Mathematics
Simplifying Fractions
Simplifying fractions involves reducing a fraction to its simplest form by dividing the numerator and denominator by their greatest common divisor. This process ensures that the fraction is expressed in its most compact and easily understandable form. By simplifying fractions, it becomes easier to compare and perform operations with different fractions.
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11 Key excerpts on "Simplifying Fractions"
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
9–1 Simplification of Fractions Parts of a Fraction A fraction has a numerator, a denominator, and a fraction line. fraction line a b numerator denominator Quotient A fraction is a way of indicating a quotient of two quantities. The fraction a/b can be read “a divided by b.” The two ways of writing a fraction, a b and a/b, are equally valid. Ratio We also speak of the quotient of two numbers or quantities as the ratio of those quantities. Thus the ratio of x to y is x y . 9 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Simplify fractional expressions. • Multiply and divide fractional expressions. • Add and subtract fractional expressions. • Simplify complex fractional expressions. • Solve fractional equations. • Solve word problems using fractional equations. • Manipulate and work with literal equations and formulas. You already know about fractions with numbers. In algebra, however, the numbers are replaced with letters, coefficients, and even entire expressions. Many equations and formulas in science and technology are in the form of a fraction. Since the rules of working with the numerators and denominators of fractions must be applied to entire algebraic expressions, we’ll need to make much use of the factoring techniques of Chapter 8 in order to simplify them. As we work with formulas that include fractions, we must be careful: it’s very easy to make mistakes when we cross multiply. Remember that any- thing you do must be done to each term on both sides of the equation. Also, don’t be intimidated by complex fractions where a numerator or denominator might contain a fraction; use your skills and take it one step at a time. Not all of this material is new to us. Some was covered in Chapter 2, and we solved simple fractional equations in Chapter 3. Fractions and Fractional Equations
- Robert Brechner, Geroge Bergeman(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
P E R F O R M A N C E O B J E C T I V E S SECTION I: Understanding and Working with Fractions 2-1: Distinguishing among the various types of fractions (p. 33) 2-2: Converting improper fractions to whole or mixed numbers (p. 34) 2-3: Converting mixed numbers to improper fractions (p. 35) 2-4: Reducing fractions to lowest terms a. Reducing fractions by inspection (p. 36) b. Reducing fractions by using the greatest common divisor (p. 37) 2-5: Raising fractions to higher terms (p. 38) SECTION II: Addition and Subtraction of Fractions 2-6: Determining the least common denominator (LCD) of two or more fractions (p. 41) 2-7: Adding fractions and mixed numbers (p. 42) 2-8: Subtracting fractions and mixed numbers (p. 44) SECTION III: Multiplication and Division of Fractions 2-9: Multiplying fractions and mixed numbers (p. 51) 2-10: Dividing fractions and mixed numbers (p. 53) Gorodenkoff/Shutterstock.com C H A P T E R 2 Fractions Copyright 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION I • UNdErSTaNdINg aNd WOrkINg WITh FraCTIONS 33 U NDERSTANDING AND W ORKING WITH F RACTIONS Fractions are a mathematical way of expressing a part of a whole thing. The word fraction comes from a Latin word meaning “break.” Fractions result from breaking a unit into a number of equal parts. This concept is used quite commonly in business. We may look at sales for 1 2 the year or reduce prices by 1 4 for a sale. A new production machine in your company may be 1 3 4 times faster than the old one, or you might want to cut 5 3 4 yards of fabric from a roll of material.
