Mathematics
Ratio
A ratio is a comparison of two quantities, typically expressed as a fraction. It represents the relationship between the two quantities and can be used to compare sizes, amounts, or values. Ratios are often used in mathematical problems, such as in proportions and in solving for unknown quantities.
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12 Key excerpts on "Ratio"
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 - ePubA Focus on Ratios and Proportions
Bringing Mathematics Education Research to the Classroom
- Marjorie M. Petit, Robert E. Laird, Matthew F. Wyneken, Frances R. Huntoon, Mary D. Abele-Austin, Jean D. Sequeira(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
kx and similarity.This chapter includes discussions on:- the meaning of a Ratio,
- two different interpretations of Ratios,
- understanding rates,
- the different language and notation used to communicate Ratios and rates that may interfere with student understanding,
- the meaning of a proportion and what is meant by a proportional relationship.
Figure 1.1 How multiplicative concepts develop across mathematics curriculumRatios
A Ratio is a multiplicative comparison of two or more quantities or measures. These comparisons can be part-to-part, such as 8 students to 2 adults or 2 cups of sugar to 5 cups of flour, or they can represent part-to-whole situations like 8 girls to 24 students in a class. Regardless of the situation, Ratio comparisons are always related multiplicatively. For example, in the Ratio 8 students to 2 adults, there are 4 times as many students as adults. One can describe this relationship in several ways such as:- 4 students for every 1 adult,
- 1 adult for every 4 students,
- the number of adults is 1 4the number of students.
Two Interpretations of a Ratio
Two related but different ways to form a Ratio are shown in Figure 1.2 . The first is by joining or composing two quantities in a way that “preserves a multiplicative relationship,” and the second is multiplicatively comparing two quantities (Lobato, Ellis, Charles, & Zbiek, 2014, p. 18).Figure 1.2 Two types of RatiosRatios as Joining Two Quantities
As introduced already, one interpretation of a Ratio is the joining of two quantities in a way that preserves a multiplicative relationship. In this interpretation, a certain number of one quantity together with a certain number of another quantity creates a composed unit (Beckmann, 2014). So we can think of this Ratio interpretation as a Ratio as a composed Unit. Figure 1.3
- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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 5 • Ratio, Proportion, and Measurement 424 The figure on the right shows how the 4-to-1 rule was used to properly position the legs of a ladder 3 feet from the base of a 12-foot-high wall. We can write a Ratio comparing the ladder’s height to its distance from the wall. 12 feet 3 feet 5 12 feet 3 feet 5 12 3 Remove the common units of feet. Since this Ratio satisfies the 4-to-1 rule, the two Ratios 4 1 and 12 3 must be equal. Therefore, we have 4 1 5 12 3 Equations like this, which show that two Ratios are equal, are called proportions. Proportion A proportion is a statement that two Ratios (or rates) are equal. Some examples of proportions are n 1 2 5 3 6 Read as “1 is to 2 as 3 is to 6.” n 3 waiters 7 tables 5 9 waiters 21 tables Read as “3 waiters are to 7 tables as 9 waiters are to 21 tables.” Write each statement as a proportion. a. 22 is to 6 as 11 is to 3. b. 1,000 administrators is to 8,000 teachers as 1 administrator is to 8 teachers. Strategy We will locate the word as in each statement and identify the Ratios (or rates) before and after it. WHY The word as translates to the 5 symbol that is needed to write the statement as a proportion (equation). Solution a. This proportion states that two Ratios are equal. 22 is to 6 as 11 is to 3 Recall that the word “to” is used to separate the numbers being compared. 22 6 5 11 3 b. This proportion states that two rates are equal.
