Direct and Inverse proportions
Direct proportion refers to a relationship where two quantities change in the same direction, meaning as one increases, the other also increases. Inverse proportion, on the other hand, describes a relationship where one quantity increases as the other decreases, and vice versa. These concepts are fundamental in understanding the relationships between variables in mathematical equations and real-world scenarios.
3 Key excerpts on "Direct and Inverse proportions"
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Proportional relationships involve some of the simplest forms of covariation. That is, two quantities are linked to each other in such a way that when one changes, the other one also changes in a precise way with the first quantity, and there exists a third quantity that remains invariant (i.e., it doesn’t change). The quantity that doesn’t change is called the constant of proportionality. In a direct proportion, the direction of change in the related quantities is the same; both increase or both decrease. As you analyze a situation, you might write this as ↑↑ or ↓↓. But a critical aspect of direct proportion is that both quantities increase by the same factor. That is, if one doubles, the other doubles. If one becomes five times as great, the other becomes five times as great. Then we say that “y is directly proportional to x” or that “y varies as x.” Just because two related quantities both increase or both decrease does not mean that they are directly proportional. For example, as a person’s age increases, his height increases ↑↑, but age and height are not directly proportional because they do not increase by the same factor. Here is a problem involving quantities that are directly proportional: • If a box of detergent contains 80 cups of powder and your washing machine recommends 1 1 4 cups per load, how many loads can you do with one box? Think: 1 1 4 cups for 1 load. The more loads I do, the more cups of detergent I need: ⇈ for 4 loads I will need 5 cups for 32 loads I will need 40 cups for 64 loads I will need 80 cups We can double both quantities (cups and loads), or quadruple both quantities, or take 8 times both quantities, but the two quantities always maintain the same relationship to each other. Notice that the number of cups (c) is always 1 1 4 times the number of loads (d). Symbolically, c = 1 1 4 d. 5 is 1 1 4 times 4; 40 is 1 1 4 times 30; 80 is 1 1 4 times 64...
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)
...This equation is also commonly linked to both ratio tables and the coordinate plane. The equation for a proportional relationship is y = kx. One can describe this in a number of ways. Two examples are listed here. y is proportional to x, or y varies proportionally with x. To determine the value of y, one multiplies x by k. Note: The CCSSM uses the equation y = cx. One can use k or c or other letters to indicate a constant of proportionality. Perhaps the terms of this equation— y, k and x —are more clearly understood through examples. Figure 2.14 provides some examples of where these terms are found in a ratio table and in a coordinate plane. Figure 2.14 y, k and x in a ratio table and a coordinate plane Each of these terms of the equation for a proportion y, k and x is referred to in different ways, resulting in a variety of different names that are impacted by the representation or the specific context. Table 2.4 provides a few examples of the different ways that each of these terms are named, represented or referred to. This is certainly not an exhaustive list, but it provides a picture of the number of different ways these terms are interpreted. Table 2.4 Some ways y, k and x can be represented in a coordinate plane and a ratio table Common Names for k and v Terms Names/Ways It Is Referred To/Represented y •Quantity associated with the vertical axis on a line graph •Quantity in the rightmost column in a ratio table •Quantity in the bottom row of a horizontal table •Dependent variable •Output k • Unit rate • Slope of the line on a line graph • Constant of proportionality • Constant rate • Rate of change x • Quantity associated with the horizontal axis on a line graph • Quantity in the leftmost column in a ratio table • Quantity in the top row of a horizontal table • Independent variable • Input From this table, we begin to see the complexity of these terms, as each is interpreted differently depending on the context...
- 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...
