals
Multiplying Decimals
Dividing Decimals
Chapter 11: Word Problems with Decimals
Solving Word Problems by Adding and Subtracting Decimals
Multiplying Decimals to Solve Word Problems
Solving Word Problems that Involve Decimal Division
Part 4: Percents
Chapter 12: Probing Percents
Understanding Percentages
Calculating Common Percents in Your Head
Switching Around Numbers to Calculate Percents
Changing Percents to Decimals, and Vice Versa
Converting Between Percents and Fractions
Chapter 13: Solving Percentage Problems
Calculating Percents
Meeting Three Types of Percentage Problems
Solving Percentage Problems by Building Equations
Chapter 14: Word Problems with Percents
Solving Straightforward Percentage Word Problems
Solving More Difficult Percentage Word Problems
Working Simple Problems That Involve Percent Increase and Decrease
Solving Tricky Percent Increase and Decrease Word Problems
Part 5: The Part of Tens
Chapter 15: Ten Important Points for Working with Fractions
Simplifying Fractions
Increasing the Terms of Fractions
Changing Mixed Numbers to Improper Fractions
Knowing When to Find a Common Denominator
Knowing When You Don’t Need a Common Denominator
Canceling Factors Before Multiplying Fractions
Using Keep-Change-Flip Only When Dividing Fractions
Using Keep-Change-Flip Before Canceling Factors
When to Change Mixed Numbers to Improper Fractions
When Not to Change Mixed Numbers to Improper Fractions
Chapter 16: Ten Real-World Uses for Fractions
Cooking and Baking
Construction and Building
Money and Finance
Sports
Telling Time
Music and Visual Art
Gambling
Nursing and Medicine
Health and Fitness
Travel and Fuel Efficiency
Index
About the Author
Connect with Dummies
End User License Agreement
List of Tables
Chapter 2
TABLE 2-1 Defining Types of Numbers
Chapter 3
TABLE 3-1 Combining Tests for Divisibility
TABLE 3-2 Using Factor Pairs to Define a Number
TABLE 3-3 The 25 Prime Numbers Less than 100
TABLE 3-4 Prime Factorizations of Numbers from 2 to 25
Chapter 4
TABLE 4-1 Fractions and Their Reciprocals
Chapter 9
TABLE 9-1 U.S. Coins as Decimals and Fractions
TABLE 9-2 Place-Value Chart for the Number 345.678
Chapter 12
TABLE 12-1 Common Percents and Equivalent Decimals and Fractions
List of Illustrations
Chapter 2
FIGURE 2-1: The set of integers from −5 to 5 on the number line.
Chapter 4
FIGURE 4-1: Subdivided circles that represent fractions through .
FIGURE 4-2: Circles that represent other fractional values that are less than 1...
FIGURE 4-3: Segmenting a number line.
FIGURE 4-4: Segmenting a number line with various equal-sized divisions.
FIGURE 4-5: Visual models of equivalent fractions.
FIGURE 4-6: More visual models of equivalent fractions.
FIGURE 4-7: Equivalent fractions on a number line that has eight equal parts.
FIGURE 4-8: Equivalent fractions on a number line that has ten equal parts.
FIGURE 4-9: Sets of circles representing improper fractions.
FIGURE 4-10: Improper fractions on number lines.
FIGURE 4-11: Putting mixed numbers into circles.
FIGURE 4-12: Representing mixed numbers by using number lines.
Chapter 9
FIGURE 9-1: Decimals from 0.1 to 0.9 expressed as shaded circles.
FIGURE 9-2: Decimals from 0.1 to 0.9 on a number line.
Guide
Cover
Table of Contents
Title Page
Copyright
Begin Reading
Index
About the Author
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Fractions For Dummies®
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Library of Congress Control Number: 2024950292
ISBN 978-1-394-29687-3 (pbk); ISBN 978-1-394-29689-7 (ePDF); ISBN 978-1-394-29688-0 (ePUB)
Introduction
As a math teacher and tutor, I hear a lot about how many students find working with fractions difficult. So, I’m really glad that in this book, Fractions For Dummies, I have an opportunity to focus solely on fractions, fractions, and more fractions — not to mention the closely related topics of decimals and percentages.
I’ve done my best to take a really slow and steady approach to fractions. Part 1 of this book is entirely devoted to an overview of fractions, followed by a foundation of basic math skills to support you.
In Part 2, you can find five full chapters devoted solely to the most important core of fractions skills. Simplifying fractions, finding a common denominator, converting between improper fractions and mixed numbers, as well as adding, subtracting, multiplying, and dividing fractions — it’s all there!
