Ratios, Proportions and Rates

Understanding and Simplifying Ratios

A ratio is a mathematical comparison of two quantities by division. Ratios can be written in three equivalent forms: with the word 'to' (a to b), with a colon (a:b), or as a fraction (a/b). The order of quantities in a ratio matters and must match the order specified in the problem statement. For example, if a question asks for the ratio of red marbles to blue marbles, placing the blue count first produces an incorrect statement.

A ratio is in simplest form when its terms are whole numbers that share no common factors other than 1. To simplify a ratio, divide each term by their greatest common factor (GCF). When comparing quantities of the same measurement type, both numbers must be converted to identical units before simplifying. For instance, comparing a length given in inches to a length given in feet requires converting both values to inches or both values to feet before dividing out common factors.

Ratios typically appear in two varieties: part-to-part and part-to-whole. A part-to-part ratio compares one distinct subcategory to another distinct subcategory within a group. In contrast, a part-to-whole ratio compares a specific subcategory to the entire collection of items. If a and b represent all the groups in a whole, the ratio a:b has a + b total ratio parts. Under these conditions, the fraction of the whole represented by a is a/(a + b).

To divide a total quantity into a given ratio, divide the total quantity by the sum of the ratio parts to find the value of one part. Multiplying the value of a single part by each individual term in a ratio gives the actual quantity for each group. An extended ratio compares three or more quantities, such as a:b:c for three quantities. Just like two-part ratios, the sum of the parts in an extended ratio a:b:c equals a + b + c, which you can use to determine the size of any individual share.