Fractional Ratio
A fractional ratio is a comparison of two quantities expressed as a fraction. It represents the relationship between a part and the whole, where the numerator represents the part and the denominator represents the whole. Fractional ratios are commonly used in mathematics and are essential for understanding proportions and solving problems involving relative quantities.
8 Key excerpts on "Fractional Ratio"
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Understanding a fraction as a number entails realizing, for example, that 1 4 refers to the same relative amount in each of the following pictures. There is but one rational number underlying all of these relative amounts. Whether we call it 1 4, 4 16, 3 8, or 2 8 is not as important as the fact that a single relationship is conveyed. When we consider fractions as single numbers, rather than focusing on the two parts used to write the fraction symbol, the focus is on the relative amount conveyed by those symbols. Regardless of the size of the pieces, their color, their shape, their arrangement, or any other physical characteristic, the same relative amount and the same rational number is indicated in each picture. (Psychologically, for the purpose of instruction, when fractions are connected with pictorial representations, which fraction name you connect with which picture is an important issue. For example, you would not call the first picture 2 8 or 4 16.) Fractions, Ratios, and Rates A ratio is an ordered pair that conveys the relative sizes of two quantities. Under this definition, a part–whole fraction is a ratio; however, every ratio is not a part–whole comparison. A ratio may compare measures of two parts of the same set (a part–part comparison) or the measures of any two different quantities (e.g. cups of juice concentrate to cups of water). Often, ratios are expressed as “per” quantities. For example, miles per hour and candies per bag are both ratios or comparisons of unlike quantities. The ratio–rate distinction is a bit more complex. When we are discussing a very particular situation, we use a ratio...
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)
...In this example, there is a fractional number of miles. The big idea here is that 3 4 and 2 5 are numbers that tell us the amount of miles Judi ran, whereas we saw above that each number in a ratio requires its own label. A fraction, on the other hand, describes one quantity. Examples include: 2 5 miles, 7 8 pizza, 4 5 hour and, 1 2 of the class. Unlike with ratios, switching the position of the numerator and denominator in a fraction results in a different number. For example, 3 4 m i l e ≠ 4 3 m i l e. Sometimes Ratios and Fractions Act in Similar Ways Despite these general differences between fractions and ratios, there are times when these two interpretations of a b act in similar ways and times when they act quite differently. This may be another reason students struggle with the differences between ratios and fractions. In some situations, for example, a part-to-whole ratio can meaningfully be interpreted as a fraction. One can restate the part-to-whole ratio example 4 r e d c h a i r s 19 t o t a l c h a r s, as a fraction. That is, “4 19 of the chairs in the classroom are red.” In addition, we can use the same procedure to identify equivalent fractions and equivalent ratios. Figure 1.11 provides an example. Figure 1.11 Equivalent ratios and equivalent fractions Although we use the same procedure, multiplying or dividing each quantity in a ratio or fraction by the same number, we interpret the meaning of these two situations differently. In the case of the ratio, 12 students 2 coaches is equivalent to 24 students 4 coaches because the multiplicative relationship between the quantities is the same in each ratio. That is, in each situation, there are 6 students for each coach. In the fraction example, 3 4 = 6 8 means that the two fractions represent the same number. They share the same point on the number line...
eBook - ePubTeaching Mathematics in Primary Schools
Principles for effective practice
- Robyn Jorgensen(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Ratio and proportion are two concepts that present a challenge to both teachers and students. Proportional reasoning is fundamental to understanding many topics in the middle-school curriculum, such as the geometry of plane shapes, trigonometry and, particularly, percentages, as well as rate, ratio and proportion applications. As Lesh and colleagues (1988) state, 'Proportional reasoning is the capstone of children's elementary school arithmetic and the cornerstone of all that is to follow' (pp. 93-4). Traditionally, ratio and proportion were considered topics for upper-primary and secondary grades. However, teachers at all year levels need to provide students with rich experiences and appropriate activities in order for them to develop understanding of these topics, so that students will later be able to apply this knowledge in a meaningful way. Teachers of early primary classes must ensure that the seeds of the concepts are sown in their students' minds in the early years, and as students move through the grades, all teachers must build on and extend this knowledge. Ratio and fractions Ratios are commonly described as fractions. In the true mathematical sense, fractions are ratios in that they are rational numbers that can be expressed in the form a/b (provided that b does not equal zero). However, the general meaning of fraction, of being part of a whole (as described in the previous section), interferes with the conceptual meaning of ratio. Continuing with the previous example, it was stated that the ratio of boys to girls in the class is two to three—that is, for every two boys there are three girls. When this is considered in the context of the Teaching Idea Ratio language through pattern For every 2 black there is 1 yellow For every 3 circles there is 1 square whole class, there is a base unit of five children, of whom two are boys and three are girls. The boys and girls situation describes the two parts of the whole unit of children...
