Mathematics
Ratio Fractions
Ratio fractions are expressions that compare two quantities using a fraction format. They are used to represent the relationship between two numbers or quantities. In a ratio fraction, the numerator and denominator can both be whole numbers, and the ratio can be simplified to its simplest form.
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eBook - PDF- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. CHAPTER 6 • Ratio, Proportion, and Measurement 542 OBJECTIVE 1 Write ratios as fractions. Ratios give us a way to compare two numbers or two quantities measured in the same units. OBJECTIVES 1 Write ratios as fractions. 2 Simplify ratios involving decimals and mixed numbers. 3 Convert units to write ratios. 4 Write rates as fractions. 5 Find unit rates. 6 Find the best buy based on unit price. SECTION 6.1 Ratios and Rates Ratios are often used to describe important relationships between two quantities. Here are three examples: To prepare fuel for an outboard marine engine, gasoline must be mixed with oil in the ratio of 50 to 1. To make 14-karat jewelry, gold is combined with other metals in the ratio of 14 to 10. In this drawing, the eyes-to-nose distance and the nose-to-chin distance are drawn using a ratio of 2 to 3. 2 3 Ratios A ratio is the quotient of two numbers or the quotient of two quantities that have the same units. There are three ways to write a ratio. The most common way is as a fraction. Ratios can also be written as two numbers separated by the word to, or as two numbers separated by a colon. For example, the ratios described in the illustrations above can be expressed as: 50 1 , 14 to 10, and 2 : 3 n The fraction 50 1 is read as “the ratio of 50 to 1.” A fraction bar separates the numbers being compared. n 14 to 10 is read as “the ratio of 14 to 10.” The word “to” separates the numbers being compared. n 2 : 3 is read as “the ratio of 2 to 3.” A colon separates the numbers being compared. Writing a Ratio as a Fraction To write a ratio as a fraction, write the first number (or quantity) mentioned as the numerator and the second number (or quantity) mentioned as the denominator. Then simplify the fraction, if possible. Copyright 2019 Cengage Learning.
- 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 - PDFMaking Sense of Number
Improving Personal Numeracy
- Annette Hilton, Geoff Hilton(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
In this chapter we will use familiar examples to support your understanding of the concepts that underpin them. The chapter begins with ratio. RATIO A quick Google search on the meaning of the term ratio will convince you that there are multiple definitions, and this is likely to be a problem for anyone who is uncertain or lacking in confidence when it comes to these ideas. The definitions vary and include ‘a comparison of quantities’ and ‘the number of times one number is contained in another’. These are not particularly helpful without further explanation or examples. In this chapter we will use the definition from the Australian Curriculum: A ratio represents a comparative situation – it is a comparison between quantities with the same units of measure (ACARA, 2020b). When we refer to units of measure we mean both parts of the ratio deal with animals or people or plants or liquids, and so on. For example, if we read that the ratio of teachers to students in Australian government schools is 1:14, this means that for every 1 teacher there are 14 students (this ratio deals with the common measure of people). As we proceed through the chapter, the definition and application of ratio will be expanded through examples. In the next section we will look at some of the basics of ratio and common representations. Representing ratio As you would have noticed from the margin definition, ratios are mathematical expressions that use the symbol ‘:’ between the quantities being compared. For example, the ratio of A ratio is a comparison between quantities with the same units of measure: for example, 3 cats to 4 dogs or 3:4. Chapter 8 Ratio, rate and scale 135 dark-leafed lettuce to light-leafed lettuce in Figure 8.1 is 2:3. This means that for every 2 dark-leafed lettuce there are 3 light-leafed lettuce. Figure 8.1 The ratio of dark-leafed lettuce to light-leafed lettuce is 2:3 The ratio in Figure 8.1 is read as ‘2 is to 3’ and represents the relationship between the two parts.