eBook - PDF- Charles P. McKeague(Author)
- 2011(Publication Date)
- XYZ Textbooks(Publisher)
DEFINITION The Meaning and Properties of Fractions 144 Chapter 2 Fractions 1: Multiplication and Division EXAMPLE 1 Name the numerator and denominator for each fraction. a. } 3 4 } b. } 5 a } c. } 7 1 } SOLUTION In each case we divide the numerator by the denominator: a. The terms of the fraction } 3 4 } are 3 and 4. The 3 is called the numerator, and the 4 is called the denominator. b. The numerator of the fraction } 5 a } is a. The denominator is 5. Both a and 5 are called terms. c. The number 7 may also be put in fraction form, because it can be written as 7 _ 1 . In this case, 7 is the numerator and 1 is the denominator. CLARIFICATION 1: The fractions } 3 4 }, } 1 8 }, and } 1 9 0 } are all proper fractions, because in each case the numerator is less than the denominator. CLARIFICATION 2: The numbers } 9 5 }, } 1 1 0 0 }, and 6 are all improper fractions, be- cause in each case the numerator is greater than or equal to the denominator. (Remember that 6 can be written as } 6 1 }, in which case 6 is the numerator and 1 is the denominator.) Fractions on the Number Line We can give meaning to the fraction } 2 3 } by using a number line. If we take that part of the number line from 0 to 1 and divide it into three equal parts, we say that we have divided it into thirds (see Figure 2). Each of the three segments is } 1 3 } (one third) of the whole segment from 0 to 1. DEFINITION For the fraction } b a }, a and b are called the terms of the fraction. More specifically, a is called the numerator, and b is called the denominator. DEFINITION Video Examples Section 2.1 DEFINITION A proper fraction is a fraction in which the numerator is less than the denominator. If the numerator is greater than or equal to the denominator, the fraction is called an improper fraction. DEFINITION Note There are many ways to give meaning to fractions like } 2 3 } other than by using the num- ber line. One popular way is to think of cutting a pie into three equal pieces, as shown below.
eBook - ePubThe Everything Guide to Pre-Algebra
A Helpful Practice Guide Through the Pre-Algebra Basics - in Plain English!
- Jane Cassie(Author)
- 2013(Publication Date)
- Everything(Publisher)
Chapter 5 FractionsBy clarifying the rules and process for how you write, alter, and manipulate fractions, you’ll be getting ready to start adding variables into your math problems. The most important part of pre-algebra is clarifying the rules that you already know how to follow, so that you can still follow the rules when the numbers start getting replaced with variables. Fractions won’t just show up by themselves anymore—they’ll be mixed into math problems that test all different kinds of material.Introduction to Fractions
There are three big ways that mathematicians represent non-integer numbers: decimals, fractions, and percentages. Let’s say you have one orange, and you put a knife right in the middle and slice. You take a piece, and your friend takes a piece. You could say, “I have half of the orange.” Or, your friend could say, “I have 50 percent of the orange.” Or, you could say, “I have .5 oranges.” All of these sentences mean the same thing.Fractions are pretty much everyone’s least favorite part of math, but they don’t have to be. The key is to take the time to understand what they mean instead of just trying to memorize the rules you are supposed to follow. Once you understand why the rules are what they are, they are much easier to remember.Numerator and Denominator
A fraction is just one number on top of another number. The top number is called the numerator , and the bottom number is called the denominator . These words are specific to fractions—they don’t mean anything except “top number” and “bottom number.”The bottom number tells you how many pieces of something make up the whole. For example, when you cut that orange in half, there are two pieces, so the denominator would be 2. The top number tells you how many pieces you have. In the same example, the numerator would be 1, because you only have one piece of the orange.
eBook - ePubQuick Arithmetic
A Self-Teaching Guide
- Robert A. Carman, Marilyn J. Carman(Authors)
- 2002(Publication Date)
- Trade Paper Press(Publisher)
dividing the top and bottom of the fraction by the same number. It is illegal to subtract a number from top and bottom.Reduce the following fractions to lowest terms:The answers are in 14 .13Lowest Terms Very often in working with fractions you will be asked to reduce a fraction to lowest terms. This means to replace the fraction with the simplest fraction in its set of equivalent fractions. For example, to reduce to its lowest terms, you would replace it with . The two fractions and and equivalent, and is the simplest equivalent fraction to because its numerator (1) and denominator (2) are the smallest whole numbers of any in the setHow can you find the simplest equivalent fraction? For example, how would you reduce to lowest terms?First, factor numerator and denominator.Second, identify and eliminate common factors.In effect, we have divided both top and bottom of the fraction by 2 × 3 = 6, the common factor.Your turn. Reduce to lowest terms. Look in 12 for the answer.14CAUTION Remember that cancellation is division by a common factor. Now check your understanding by reducing the fraction 6/3 to lowest terms. Check your answer in 15 .15LEARNING HELP Any whole number may be written as a fraction by using a denominator equal to 1. The number 1 can be written as any fraction whose numerator and denominator are equal.If you were offered your choice between ¾ of a certain amount of money and of the same amount of money, which would you choose? Which is the larger fraction, ¾ or ? Can you decide? Try. Renaming the fractions would help. The answer is in 16 .16 To compare two fractions, rename each by changing them to equivalent fractions with the same denominator.Now compare the new fractions: is greater than .LEARNING HELP 1. The new denominator is the product of the original denominators (24 = 8 × 3).2.
