Mathematics
Fractions in Expressions and Equations
Fractions in expressions and equations involve using numerical values in the form of fractions to represent parts of a whole within mathematical operations. This includes adding, subtracting, multiplying, and dividing fractions within algebraic expressions and equations. Understanding how to work with fractions in these contexts is essential for solving a wide range of mathematical problems.
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7 Key excerpts on "Fractions in Expressions and Equations"
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
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
NCTM Standard All students should develop understanding of fractions as parts of unit wholes, as a part of a collection, as locations on number lines, and as divisions of whole numbers. Reflection from Research Students need to develop not only a conceptual understanding of fractions, but also an under- standing of the proper notation of fractions (Brizuela, 2006). Fractions A fraction is a number that can be represented by an ordered pair of whole num- bers a b (or a b / ), where b ≠ 0. In set notation, the set of fractions is F a b a b b = ≠ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ | . and are whole numbers, 0 D E F I N I T I O N 6 . 1 On a piece of paper explain, using pictures, why 3 4 -and 6 8 are equivalent. Compare and contrast your explanation with that of your peers. Fractions can also be represented on a number line. The fraction 1 b is located on a number line by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Each part is of length 1 b and the number 1 b is located at the right endpoint of the part with its left endpoint at 0 as shown in Figure 6.4. Figure 6.4 Common Core – Grade 3 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Section 6.1 The Set of Fractions 211 The fraction a b is located on the number line by starting at 0 and marking off a copies of the length determined by the fraction 1 b end to end as shown in Figure 6.5. The right endpoint of the ath copy is the location of the fraction a b . 0 1 Figure 6.5 Before proceeding with the computational aspects of fractions as numbers, it is instructive to comment further on the complexity of this topic—namely, viewing fractions as numerals and as numbers. Recall that the whole number three was the attribute common to all sets that match the set { , , } a b c .
- Mark Zegarelli(Author)
- 2022(Publication Date)
- For Dummies(Publisher)
CHAPTER 10 Understanding Fractions 179 The result is that each fraction now has a new number written underneath it. The larger num- ber is below the larger fraction. You can use cross-multiplication to rewrite a pair of fractions as two new fractions with a com- mon denominator: 1. Cross-multiply the two fractions to find the numerators of the new fractions. 2. Multiply the denominators of the two fractions to find the new denominators. When two fractions have the same denominator, the one with the greater numerator is the greater fraction. Q. Which fraction is greater: 5 8 or 6 11 ? A. 5 8 . Cross-multiply the two fractions: Because 55 is greater than 48, 5 8 is greater than 6 11 . Q. Which of these three fractions is the least: 3 4 , 7 10 , or 8 11 ? A. 7 10 . Cross-multiply the first two fractions: Because 28 is less than 30, 7 10 is less than 3 4 , so you can rule out 3 4 . Now compare 7 10 and 8 11 similarly: Because 77 is less than 80, 7 10 is less than 8 11 . Therefore, 7 10 is the least of the three fractions. 19 Which is the greater fraction: 1 5 or 2 9 ? 20 Find the lesser fraction: 3 7 or 5 12 . 21 Among these three fractions, which is greatest: 1 10 , 2 21 , or 3 29 ? 22 Figure out which of the following fractions is the least: 1 3 , 2 7 , 4 13 , or 8 25 . 180 UNIT 4 Fractions Working with Ratios and Proportions A ratio is a mathematical comparison of two numbers, based on division. For example, suppose you bring 3 shirts and 5 ties with you on a business trip. Here are a few ways to express the ratio of shirts to ties: 3:5 3 to 5 3 5 A good way to work with a ratio is to turn it into a fraction. Be sure to keep the order the same: The first number goes on top of the fraction, and the second number goes on the bottom. You can use a ratio to solve problems by setting up a proportion equation — that is, an equation involving two ratios.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Divide whole numbers by a unit fraction and unit fractions by whole numbers. Solve problems involving multiplication and division with these types of numbers. Grade 6 Interpret and compute quotients of fractions, and solve word problems involving division of frac- tions by fractions, e.g., by using visual fraction models and equations to represent the problem. Source: Copyright 2010 by the National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. MULTIPLYING AND DIVIDING FRACTIONS 3 4 3 groups of 4 is 12 3 groups of 4 fifths is 12 fifths 3 4 5 If we symbolize three groups of four as 3 4, it makes sense to symbolize three groups of 4 5 as 3 4 5 . Then we can consider 3 4 as repeated addition: 4 4 4 12. We can approach 3 4 5 in the same way: 4 5 1 4 5 1 4 5 5 12 5 Multiplying and Dividing Fractions 259 5 4 4 5? You may have to return to the area model for multiplication as we will in the next section. Fraction Times a Fraction Consider this problem: You own 3 4 of an acre of land, and 5 6 of this is planted in trees. What part of the acre is planted in trees? Children need to understand why this is a multiplication problem. Showing them pictures like the ones below may help them see how this problem is related to the area of a rectangle and thus to multiplication. In the first picture, the acre is partitioned into fourths and the amount you own ( 3 4 ) is shaded. After students have solved problems like this one with pictures, see if they can solve them without pictures, maybe with repeated addition. Be sure to place a strong emphasis on the words. For example, in discussing the problem 5 2 3 , have them listen carefully as you read: 5 2 3 is 5 groups of two-thirds, which is 5 groups of 2 thirds or 10 thirds—which is 10 3 . Fraction Times a Whole Number Again, begin with a problem—for example: You have 3 4 of a case of 24 bottles.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
