Ratios

A ratio compares two quantities by division. Ratios describe how much of one quantity there is compared with another quantity. For example, if a class has 12 students wearing sneakers and 8 students wearing sandals, the ratio of students wearing sneakers to students wearing sandals is 12 to 8.nnA ratio can be written in three common forms: 12 to 8, 12:8, or 12/8. Each form represents the same comparison. The order of the quantities matters. The ratio of sneakers to sandals is 12:8, while the ratio of sandals to sneakers is 8:12.nnRatios may compare part to part or part to whole. In the example with 12 students wearing sneakers and 8 wearing sandals, 20 students are represented altogether. The ratio of sneakers to sandals is a part-to-part ratio of 12:8. The ratio of sneakers to all students is a part-to-whole ratio of 12:20.nnEquivalent ratios describe the same relationship. To generate an equivalent ratio, multiply or divide both quantities by the same nonzero number. For example, 3:5, 6:10, and 9:15 are equivalent because both quantities in 3:5 were multiplied by the same factor.nnRatios can also be represented using tables, diagrams, double number lines, and graphs. A ratio table organizes pairs of equivalent quantities. If 2 notebooks cost $6, equivalent pairs include 4 notebooks for $12 and 6 notebooks for $18.nnRatios can compare quantities with the same units or different units. A ratio comparing 3 red markers with 5 blue markers uses the same type of unit. A comparison such as 120 miles in 2 hours uses different units and is called a rate. Rates and unit rates are examined in the next lesson.nnTo interpret a ratio correctly, identify the two quantities, preserve their order, and include the meaning of each number. For example, the ratio 4:7 does not provide enough information by itself. A complete interpretation might state that there are 4 fiction books for every 7 nonfiction books.nnCommon misconceptions and corrections: Students may reverse the order of a ratio, add the same number to both terms instead of multiplying, or confuse a part-to-part comparison with a part-to-whole comparison. Always match the order of the numbers to the order of the words and generate equivalent ratios by multiplying or dividing both quantities by the same nonzero factor.