Ratios, Proportions, Rates, and Scale

Understanding Ratios and Ratio Relationships

A ratio is a mathematical comparison of two numbers or quantities by division. When you compare values, you can write the comparison in three common ways: using the word 'to' (such as 3 to 4), using a colon (3:4), or writing the relationship as a fraction (3/4). Always write the terms of a ratio in the exact order requested by the question. Reversing the order of the numbers reverses the terms and changes the entire comparison.

Ratios generally describe one of two relationships: part-to-part or 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 one distinct subcategory to the total combined quantity of all categories. If you add the terms of a part-to-part ratio together, you find the total number of ratio parts. To find the whole quantity from a part-to-part ratio, you must know the value of one ratio part and whether the categories in the ratio include the entire group; if they do not, you need information about the remaining categories.

Simplifying a ratio means dividing both terms by their greatest common factor until they share no common factors other than 1. Equivalent ratios express the exact same relationship, and you can generate them by multiplying or dividing both terms by the same nonzero number. When both quantities in a ratio use the same unit of measurement, the units cancel out completely. If the quantities are given in different units of the same type, convert them to the same unit first so the units can cancel.

You can use the on-screen TI-30XS calculator to save time during calculations. Entering a ratio using the n/d fraction key and pressing enter automatically reduces it to lowest terms.