A proportional relationship exists between two variables if they have a constant ratio. In other words, if one variable changes, the other variable changes in proportion to the first. This type of relationship is also known as direct variation.

To better understand proportional relationships, consider the following example:

Suppose that you are driving at a constant speed of 50 miles per hour (mph). The distance you travel is proportional to the time you spend driving at that speed. For instance, if you drive for 2 hours, you will have traveled 100 miles. If you drive for 3 hours, you will have traveled 150 miles, and so on. In this case, the distance you travel is directly proportional to the time you spend driving.

Another way to look at proportional relationships is through their graphs. In a graph of a proportional relationship, the points will fall on a straight line that passes through the origin (0,0). The slope of this line represents the constant ratio between the two variables.

Proportional relationships can be expressed using an equation in the form y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of proportionality. In the example above, the equation that represents the relationship between the distance traveled (y) and the time spent driving (x) is y = 50x.

Proportional relationships are commonly used in mathematics and science, as they can be used to model many real-world phenomena, such as distance-time and speed-time relationships, and relationships between other physical quantities like mass and weight. Understanding proportional relationships can also be useful in everyday life, such as when making conversions between different units of measurement.

In summary, a proportional relationship is one in which two variables have a constant ratio, meaning that a change in one variable results in a corresponding change in the other variable. These relationships can be expressed using equations, graphs, and can be applied to various fields, such as science, mathematics, and even daily life.