Ratios as Fractions

12 Key excerpts on "Ratios as Fractions"

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  eBook - ePub

  A 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 curriculum

  Ratios

  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 4
    the number of students.

  Regardless of the way this relationship is communicated, there are 4 times as many students as adults.

  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 ratios

  Ratios 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

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  Prealgebra

  • 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.

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  Making 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.

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  Basic Mathematics with Early Integers

  • Charles P. McKeague(Author)
  • 2011(Publication Date)
  • XYZ Textbooks

    (Publisher)

  We can define the ratio of two numbers in terms of fractions. We handle ratios the same way we handle fractions. For example, when we said that the ratio of 10 men to 5 women was the same as the ratio 2 to 1, we were actually saying } 1 5 0 } 5 } 2 1 } Reducing to lowest terms Because we have already studied fractions in detail, much of the introductory ma- terial on ratios will seem like review. EXAMPLE 1 Express the ratio of 16 to 48 as a fraction in lowest terms. SOLUTION Because the ratio is 16 to 48, the numerator of the fraction is 16 and the denominator is 48: } 1 4 6 8 } 5 } 1 3 } In lowest terms Notice that the first number in the ratio becomes the numerator of the fraction, and the second number in the ratio becomes the denominator. DEFINITION The ratio of two numbers is a fraction, where the first number in the ratio is the numerator and the second number in the ratio is the denominator. In symbols: If a and b are any two numbers, then the ratio of a to b is } b a }. (b ? 0) DEFINITION Video Examples Section 5.1 Ratios ©iStockPhoto.com 296 Chapter 5 Ratio and Proportion EXAMPLE 2 Give the ratio of 2 __ 3 to 4 __ 9 as a fraction in lowest terms. SOLUTION We begin by writing the ratio of 2 __ 3 to 4 __ 9 as a complex fraction. The numerator is 2 __ 3 , and the denominator is 4 __ 9 . Then we simplify. } 2 3 } } } 4 9 } 5 } 2 3 } ? } 9 4 } Division by 4 __ 9 is the same as multiplication by 9 __ 4 5 } 1 1 8 2 } Multiply 5 } 3 2 } Reduce to lowest terms EXAMPLE 3 Write the ratio of 0.08 to 0.12 as a fraction in lowest terms. SOLUTION When the ratio is in reduced form, it is customary to write it with whole numbers and not decimals. For this reason we multiply the numerator and the denominator of the ratio by 100 to clear it of decimals. Then we reduce to lowest terms. 0.08 } 0.12 5 } 0 0 .

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  Mathematics for Elementary School Teachers

  • Tom Bassarear, Meg Moss(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  For example, we hear on television that a paper towel absorbs “50% more liquid than the leading brand” as opposed to “in tests it absorbed 4 more ounces of water than the other brand.” Multiplicative comparisons provide a context that helps us to know how much more. Ratio and Proportion SECTION 5.1 What do you think? l Are all ratios fractions? Are all fractions ratios? Why or why not? l What do we mean by proportional reasoning? l How do the concepts of ratio and proportion give us tools for making comparisons? Section 5.1 Ratio and Proportion 249 When we set two ratios equal to each other, we have a proportion. Above we said that 35 miles per 1 gallon is the same as 70 miles per 2 gallons. Writing this as a proportion would look like: 35 miles 1 gallon 70 miles 2 gallons 5 Other examples include: 12 students : 1 teacher 72 students 5 : 6 teachers, or 55 miles 1 hour 110 miles 2 hours 5 . That is, : : a b c d 5 if a b c d 5 and b ± 0, d ± 0. n n Ratios, Rates, and Proportions in Mathematics and Real Life Ratios and rates pervade mathematics. Ratio is inherent in the concept of place value: the ratio of the value of each place to the value of the place to its right is 10 :1. The concept of similarity involves equal ratios. When we make graphs, we use proportions. When we make scale drawings and scale models, we obey proportions. The essence of probability involves ratios. Ratios, rates, and proportions show up in all kinds of real-world contexts. • Banking: When you apply for a mortgage, the bank applies a ratio called the 28% rule: If the ratio of fixed monthly payments (mortgage, property tax, car payments, and so forth) to monthly income is more than 28 :100, the bank is not likely to make the loan. • Botany: If we represent the number of complete turns made around the stem by t and rep- resent the number of leaves between the two points as n, then the fraction t n is called a divergency constant for that species (Figure 5.1).