eBook - PDFIntroductory Mathematics
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
The first is when the denominator of a fraction is 1. In this case, if we let a represent any number, then a __ 1 = a The second situation occurs when the numerator and the denominator of a fraction are the same nonzero number. a __ a = 1 EXAMPLE 5 Simplify each fraction. a. 24 __ 1 b. 24 __ 24 c. 48 __ 24 d. 72 __ 24 Solution In each case, we divide the numerator by the denominator. a. 24 __ 1 = 24 b. 24 __ 24 = 1 c. 48 __ 24 = 2 d. 72 __ 24 = 3 Comparing Fractions We can compare fractions to see which is larger or smaller when they have the same denominator. EXAMPLE 6 Write each fraction as an equivalent fraction with denominator 24. Then write them in order from smallest to largest. 5 __ 8 5 __ 6 3 __ 4 2 __ 3 Solution We begin by writing each fraction as an equivalent fraction with denominator 24. 5 __ 8 = 15 __ 24 5 __ 6 = 20 __ 24 3 __ 4 = 18 __ 24 2 __ 3 = 16 __ 24 3. Write 2 _ 3 as an equivalent fraction with a denominator of 15x. PROPERTY Division Property for Fractions If a, b, and c are integers and b and c are not 0, then it is always true that a __ b = a ÷ c _____ b ÷ c In words: If the numerator and the denominator of a fraction are divided by the same nonzero number, the resulting fraction is equivalent to the original fraction. 4. Write 20 __ 25 as an equivalent fraction with a denominator of 5. Answers 3. 10x ___ 15x 4. 4 _ 5 5. a. 15 b. 1 c. 3 d. 10 6. 1 _ 3 < 5 __ 12 < 5 _ 9 < 5 _ 6 5. Simplify each fraction. a. 15 __ 1 b. 15 __ 15 c. 45 __ 15 d. 150 ___ 15 6. Write each fraction as an equivalent fraction with denominator 36. Then write the original fractions in order from smallest to largest. 5 _ 9 , 1 _ 3 , 5 _ 6 , 5 __ 12 Chapter 2 Multiplication and Division of Fractions 86 Now that they all have the same denominator, the smallest fraction is the one with the smallest numerator and the largest fraction is the one with the largest numerator.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
7.1 Simplifying Rational Expressions 519 EXAMPLE 8 Simplify: a. b. Strategy We will begin by factoring the numerator and denominator. Then we look for common factors, or factors that are opposites, and remove them. Why We need to make sure that the numerator and denominator have no common factor (or opposite factors) other than 1. When this is the case, then the rational expression is simplified. Solution a. To prepare to simplify, factor the numerator, and factor the denominator. Since and are opposites, simplify by replacing with the equivalent fraction . This removes the factor . In the numerator, can be written as . In the denominator, . This result may be written in several other equivalent forms. The symbol in can be written in the front of the fraction, and the parentheses can be dropped. The symbol in represents a factor of . Distribute the multiplication by in the numerator. The symbol in can be applied to the denominator. However, we don’t usually use this form. b. The binomials and are not opposites because their first terms do not have opposite signs. Thus, does not simplify. t 8 t 8 t 8 t 8 ( y 1) ( y 1) 3 y 1 3 1 1 ( y 1) ( y 1) 3 y 1 3 ( y 1) ( y 1) 3 y 1 3 3 1 3 ( y 1) 1 ( y 1) ( y 1) 3 y 1 1 y 1 1 1 y 1 1 y 1 y y 1 ( y 1)( y 1 1 ) 3(1 y 1 ) y 2 1 3 3 y ( y 1)( y 1) 3(1 y ) t 8 t 8 y 2 1 3 3 y Self Check 8 Simplify: a. b. 2 x 3 2 x 3 m 2 100 10 m m 2 Problem 63 Now Try Caution A symbol in front of a fraction may be applied to the numerator or to the denominator, but not to both: y 1 3 ( y 1) 3 STUDY SET SECTION 7.1 VOCABULARY Fill in the blanks. 1. A quotient of two polynomials, such as , is called a expression. 2. To simplify a rational expression, we remove common of the numerator and denominator. 3. Because of the division by 0, the expression is . 4. The binomials and are called , because their terms are the same, except that they are opposite in sign. 15 x x 15 8 0 x 2 x x 2 3 x CONCEPTS 5. When we simplify , the result is .
