Final answer:
A ratio compares two quantities and can be expressed in various forms, such as 3 to 5, 3:5, or 3/5 for the relationship between loads of dirt and pear trees. They are used to understand relationships and to create proportions when ratios are equivalent. Unit rates and unit scales are practical applications of ratios.
Step-by-step explanation:
A ratio is a way to compare two quantities. It can be written in different forms such as fractions, with a colon, or with the word "to." For example, if Mr. Patterson ordered 3 loads of dirt for every 5 pear trees, the ratio can be expressed as 3 to 5, 3:5, or as a fraction 3/5.
A ratio like this tells us the relative size or quantity of one thing in comparison to another. This is particularly useful in understanding scale distances, dimensions, and proportional relationships such as unit rates and unit scales.
When two ratios are equivalent, we call them a proportion. For instance, 1/2 is equivalent to 3/6, and that forms a proportion. A unit rate compares two measurements in which one of them is 1, like 55 miles per hour or (55/1) miles/hour.
Similarly, a unit scale compares the actual dimensions to the dimensions of a model or drawing, like a map where 1 inch represents 100 feet. In such cases, the ratio would be 1 inch/100 ft. Ratios and proportions are foundational concepts in Mathematics and are used in many different contexts.