Understanding Proportional Relationships in Mathematics

Study Notes

Proportional Relationships

What is a Proportion?

A proportion is a relationship between two ratios where the values of each ratio are equal. In mathematics, proportions are represented using mathematical symbols such as an equals sign, > or < to denote the comparison between the ratios. If the value of the denominator of one ratio is equal to the numerator of another ratio, we say they have a proportional relationship.

For example, if two people walk a certain distance in different amounts of time, and those times are proportional to their strides, it means they are walking with proportional relationships. This is also known as the rate. Another example could be when the temperature in three cities changes by the same percentage, indicating a proportional relationship.

Understanding Proportional Relationships

Calculating Proportions

To calculate proportions, divide both numbers by the common factor. This is sometimes called cross-multiplying. For instance, if a rectangle has a length of 3 and a width of 2, the ratio between the length and width is 3:2, or 3 to 2. If the width is multiplied by 5, the length will be multiplied by 5 as well.

Differences Between Ratios and Proportions

Ratios are used to compare two values, while proportions are used to determine the relative sizes of two values. For instance, the ratio of 2 to 3 can be written as 2/3, which is a proportion. However, 2/3 is not a proportion of 3/2, and 3/2 is not a proportion of 2/3.

Proportions and Equations

Proportions can be represented by equations, which often involve variables. For instance, the equation y = kx represents a proportional relationship between x and y, where k is a constant of proportionality. This means that if x increases by a certain factor, y will increase the same amount as well.

Verifying Proportions

To verify whether two ratios have a proportionally equal identity, find the first common multiple of the two denominators. If the product of these two numbers is equivalent to the product of any other common multiple of the two denominators, then the two ratios are proportional.

Proportions and Functions

Proportions can also be applied to functions, which map inputs to outputs according to specific rules. For example, the function f(x) = x^2 maps every real number to its square, so the function f(a + b) = f(a) + f(b). Therefore, this relationship shows that the area of squares increases in proportion to the side lengths involved.

Description

Learn about proportions in mathematics and how they are used to compare ratios. Explore the calculation of proportions, differences between ratios and proportions, representing proportions with equations, verifying proportions, and applying proportions to functions.