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...6 RATIOS AND PROPORTIONS WHAT YOU WILL LEARN • The concept of a ratio and how to use ratio language to describe a ratio relationship between two quantities • How to use ratio and rate reasoning to solve real-world and mathematical problems • How to recognize and represent proportional relationships between quantities • How to compute unit rates • How to use proportional relationships to solve multi-step ratio and percent problems SECTIONS IN THIS CHAPTER • What Is a Ratio? • What Is a Rate? • Comparing Unit Rates • What Is a Proportion? • Solving Proportions • Similar Figures • Scale Drawings • Word Problems DEFINITIONS Extremes of a proportion The two outermost terms in the ratio of a proportion. Map scale A key that provides equivalence between a distance on a map and the associated real-world distance. Means of a proportion The two middle terms in the ratios of a proportion. Proportion An equation that states that two ratios are equivalent. Rate A ratio that compares quantities of different. units. Ratio A comparison of two numbers or two like quantities by division. Scale (1) The ratio of the size of an object in a representation (drawing) of the object to the actual size of the object; the ratio of the distance on a map to the actual distance (e.g., the scale on a map is 1 inch:10 miles); (2) an instrument used to measure an object’s mass. Scale drawing A proportionally correct drawing (enlargement or reduction) of an object or area. Similar triangles Triangles that have the same shape but not necessarily the same size; corresponding sides are in proportion and corresponding angles are congruent. EXAMPLE: Unit price The price of one item or one unit (e.g., $0.15 per pound). 6.1 What Is a Ratio? A ratio is a comparison of two numbers or two like quantities by division. Ratios can be written in a variety of ways. The main idea is comparison. Think back to a time when your parents were trying to win an argument with you...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 2 Fractions, decimals and percentages Why it is important to understand: Fractions, decimals and percentages Engineers use fractions all the time, examples including stress to strain ratios in mechanical engineering, chemical concentration ratios and reaction rates, and ratios in electrical equations to solve for current and voltage. Fractions are also used everywhere in science, from radioactive decay rates to statistical analysis. Calculators are able to handle calculations with fractions. However, there will be times when a quick calculation involving addition, subtraction, multiplication and division of fractions is needed. Real life applications of ratio and proportion are numerous. When you prepare recipes, paint your house, or repair gears in a large machine or in a car transmission, you use ratios and proportions. Trusses must have the correct ratio of pitch to support weight of roof and snow, cement must be the correct mixture to be sturdy, and doctors are always calculating ratios as they determine medications. Almost every job uses ratios one way or another: ratios are used in building & construction, model making, art & crafts, land surveying, die and tool making, food and cooking, chemical mixing, in automobile manufacturing and in airplane and parts making. Engineers use ratios to test structural and mechanical systems for capacity and safety issues. Millwrights use ratio to solve pulley rotation and gear problems. Operating engineers apply ratios to ensure the correct equipment is used to safely move heavy materials such as steel on worksites. It is therefore important that we have some working understanding of ratio and proportion. Engineers and scientists also use decimal numbers all the time in calculations...