After that, Parts 3 and 4 cover decimals and percentages. And at the end, just for fun, Part 5 gives you two top-ten lists about fractions that you may find useful.
About This Book
This book provides clear instruction related to working with fractions and the associated topics of decimals and percentages. Each chapter contains multiple sections, each providing
A clear explanation
of the topic under focus
A variety of example problems,
each with a step-by-step solution
Practice problems,
with solutions at the end of the section
I introduce the topics in an order that I believe can guide you to the best understanding of fractions, decimals, and percentages. But don’t feel under any obligation to follow that path from start to finish! Feel free to skip around to whatever topic you want to work on.
Each chapter includes multiple cross-references so that whenever I use a word, phrase, or concept that you may not be familiar with, you can flip to that section to get an explanation. Cross-references tell you where in the book I introduce or go into detail about the item in question.
Foolish Assumptions
My first assumption is that you want or need to know more about fractions, decimals, and percentages, perhaps because you are
A student
who wants to get a solid understanding of fractions, either to get through your homework or to do well on an upcoming test.
An adult
who needs to get clarity on some aspects of fractions that you feel a little shaky about, so that math doesn’t hold you back from your educational or career plans.
The parent, relative, or friend
of somebody in one of the two groups above, and you want to support them in conquering fractions.
Whatever the reason you’re reading this book, I try to give you as much help as possible. As long as you have a basic understanding of addition, subtraction, multiplication, and division by using whole numbers, you should be good to go. Here’s a quick test to see if you’re ready:
If you can answer these four questions (the answers are 12, 7, 35, and 5), consider yourself ready to go.
Icons Used in This Book
Throughout the book, you’ll see the following four icons, which can help you navigate your way.
This icon highlights important information that you need to understand and remember. Make sure to get clear on this information before moving on. You can also use these icons when studying for a test.
Tips include easier and usually quicker ways to solve problems. Refer to them to help you move through the practice problems more easily.
This icon warns you of common mistakes that students make if they’re not careful. Take note and proceed with caution!
Practice problems give you a chance to try out the skills you just worked on, to see whether you fully understand them. If you get stuck, flip to the end of the chapter, where the answer awaits — in some cases, accompanied by a step-by-step in the solution.
Beyond the Book
Fractions For Dummies isn’t just a book. It’s also a passport to other resources available on the For Dummies website (www.dummies.com); be sure to check out the Cheat Sheet, where you can find extra information to help you embrace the joy of fractions. To access the Cheat Sheet, go to www.dummies.com and type the words “Fractions For Dummies Cheat Sheet” in the Search text box.
Additionally, I have several YouTube playlists that you may find helpful for working with fractions. You can find these at the following links:
Math – Fractions 1:
https://tinyurl.com/5yh7dcpc
Math – Fractions 2:
https://tinyurl.com/y9x6nmn7
Math – Fractions 3:
https://tinyurl.com/4vkvj2cj
Where to Go from Here
You can use this book in a variety of ways. If you’re reading this without feeling that you’re under immediate pressure to do a homework assignment or study for a test, feel free to start at the beginning and work your way forward.
Chapter 1 gives you an overview of fractions, decimals, and percentages. And Chapters 2 and 3 provide a variety of skills that I believe can help you succeed in all sorts of ways.
But if you need to know something now — or preferably yesterday — to keep up with your current math class, go ahead and refer to the Index or Table of Contents to skip directly to that topic. In every chapter, I do my best to orient you to the topic at hand, providing plenty of worked-through examples so that you can see not only what you need to know, but how to do it yourself.
No matter what order you choose to read this book in, I promise you that on every page, you can find something that can help you along in your understanding of fractions.
Part 1
An Overview of Fractions, Decimals, and Percents
IN THIS PART …
Introduce yourself to fractions, decimals, and percents.
Refresh on the basic math that you need to succeed when studying fractions.
Work with factors and multiples as an important building block for fractions.
Chapter 1
What Are Fractions, Decimals, and Percents?
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.
Taking a First Look at Fractions
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.
Understanding how fractions work
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:
Numerator:
The top number in a fraction (3 in the chocolate bar example), which shows how many of the equal parts you’re specifically working with (in this case, you ate them)
Denominator:
The bottom number in a fraction (4 in the chocolate bar example), which shows the total number of equal parts that the whole is divided into
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).
Making sense of equivalent fractions
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.)
Understanding improper fractions
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.
