Final answer:
Two ratios are equivalent if they form a proportion when set equal to each other. Graphically, if you plot ratios as points and they lie on the same line passing through the origin, they are equivalent. Alternatively, comparing the ratios to form unit rates can also determine equivalency.
Step-by-step explanation:
To show that two ratios are equivalent, one can use either an equation or a graph. Starting with an equation, you would set two ratios equal to each other to form a proportion. For example, the proportion 1/20 = 1/5.5 indicates that these two ratios are equivalent when they have been simplified or when both sides are multiplied by the same number to achieve equality. To check if the ratios are equivalent, you can cross-multiply and see if the products are equal.
Graphically, if you plot each ratio as a point on a coordinate plane, with the first number of the ratio on the x-axis and the second number on the y-axis, equivalent ratios will lie on the same straight line that passes through the origin (0,0). This line represents a constant rate of change or slope, which is characteristic of equivalent ratios.
For example, if we take two length ratios and two width ratios, say 4 inches to 2 inches and 6 feet to 3 feet respectively, and we set each pair equal to form two proportions (e.g. 4 inches/2 inches = 6 feet/3 feet), we can also graph these ratios. If the points (4,2) and (6,3) lie on the same line that passes through the origin, then the ratios are equivalent.
In addition to forming proportions, ratios can be converted into unit rates or unit scales to establish equivalence. A unit rate is when one of the measures in the ratio is 1. If you have a unit rate and you want to find out whether another ratio is equivalent to it, you can create a ratio that compares the other measure to the unit rate and see if they form a proportion.