- Mark Zegarelli(Author)
- 2022(Publication Date)
- For Dummies(Publisher)
(See the nearby sidebar for why this works.) » When the numerator of one fraction and the denominator of the other are divisible by the same number, factor this number out of both. In other words, divide the numerator and denominator by that common factor. (For more on how to find factors, see Chapter 9.) For example, suppose you want to multiply the following two numbers: 5 13 13 20 × . You can make this problem easier by canceling out the number 13, as follows: 5 13 13 20 5 20 13 13 1 1 You can make it even easier by noticing that 20 and 5 are both divisible by 5, so you can also factor out the number 5 before multiplying: CHAPTER 11 Fractions and the Big Four Operations 193 5 13 13 20 1 1 4 1 Now, multiply across to complete the problem: = 1 4 ONE IS THE EASIEST NUMBER With fractions, the relationship between the numbers, not the actual numbers themselves, is most important. Understanding how to multiply and divide fractions can give you a deeper understand- ing of why you can increase or decrease the numbers within a fraction without changing the value of the whole fraction. When you multiply or divide any number by 1, the answer is the same number. This rule also goes for fractions, so 3 8 1 3 8 1 3 8 5 13 1 5 13 1 5 13 67 70 1 67 70 1 and 3 8 and 5 13 and 67 70 67 70 And as I discuss in Chapter 10, when a fraction has the same number in both the numerator and the denominator, its value is 1. In other words, the fractions 2 2 3 3 , , and 4 4 are all equal to 1. Look what happens when you multiply the fraction 3 4 by 2 2 : 3 4 2 2 3 2 4 2 6 8 The net effect is that you’ve increased the terms of the original fraction by 2. But all you’ve done is multiply the fraction by 1, so the value of the fraction hasn’t changed.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Most people who have degrees in mathematics were students who could learn mathematics on their own. This doesn’t mean that you must learn it all on your own, or that you study alone, or that you don't ask questions. It means that you know your resources, both internal and external, and you can count on those resources when you need them. Attaining this goal gives you independence and puts you in control of your success in any math class you take. © clu/iStockPhoto This is the last chapter in which we will mention study skills. You know by now what works best for you and what you have to do to achieve your goals for this course. From now on, it is simply a matter of sticking with the things that work for you and avoiding the things that do not. It seems simple, but as with anything that takes effort, it is up to you to see that you maintain the skills that get you where you want to be in the course. 415 6.1 Learning Objectives In this section, we will learn how to: 1. Evaluate a rational expression. 2. Determine when a rational expression is undefined. 3. Reduce a rational expression to lowest terms. 4. Reduce a rational expression containing factors that are opposites. Evaluating and Reducing Rational Expressions Introduction We will begin this section with the definition of a rational expression. Recall from Chapter 1 that a rational number is any number that can be expressed as the ratio of two integers: Rational numbers = a __ b a and b are integers, b ≠ 0 We define a rational expression in a similar fashion. A rational expression is any expression that can be written in the form P __ Q where P and Q are polynomials and Q ≠ 0. rational expression DEFINITION Some examples of rational expressions are 2 x − 3 ______ x + 5 x 2 − 5 x − 6 _________ x 2 − 1 a − b _____ b − a Evaluating Rational Expressions To evaluate a rational expression means to find its value when any variables in the expression are replaced by specific numbers.