eBook - ePubMaking Sense of Mathematics for Teaching Grades 6-8
(Unifying Topics for an Understanding of Functions, Statistics, and Probability)
- Edward C. Nolan, Juli K. Dixon(Authors)
- 2016(Publication Date)
- Solution Tree Press(Publisher)
This builds on earlier understandings around multiplication, division, and fractions. Ratio problems also include determining the unit rate as a relationship of an amount of a first quantity to one unit of a second quantity. For example, if a mixture in science is described as 8 grams of salt to 4 ounces of water, the Ratio is 8 grams to 4 ounces, which is the same as 2 grams for every ounce, or the unit rate of 2 grams of salt per 1 ounce of water. This process can be shown numerically as = = 2. The unit rate is important because it supports the understanding of equivalent Ratios, the advantage of making comparisons, and the ability to solve problems. Students also learn about percent in grade 6, which is a quantity represented as a Ratio out of 100. Students solve problems including finding the whole when given a part of the whole and the percent, such as finding the number of students in a class if eleven students make up 50 percent of the class. Students determine equivalent relationships involving Ratios as well as reason with a constant value that is multiplied or divided between each of the Ratio pairs in tables, bar models, double number line diagrams, equations, and graphs. Grade 7 Students in grade 7 expand their understanding of Ratio as a way to analyze proportional relationships and direct variation (equations of the form y = kx) and use these understandings to solve problems. Students expand their understanding of unit rate to include computations that include Ratios of fractions, such as determining that a person who walks ¼ of a mile in ½ of an hour can be described as walking ½ of a mile per hour. Students explore proportional reasoning in grade 7, including distinguishing proportional relationships from nonproportional relationships, in equations, tables, and graphs. Students build on their understanding of unit rate in tables and graphs, as well as with equations, diagrams, and verbal descriptions
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)
Ratio tables provide an excellent model for exploring patterns involving Ratios. A proportion provides a relationship between two or more Ratios. Together with proportion, Ratios provide an opportunity to practise many computational skills as well as strengthen problem-solving skills. Proportions provide a way to find answers to problems where the numbers are relational. These relationships may be considered additive (absolute) or multiplicative (relative). Children often apply additive procedures to proportions that require multiplicative thinking, particularly in dealing with like quantities. The ability of children to apply multiplicative thinking or proportional reasoning is an important instructional goal and is one of the reasons proportional reasoning has been called the capstone of primary school mathematics. The use of the double number line is a powerful model for helping children engage in proportional reasoning. Proportional reasoning is complex, both in terms of the mathematics and of the developmental experiences it requires. Yet it is an important skill for children to gain because it facilitates algebraic thinking. Ratios and proportions also provide a natural means of studying percentages, which has a comparison base of 100. Because few mathematical topics have a more practical usage than percentages, it is essential that meaningful and systematic development of percentages be provided. Instruction should use concrete models to support foundational percentage concepts. Key among those concepts are benchmarks of 100% and 50% as well as 90%, 10% and 1%. These benchmarks provide anchors for children to solve a wide range of problems involving percentages and to gauge the reasonableness of their answer. 464 Helping children learn mathematics THINGS TO DO: FROM WHAT YOU’VE READ 1 Given the importance of context, provide a real-life example of a Ratio and a proportion.
eBook - ePubFostering Children's Mathematical Power
An Investigative Approach To K-8 Mathematics Instruction
- Arthur Baroody, Arthur J. Baroody, Jesse L.M. Wilkins, Ronald T. Coslick(Authors)
- 1998(Publication Date)
- Routledge(Publisher)
- investigate relationships among fractions, decimals, and percents” (p. 87).
12•1 RatiosAs Figure 12.1 implies, an accurate understanding of Ratios is a key aspect of mathematical literacy. The aim of this unit is to foster an explicit understanding of Ratios (Subunit 12•1•1) and how teachers can help students to do the same (Subunit 12•1•2). (Continued on page 12-4.)12•1•1 Mathematics & Learning: Understanding Ratios In this subunit, we touch on the uses of Ratios, define Ratios, and note how they differ from a part-of-the-whole meaning of common fractions.Figure 12.1: A Misunderstood RatioFRANK AND ERNEST reprinted with the permission of Bob Thaves.☞ Try Probe 12.1 (page 12-5), Investigation 12.2 (page 12-6), and Probe 12.2 (page 12-7).Investigation 12.2: Explicitly Understanding Ratios♦ Identifying Ratios ♦ 5-8 ♦ Groups of four + class discussionThe aim of this investigation is to help students explicitly define the concept of Ratio. To see what is involved and to perhaps clarify your own concept of Ratio, try the activity along with your group. Discuss your conclusions with your class.Instructions: Which of the following involve a Ratio? If the comparison involves a Ratio, indicate with a R; otherwise, write NA._____ 1. Double the Dough. A father, anxious to teach his son Günter the virtue of thrift, promised to match dollar for dollar the amount the boy put in a savings account. To help his son appreciate the value of this deal, the father drew up the table below to illustrate the relationship between the amount of the boy’s contribution and the total deposit:_____ 2. Mounting Admissions Costs. A day’s admission charge to Walter Dizzy’s Wonder- Why-You-Ever-Came World was $50 for one person. For each additional family member the charge was reduced by $5. The sign above the ticket booth summarizes the relationship between the number of family members and the admissions costs:_____ 3. Relationship Between the Side and Area of a Square
- Mary M. Hatfield, Nancy Tanner Edwards, Gary G. Bitter, Jean Morrow(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
What are the next steps you need to do to help all the students continue to grow? 8. Share your thoughts with others. See what your com- bined knowledge might be able to create. SUMMARY Percent, Ratio, proportion, and rate involve a multiplicative relationship and a constant. Real-world contexts and the use of manipula- tives are important in the development of con- ceptual understanding of percents. Computer spreadsheets and calculators are valuable tools in the teaching of percents. Ratios are an extension from common frac- tions and may use the same symbolic notation. Two of the most frequently used relation- ships of Ratios in problem solving are the part-part and the factor-factor-product rela- tionships. Proportions are equivalent Ratios. Three common approaches to solving propor- tions are the rule of 3 or cross-product method, the unit rate method, and the factor of change method. The rule of 3 or cross-product method is the most common approach to solving proportions and is based on the fact that three of the four variables in a proportion are known. FIGURE 11.18 Shading percent pictures by student A, age 8. FIGURE 11.19 Shading percent pictures by student B, age 9. Student C Student D Student E FIGURE 11.20 Shading percent pictures by adults. 316 CHAPTER 11 Percent, Ratio, Proportion, and Rate The unit rate method of solving proportions involves enlarging or decreasing the original measure until the desired proportion is reached. The factor of change method involves main- taining the relationship among factors even when a factor is changed. Rate refers to a Ratio in which the comparisons involve two units uniquely different from one another. Students with special needs will benefit from a scaffolding approach and the use of real-world contexts. PRAXIS II TM -STYLE QUESTIONS Multiple Choice Format 1. The method shown below is an example of an appli- cation of a mathematical concept,__________, used in ancient__________.