Seeing how improper fractions and mixed numbers relate
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.
Seeing Why Decimals Are So User-Friendly
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.
Understanding decimals that have one decimal place
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 decimal number 0.9, the digit 9 is in the tenths place, where it represents 9 out of 10 equal parts of a whole. So, 0.9 is the same as of a whole.
Here are a few real-world examples to illustrate how to interpret decimals:
Measuring weight:
If you have a bag of nails that weighs 2.3 kilograms, the digit 3 in the tenths place represents of a kilogram (or 300 grams because one kilogram contains 1000 grams). Thus, the bag of nails weighs 2 kilograms, plus an additional of a kilogram.
Measuring length:
If you have a piece of ribbon that’s 3.4 meters long, the digit 4 in the tenths place means of a meter (or 40 centimeters because one meter contains 100 centimeters). Thus, the ribbon is 3 meters plus another of a meter in length.
Cooking:
If a recipe calls for 1.5 cups of flour, the 5 in the tenths place represents of a cup, which equals half a cup (because 5 is half of 10). So, 1.5 cups is equal to 1 cup plus an extra half a cup of flour.
The preceding examples show you that decimals use a digit’s place value to represent a fraction by applying a format that’s similar to whole numbers. This similarity means that you can make calculations and comparisons more easily with decimals than with fractions.
Understanding decimals that have two decimal places
After understanding decimals that have one decimal place — the tenths place, which breaks a whole into 10 equal parts (see the preceding section) — you can move on to decimals that have two decimal places. These decimals break a whole into 100 equal parts, which are called hundredths.
When you have two decimal places, the second digit represents hundredths. For example, 0.01 means 1 out of 100 equal parts of a whole. Similarly, 0.25 means 25 out of 100 equal parts. Think of it like cutting something into 100 tiny pieces and counting how many pieces you have.
Here are some real-world examples of what decimals that have two places represent:
Counting money:
In the United States, people most commonly use decimals that have two places when dealing with money. You can break down every U.S. dollar into 100 cents, where each cent is represented as the decimal $0.01. So, if you have $0.25, you have of a dollar, or 25 cents.
Measuring length:
Another common use for decimals involves measuring length by using the metric system. For example, imagine that you’re measuring the thickness of a short stack of paper, like your term paper, and it measures 0.08 centimeters. This means that the paper is of a centimeter thick, or 8 out of 100 equal parts of a centimeter.
Measuring weight:
Metric weight measurements commonly use decimals. For example, if you weigh a small object and find that it weighs 0.35 kilograms, this indicates that the object weighs of a kilogram, or 35 out of 100 equal parts of a kilogram.
Understanding fraction-decimal equivalence
Fractions (flip back to the section “Taking a First Look at Fractions,” earlier in this chapter) and decimals are two contrasting but mathematically equivalent ways of representing parts of a whole. Although they might look different, they both express the same idea: how much of something you have relative to its whole.
And because fractions and decimals express the same ideas, you can convert between them. The following tells you how to convert
From fractions to decimals:
Simply divide the numerator by the denominator. For instance, to convert the fraction to a decimal, divide 3 by 5, which equals 0.6.
From decimals to fractions:
Write the decimal as a fraction that has a denominator based on its place value. For example, you can write 0.46 as , which simplifies to . (You can find out the nuances of how to simplify a fraction in
Chapter 4
.)
Because fractions and decimals are just two sides of the same coin, you can switch between these forms as needed, making it easier to work with parts of a whole in different contexts.
Putting Percents into Perspective
You can describe parts of a whole as fractions (discussed in the section “Taking a First Look at Fractions,” earlier in this chapter) and decimals (check out the section “Seeing Why Decimals Are So User-Friendly,” earlier in this chapter); I want to introduce a third way to describe them. You can use percents, in the same way that you can use fractions and decimals, to express what part of a whole you have.
For example, you can express a part of a whole as a fraction (for example, ) or an equivalent decimal (0.5), and you can also express the equivalent amount as a percent (50%). Percents are a useful and often intuitive way to communicate proportions, especially in everyday contexts such as sales discounts, real estate and investment calculations, and statistics and probability.
In the following sections, you can explore how to understand, calculate, and use percents in various situations. And you can make connections between fractions, decimals, and percents while you proceed so that you can see how you can use percents as a powerful tool for comparing and understanding parts of a whole.
Making sense of percents
The word percent comes from the Latin phrase per centum, which literally means per hundred. So that can clue you in that percents tell you how many parts you’re dealing with specifically, out of 100 equal parts.