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 - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
CHAPTER 13 Ratio, proportion and percentages: meanings and applications LEARNING OUTCOMES 13.1 Understanding how ratio and proportion are related and how they are different. 13.2 Developing ideas that use children’s intuition and reasoning to progress their thinking about proportions. 13.3 Generating teaching ideas around models that help develop the concept of percentages. ‘Although children are only 24 percent of the population, they’re 100 percent of our future and we cannot afford to provide any child with a substandard education.’ Ed Markey (American lawyer and politician) Chapter 13 concept map Part–part ratio Part–whole ratio Whole–part ratio Relates two quantities in a multiplicative relationship Part and whole measured as hundredths Equality between ratios Equivalent fractions Fraction with 100 denominator Can be expressed as Fractions Ratio Proportion Percentage Introduction ‘Twice around your thumb is equivalent to once around your wrist.’ ‘Family income this year increased by 10%.’ ‘Ian did only half the work Angela did.’ ‘Her salary is three times my salary.’ ‘The cost of living tripled during the last eight years.’ ‘Your chances of winning the lottery are less than one in a million.’ ‘I can purchase a 360 mL bottle of water for $1 or a 400 mL bottle for $1.50. Which bottle is the better buy?’ These statements show ratio, proportion and percentage in action. They demonstrate that much of quantitative thinking is relational. In such thinking, what is important is the multiplicative relationship between numbers, rather than the actual numbers themselves. Multiplicative thinking is described by Hurst and Hurrell (2014, p. 13) as possibly ‘the “biggest” of the “big number ideas”’. Unfortunately, children can struggle with developing multiplicative thinking and this can impact on their development of other mathematical understandings such as those involving place value.
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.
- Mary M. Hatfield, Nancy Tanner Edwards, Gary G. Bitter, Jean Morrow(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
11 Percent, Ratio, Proportion, and Rate WHERE’S THEMATHEMATICS? THE MATHEMATICAL STRUCTUREOF PERCENT, RATIO, PROPORTION, ANDRATE The study of ratio, proportion, percent, and rate largely involves activities for grade 5 through grade 8. From decimals, students proceed to the study of ratio and percent, two other focal points in the study of rational numbers. Proportion is an exten- sion of the comparisons developed when studying ratio. Rate is another way of comparing relation- ships. All four concepts have many real-world applications and frequently become topics of word problems in elementary and middle school pro- blem solving. If you would like to review the defini- tions and some algebraic examples, go to ‘‘Check Your Knowledge’’ (Chapter 11, on the Web site). Figure 11.1 gives a partial listing of the NCTM standards related to percent, ratio, proportion, and rate. For a more complete listing, check the NCTM Web site at http://standards.nctm.org/ index.htm. You may be asking, ‘‘Why are these four math- ematical concepts placed together in a chapter?’’ It is of interest that they are related. Let’s look at the following word problem as an example. The pro blem: Mateo rewards himself with three bite-size candies every time he gets a hit in baseball. Figure 11.2 shows how rewording can change the problem from one concept to another. All these problems have a multiplicative rela- tionship and a constant. In this example, the con- stant is the 3. The number of candies is three times the hits or c ¼ 3h. We will see how that under- standing is developed in each of the four concepts. In a school setting, each concept, although related, is initially presented separately, so we present them separately in this chapter as well. Chapter 11 At the Start ~~~~ KnowWhat ChildrenWill Be Doing .