OBJECTIVES 1. Evaluate: a. b. 3. Factor: 5. Factor: 2. Simplify: a. b. 4. Factor: 6. Evaluate: 6 6 a 2 16 5 45 18 24 5 x 2 23 x 10 12 x 8 7 0 0 7 ARE YOU READY? The following problems review some basic skills that are needed when simplifying rational expressions. Fractions that are the quotient of two integers are rational numbers. Examples are and . Fractions such as , , and that are the quotient of two polynomials are called rational expressions. Evaluate Rational Expressions. Rational expressions can have different values depending on the number that is substituted for the variable. 2 a 2 8 a a 2 6 a 8 x x 2 3 2 y 9 5 1 2 A rational expression is an expression of the form , where and are polynomials and does not equal 0 . B B A A B Rational Expressions EXAMPLE 1 Evaluate for and for . Strategy We will replace each in the rational expression with the given value of the variable. Then we will evaluate the numerator and denominator separately, and simplify, if possible. Why Recall from Chapter 1 that to evaluate an expression means to find its numerical value, once we know the value of its variable. x x 0 x 3 2 x 1 x 2 1 Copyright 201 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. Solution For : For : 7 10 6 1 9 1 2 x 1 x 2 1 2( 3 ) 1 ( 3 ) 2 1 x 0 x 3 7.1 Simplifying Rational Expressions 513 1 0 1 0 1 2 x 1 x 2 1 2( 0 ) 1 ( 0 ) 2 1 Self Check 1 Evaluate for and for x 2. x 7 2 x 1 x 2 1 Problems 13 and 21 Now Try Find Numbers That Cause a Rational Expression to Be Undefined. The fraction bar in a rational expression indicates division.
eBook - PDFIntermediate Algebra
Concepts and Graphs 2E
- Charles P. McKeague(Author)
- 2019(Publication Date)
- XYZ Textbooks(Publisher)
Note that the important thing about the fractions in this example is that they each have a denominator of 9. If they did not have the same denominator, we could not have written them as two terms with a factor of 1 _ 9 in common. Without the 1 _ 9 common to each term, we couldn’t apply the distributive property. Without the distributive property, we would not have been able to add the two fractions in this form. In the following examples, we will not show all the steps we showed in Example 1. The steps are shown in Example 1 so you will see why both fractions must have the same denominator before we can add them. In practice, we simply add numerators and place the result over the common denominator. We add and subtract rational expressions with the same denominator by combining numerators and writing the result over the common denominator. Then we reduce the result to lowest terms, if possible. Example 2 shows this process in detail. If you see the similarities between operations on rational numbers and operations on rational expressions, this chapter will look like an extension of rational numbers rather than a completely new set of topics. EXAMPLE 2 Add x _____ x 2 − 1 + 1 _____ x 2 − 1 . SOLUTION Because the denominators are the same, we simply add numerators: x _____ x 2 − 1 + 1 _____ x 2 − 1 = x + 1 _____ x 2 − 1 Add numerators. = x + 1 ___________ (x − 1)(x + 1) Factor denominator. = 1 _____ x − 1 Divide out common factor x + 1. 5.3 Addition and Subtraction of Rational Expressions 431 Our next example involves subtraction of rational expressions. Pay careful attention to what happens to the signs of the terms in the numerator of the second expression when we subtract it from the first expression. EXAMPLE 3 Subtract 2x − 5 ______ x − 2 − x − 3 _____ x − 2 .
eBook - PDF- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Keep in mind that listing the restrictions at the beginning of a problem does not replace check-ing the potential solutions. In Example 4, the answer 11 needs to be checked in the original equation. Solve a a 2 2 1 2 3 5 2 a 2 2 . Solution a a 2 2 1 2 3 5 2 a 2 2 , a ? 2 3( a 2 2) a a a 2 2 1 2 3 b 5 3( a 2 2) a 2 a 2 2 b Multiply both sides by 3( a 2 2) 3( a 2 2) a a a 2 2 b 1 3( a 2 2) a 2 3 b 5 3( a 2 2) a 2 a 2 2 b 3( a ) 1 2( a 2 2) 5 3(2) 3 a 1 2 a 2 4 5 6 5 a 5 10 a 5 2 Because our initial restriction was a ? 2 , we conclude that this equation has no solution. Thus the solution set is [ . Solving Proportions A ratio is the comparison of two numbers by division. We often use the fractional form to ex-press ratios. For example, we can write the ratio of a to b as a b . A statement of equality between two ratios is called a proportion . Thus if a b and c d are two equal ratios, we can form the propor-tion a b 5 c d ( b ? 0 and d ? 0 ). We deduce an important property of proportions as follows: a b 5 c d , b ? 0 and d ? 0 bd a a b b 5 bd a c d b Multiply both sides by bd ad 5 bc Cross-Multiplication Property of Proportions If a b 5 c d ( b ? 0 and d ? 0 ), then ad 5 bc . EXAMPLE 5 Classroom Example Solve x x 1 3 1 3 2 5 2 3 x 1 3 . Copyright 2013 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. Chapter 4 • Rational Expressions 220 We can treat some fractional equations as proportions and solve them by using the cross-multiplication idea, as in the next examples.
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