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 - PDF- Alan Tussy, Diane Koenig(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Recall that an equation is a statement indicating that two expressions are equal. All equations contain an 5 symbol. Some examples of equations are: 4 1 4 5 8, 15.6 2 4.3 5 11.3, 1 2 ? 10 5 5, and 216 4 8 5 22 Each of the equations shown above is true. Equations can also be false. For example, 3 1 2 5 6 and 240 4 (25) 5 28 are false equations. In this section, we will work with equations that state that two Ratios (or rates) are equal. OBJECTIVE 1 Write proportions. Like any tool, a ladder can be dangerous if used improperly. When setting up an extension ladder, users should follow the 4-to-1 rule: For every 4 feet of ladder height, position the legs of the ladder 1 foot away from the base of the wall. The 4-to-1 rule for ladders can be expressed using a Ratio. 4 feet 1 foot 5 4 feet 1 foot 5 4 1 Remove the common units of feet. rook76/Shutterstock.com Copyright 2019 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 6 • Ratio, Proportion, and Measurement 556 The figure on the right shows how the 4-to-1 rule was used to properly position the legs of a ladder 3 feet from the base of a 12-foot-high wall. We can write a Ratio comparing the ladder’s height to its distance from the wall. 12 feet 3 feet 5 12 feet 3 feet 5 12 3 Remove the common units of feet. Since this Ratio satisfies the 4-to-1 rule, the two Ratios 4 and 12 must be equal. Therefore, we have 4 1 5 12 3 Equations like this, which show that two Ratios are equal, are called proportions. 1 3 Proportion A proportion is a statement that two Ratios (or rates) are equal.
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 - 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.
eBook - PDF- Charles P. McKeague(Author)
- 2015(Publication Date)
- XYZ Textbooks(Publisher)
The second and third terms are called the means. a __ b = c __ d 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: 150 miles ________ 3 hours = 50 miles _____ hour = 50 miles per hour 3. If a 10-ounce package of frozen peas costs 69 ¢, then the price per ounce, or unit price, is 69 cents _________ 10 ounces = 6.9 cents _____ ounce = 6.9 cents per ounce 4. Solve: 5 ⋅ x = 40 5 ⋅ x = 40 5 ⋅ x ____ 5 = 40 ___ 5 Divide both sides by 5 x = 8 40 ÷ 5 = 8 5. The following is a proportion: 6 __ 8 = 3 __ 4 The terms 6 and 4 are the extremes, while 8 and 3 are the means. First term 88n Second term 88n m88 Third term m88 Fourth term Means 888n m888 Extremes 304 Chapter 5 Ratio and Proportion Fundamental Property of Proportions [5.4] In any proportion the product of the extremes is equal to the product of the means. In symbols, If a __ b = c __ d then ad = bc (b ≠ 0, d ≠ 0) Finding an Unknown Term in a Proportion [5.4] To find the unknown term in a proportion, we apply the fundamental property of proportions and solve the equation that results by dividing both sides by the number that is multiplied by the unknown. For instance, if we want to find the unknown in the proportion 2 __ 5 = 8 __ x we use the fundamental property of proportions to set the product of the extremes equal to the product of the means. Using Proportions to Find Unknown Length with Similar Figures [5.6] Two triangles that have the same shape are similar when their corresponding sides are proportional, or have the same Ratio. The triangles below are similar. Corresponding Sides Ratio Side a corresponds with side d a __ d Side b corresponds with side e b __ e Side c corresponds with side f c _ f Because their corresponding sides are proportional, we write a __ d = b __ e = c _ f 6.
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