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  Helping Children Learn Mathematics

  • 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.

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  Helping Children Learn Mathematics

  • 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.

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  Practical 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.

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  Arithmetic For Parents: A Book For Grown-ups About Children's Mathematics (Revised Edition)

  A Book for Grown-Ups About Children's Mathematics

  • Ron Aharoni(Author)
  • 2015(Publication Date)
  • World Scientific

    (Publisher)

  To return to the original price, before tax, the price is divided by 1.2. The price before tax was: 300 ÷ 1 . 2 = 300 ÷ 6 5 = 300 × 5 6 = 250 . This page intentionally left blank This page intentionally left blank E. Ratios Ratios integrate in one subject much of the material taught in elementary school: division, fractions, and their connection to real-life situations. Ratio problems are also useful since reality often obeys rules of constant ratios between quantities called “direct proportion” or “proportionality.” The main tool used to handle ratio problems is “rate per one unit,” that is, how many units of type A there are per one unit of type B. For instance, how many kilometers a car travels in one hour. 181 Proportionality There are days when the greens Are seven times greener And the blue above is seventy times bluer. Rachel, Kineret Ratio Problems What is “maturity”? I wish I knew . . . “Mathematical maturity,” however, is easier to define. It means internalization of the mathematical way of thinking. Someone who has just acquired a concept will grope for its correct usage. One who internalized it will assuredly know to which situations it applies. Mathematical maturity means confidence in handling mathematical arguments. The final subject in elementary school arithmetic is ratios. Ratio here is related to proportionality; it is a relationship — how many times one quantity is greater than the other, rather than how large each quantity, which might be changeable, is in itself. No other subject is more appropriate as the grand finale to elementary school, making it a real test of mathematical maturity. It requires internalization of the concepts of division and ratio, and understanding of the connection between arithmetical operations and real life, that same connection we called “meaning.” Direct Proportion Tree A is twice as tall as tree B. How many times is the shadow of A longer than the shadow of B? Of course, the shadow is twice as long, too.

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  Visualizing Elementary and Middle School Mathematics Methods

  • Joan Cohen Jones(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  • Proportions are used in many parts of everyday life, for example, finding distances on maps, as shown here. THE PLANNER 1 • In mathematics, there are famous ratios such as the golden ratio and pi, the ratio of the circumference to the diameter of a circle. Learning About Proportion 328 • Students should learn to solve proportions using informal techniques before using cross multiplication. Teach students to use equivalent ratios, ratio tables, and other informal strategies. • When teaching cross multiplication, require that students draw pictures so they know where to place the numerical amounts in the proportion. • Proportions are used in many other areas of mathematics, such as slope, similarity, and linear functions, as illustrated in this image. Learning About Percents 333 • Percent means “out of one hundred.” If students understand fractions, then they should understand percents. Expose students to many different percent models, as illustrated. 3 4 • Proportional reasoning is formally taught in the middle grades. Students learn ratio in sixth grade and proportions in seventh grade. Students often have great difficulty learning to reason proportionally and progress through four stages in their development of proportional reasoning. Some students and adults never learn it. Learning About Ratio 324 • There are three types of ratios, but the part-whole definition is the most common, as illustrated in this photo. Students are often familiar with many real-world uses of ratio. It is helpful to teach ratio concepts with a ratio table. This helps organize information and makes it easy to identify patterns. 2 Figure 13.1 Figure 13.3 Figure 13.7 Activity 13.11 • When introducing percents, give students practice with percents in the real world, and teach them to use mental computation and estimation to solve percent problems. • Encourage students to use informal techniques to solve percent problems. Ask them to draw pictures to explain their solutions.

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  Introductory 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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  Basic College Mathematics

  • 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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