Two quantities are proportional when their ratio stays the same. For example, if y equals 2x, then y is proportional to x with a constant ratio of 2. This creates a straight line passing through the origin on a graph. Each point shows the constant relationship between x and y values.
The first method to check if quantities are proportional is to examine their ratio. Calculate y divided by x for each pair of values. If this ratio is always the same constant number, then the quantities are proportional. In our example with distance and time at constant speed, the ratio is always 60, confirming they are proportional.
The second method is to check the graph. Plot the relationship between your two quantities. If they are proportional, you will get a straight line that passes through the origin at point zero, zero. The red line shows a proportional relationship, while the blue line shows a non-proportional relationship because it doesn't pass through the origin.
The third method is to check the equation form. A proportional relationship must be written as y equals k times x, where k is a constant. The equation y equals 3x is proportional. However, y equals 3x plus 2 is not proportional because of the added constant. Similarly, y equals x squared is not proportional because x is raised to a power other than one.
To summarize, there are three reliable methods to determine if quantities are proportional. First, check if the ratio y over x remains constant for all data pairs. Second, plot the relationship on a graph and verify it forms a straight line passing through the origin. Third, express the relationship as an equation in the form y equals k times x, with no additional constants. Using any of these methods will help you identify proportional relationships in mathematics and real-world situations.