eBook - PDF- Charles P. McKeague, Kate Duffy Pawlik(Authors)
- 2014(Publication Date)
- XYZ Textbooks(Publisher)
186 Chapter 2 Fractions and Mixed Numbers As a summary of what we have done so far, and as a guide to working other problems, we now list the steps involved in adding and subtracting fractions with different denominators. The idea behind adding or subtracting fractions is really very straight-forward. We can only add or subtract fractions that have the same denominators. If the fractions we are trying to add or subtract do not have the same denominators, we rewrite each of them as an equivalent fraction with the LCD for a denominator. Here are some additional examples of sums and differences of fractions. Subtract 3 __ 5 − 1 __ 6 . Solution The LCD for 5 and 6 is their product, 30. We begin by rewriting each fraction with this common denominator: 3 __ 5 − 1 __ 6 = 3 ⋅ 6 ____ 5 ⋅ 6 − 1 ⋅ 5 ____ 6 ⋅ 5 = 18 ___ 30 − 5 ___ 30 = 13 ___ 30 Add 1 __ 6 + 1 __ 8 + 1 __ 4 . Solution We begin by factoring the denominators completely and building the LCD from the factors that result: 8 divides the LCD 6 = 2 ⋅ 3 8 = 2 ⋅ 2 ⋅ 2 4 = 2 ⋅ 2 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 = 24 4 divides the LCD 6 divides the LCD We then change to equivalent fractions and add as usual: 1 __ 6 + 1 __ 8 + 1 __ 4 = 1 ⋅ 4 ____ 6 ⋅ 4 + 1 ⋅ 3 ____ 8 ⋅ 3 + 1 ⋅ 6 ____ 4 ⋅ 6 = 4 ___ 24 + 3 ___ 24 + 6 ___ 24 = 13 ___ 24 To Add or Subtract Any Two Fractions Step 1 Factor each denominator completely, and use the factors to build the LCD. (Remember, the LCD is the smallest number divisible by each of the denominators in the problem.) Step 2 Rewrite each fraction as an equivalent fraction with the LCD. This is done by multiplying both the numerator and the denominator of the fraction in question by the appropriate whole number. Step 3 Add or subtract the numerators of the fractions produced in Step 2. This is the numerator of the sum or difference. The denominator of the sum or difference is the LCD. Step 4 Reduce the fraction produced in Step 3 to lowest terms if it is not already in lowest terms. Example 7 Example 8
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
RULE EXAMPLE 5 Note It is convenient to draw a line through the factors as we divide them out. It is especially helpful when the problems become longer. Note Students sometimes make the mistake of dividing out common terms: x 2 − 9 _________ x 2 + 5 x + 6 ≠ − 9 _____ 5 x + 6 This does not give us an equivalent expression. We can only divide out common factors, which usually requires that we factor the rational expression first. EXAMPLE 6 EXAMPLE 7 7.1 Reducing Rational Expressions to Lowest Terms 451 Reduce x − 5 ______ x 2 − 25 to lowest terms. Also, state any restrictions on the variable. SOLUTION First, we reduce the expression by dividing out common factors. x − 5 ______ x 2 − 25 = x − 5 ___________ ( x − 5)( x + 5) Factor numerator and denominator completely = 1 _____ x + 5 Divide out the common factor , x − 5 To find any restrictions on the variable, we must find any values of x that make the original expression undefined. This will be the case if x 2 − 25 = 0. x 2 − 25 = 0 ( x + 5)( x − 5) = 0 Factor x + 5 = 0 or x − 5 = 0 Zero-factor property x = − 5 or x = 5 Our restrictions are x ≠ − 5 and x ≠ 5. Ratios For the rest of this section we will concern ourselves with ratios , a topic closely related to reducing fractions and rational expressions to lowest terms. Let’s start with a definition. As you can see, ratios are another name for fractions or rational numbers. They are a way of comparing quantities. Since we also can think of a _ b as the quotient of a and b , ratios are also quotients. The following table gives some ratios in words and as fractions. EXAMPLE 8 Note Even though the rational expression in Example 8 can be reduced, the original expression is undefined for both x = − 5 and x = 5. When determining any restrictions on the variable, we must work with the original denominator prior to reducing the expression.
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