- Derek Haylock, Paul Warburton(Authors)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...UNDERSTANDING FRACTIONS AND RATIOS 11 OBJECTIVES In everyday and healthcare contexts the practitioner should be able to: interpret a fraction as a proportion of a whole unit or set recognize what must be added to a fraction to make 1 interpret a fraction as the division of one number by another interpret a fraction as a ratio recognize equivalent fractions or equivalent ratios use cancelling to simplify a fraction or a ratio express a ratio as ‘one to something’ or ‘something to one’ use mixed numbers state common equivalences between fractions and decimal numbers convert a fraction into a decimal number, or vice versa calculate a simple fraction of a number or quantity using informal mental, written or calculator methods In this chapter our focus is on explaining the different concepts and principles involved in handling quantities expressed in fraction notation. To do this we use a mixture of everyday examples and some simple but occasionally rather artificial healthcare illustrations. In some cases we use results that turn up in the middle of the kinds of drug and infusion calculations that are explained in full in Chapter 12. This is a long chapter, which for some readers may require persistence and concentration. But if you work hard at the content here you should emerge with a deeper understanding of some key concepts that will enable you to engage with the mathematical demands of healthcare practice with greater confidence. SPOT THE ERRORS Identify any obvious errors in the use of fraction notation in the following 15 statements. 1 If a bar of chocolate is divided into 4 equal pieces and 3 of them are eaten then the amount eaten is ¾ of the bar. 2 If 3 bars of chocolate are shared equally between 4 people then each gets ¾ of a bar. 3 A patient’s dosage of Drug A is reduced from 4 mg to 3 mg each day...
eBook - ePubRtI in Math
Evidence-Based Interventions
- Linda Forbringer, Wendy Weber(Authors)
- 2021(Publication Date)
- Routledge(Publisher)
...While concrete and pictorial representation is used in Tier 1, core materials often move too quickly to abstract representation before learners who struggle with mathematics are able to fully grasp the concepts (Gersten et al., 2009 ; van Garderen, Scheuermann, Poch, & Murray, 2018). In this chapter, we will discuss ways to incorporate these intensive intervention strategies when introducing rational numbers. Fractions Developing Fraction Concepts Fractions present one of the greatest challenges students encounter. National and international test results reveal that American students have consistently struggled with basic fraction concepts (NMAP, 2008, 2019; Siegler, 2017). Understanding fraction concepts is necessary to perform meaningful computations with fractions, and fractions are a pre-requisite for decimals, percent, ratio and proportion, and algebra. Knowledge of fractions in fifth grade predicts student's math achievement in high school, even after controlling for the student's IQ, knowledge of whole numbers, and family education level or income (Siegler, 2017). Even students who have not experienced previous mathematical difficulty can be challenged by fractions. For students with a history of mathematical difficulty, the problem is magnified. To understand fractions, students must master a few big ideas. First, fractional parts are formed when a whole or unit is divided into equal parts. In other words, to understand a fraction, students first need to identify the unit and then make sure it is divided into equal parts. Students who struggle with fractions sometimes miss the importance of having equal parts. The concept of unit can also confuse students, because the word has several different mathematical applications. The smallest piece in base-ten blocks is sometimes called a unit block. For fractions, the unit is the whole object, set, or length that is divided into equal parts...
- Sandra Rush(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The total purchase of yards rounds up to 11 yards. d. $3.00. The difference between 11 yards and yards is yard. At $36 per yard, this cost would be. Another way to calculate this is to find the cost of 11 yards, and subtract the cost of yards from part (b): $396 – $393 = $3. Ratios and Proportions A ratio is simply a way of comparing two numbers or expressing the relation between two quantities. A ratio can be expressed as a fraction or a decimal, and often is written with a colon between the numbers. For example, a small company employs 3 men and 7 women. Then we can say the ratio of men to women in that company is 3:7, or, both read as “3 to 7” because we are stating a ratio. We can even say there is.4286 of a man for each 1 woman, by dividing 7 into 3, although this is a little awkward to visualize. As another example, if the results of an election are a landslide in which one candidate received 5 times the number of votes of the other candidate, the ratio of votes is 1:5. However, the fractions of the votes received by the candidates are and (not for the losing candidate, as you might think). The ratio is 1:5, so the whole must be the sum, or 1 + 5 = 6. A proportion is the way to express that two ratios are equal. Again, the most common ways to write a proportion are with colons (3:7 = 6:14) or fractions, which is said as “3 is to 7 as 6 is to 14.” We will see proportions again in Chapter 5, where we discuss solutions to algebra problems such as “If the ratio of men to women in the workplace is 3 to 7, how many men are in an office with 70 women?” Maybe you figured that out already. The ratio is, which is the same as, so for 70 women, there are 30 men. If you didn’t see this right away, it’s okay—this solution actually involves algebra. Example 3.8. Two-thirds of a Spanish class have never studied Spanish before. In a class of 30 students, what is the ratio of those who have studied Spanish to those who haven’t? Answer 3.8. 2:1...