eBook - PDFPractical Algebra
A Self-Teaching Guide
- Bobson Wong, Larisa Bukalov, Steve Slavin(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
4 RATIOS AND PROPORTIONS As we said in Chapter 2, fractions have been a central part of number systems around the world for thousands of years. In this chapter, we focus on how we use fractions to compare quantities. 4.1 Expressing Ratios in Simplest Form In Figure 4.1, the distance from point A to point B ( AB ) is 6 units, and the distance from point B to point C ( BC ) is 4 units. B C A 6 4 Figure 4.1 Number line showing a 6:4 ratio. To describe the relationship between AB and BC , we can use a ratio − a quantity that indicates how many times one number contains another. We express the number of times that AB contains BC in any of the following ways: • 6 to 4 • 6:4 • 6/4 • 6 4 Reading and Writing Tip Although ratios can be written in different ways, they are pronounced as “[number] to [number].” All the ratios listed above are pronounced as “6 to 4.” When we write a ratio as a fraction, we don’t use the pronunciation used for fractions (“six-fourths” or “six over four”). The ratio of AB to BC should not be confused with the ratio of AB to the entire segment length AC . The ratio of the two parts of the segment to each other ( AB : BC ) is 6:4, but the ratio of the part AB to the whole ( AB : AC ) is 6:10. Like fractions, ratios are often expressed in simplest form, so we usually write the ratio AB : BC as 3 to 2, 3:2, 3/2, or 3 2 . To describe how many times BC contains AB , we write the ratio 2 to 3, 2:3, 2/3, or 2 3 . 79 80 PRACTICAL ALGEBRA If the quantities in a ratio have the same units, we don’t have to include these units in the final form of the ratio. For example, the ratio 30 miles 4 miles is typically written as 30 4 or 15 2 . When the quantities in a ratio have different units, we include them. In these cases, the ratio a : b represents a portion of a that corresponds to one unit of b , so 30 miles 4 trips is expressed as 15 2 miles per trip. Example 4.1 Express 8 cups to 3 cups as a ratio (with units if appropriate) in simplest form.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
By emphasizing multiplicative relationships, you will help your students take advantage of the many opportunities to use these relationships in real-world situations. The Common Core State Standards for Mathematics makes it clear that students in elementary school and mid- dle school must be able to express appropriate relation- ships using fractions, ratios, proportions, and percents. In Table 13-1, note that although the main focus on ratio, pro- portion, and percent takes place at grade 6, there is a de- velopmental progression from grade 3 to grade 6 of the necessary underlying concepts. Here is an example to get you started thinking about ratios, proportions, and percents. Consider the prices of three carpets: TABLE 13-1 • Ratio and Proportion Standards for Grade 6 from CCSSM. The full CCSSI document is available at www.corestandards.org Understand ratio concepts and use ratio reasoning to solve problems • CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” • CCSS.Math.Content.6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” • CCSS.Math.Content.6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
eBook - PDFIntroductory Mathematics
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
CHAPTER 5 Trail Guide Project 278 Chapter 5 Ratio and Proportion Supplies Needed A Internet access Olympic Rates In 2010, the Winter Olympics were held in Vancouver, British Columbia. At the conclusion of these games, Canada became the first host nation to have won the most gold medals since Norway in 1952. The following is a list of some of the sports in which athletes competed during the 2010 Olympic Games: Alpine skiing Bobsleigh Freestyle skiing Luge Ski jumping Biathlon Cross-country skiing Ice hockey Speed skating Snowboarding Working in groups, choose a sport from the above list. Research the sport, as well as the 2010 race details and results. Explain how rates and proportions can be used to describe the details of each sport’s race and results. Present your findings to the class. Supplies Needed 279 Chapter 5 Summary Chapter 5 Summary Ratio [5.1] The ratio of a to b is a _ b . The ratio of two numbers is a way of comparing them using fraction notation. Rates [5.2] A ratio that compares two different quantities, like miles and hours, gallons and seconds, etc., is called a rate. Unit Pricing [5.2] The unit price of an item is the ratio of price to quantity when the quantity is one unit. Solving Equations by Division [5.3] Dividing both sides of an equation by the same number will not change the solution to the equation. For example, the equation 5 ⋅ x = 40 can be solved by dividing both sides by 5. Proportion [5.4] A proportion is an equation that indicates that two ratios are equal. The numbers in a proportion are called terms and are numbered as follows: First term a __ b = c _ d Third term Second term Fourth term The first and fourth terms are called the extremes. The second and third terms are called the means. Means a __ b = c _ d Extremes EXAMPLES 1. The ratio of 6 to 8 is 6 _ 8 which can be reduced to 3 _ 4 . 2. If a car travels 150 miles in 3 hours, then the ratio of miles to hours is considered a rate.
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