A ratio is a comparison between a pair of numbers, and while you can usually obtain it by direct measurement, you might have to do some calculations to make it useful. These calculations are called scaling, and they can be important when you're doing something like adapting a recipe for different numbers of people. When comparing numbers in a ratio, it's important to know what they represent. The numbers may represent two parts of a whole, or one of the numbers may represent a part of a whole while the other number represents the whole itself.

## Expressing a Ratio

Mathematicians and scientists use one of three conventions to express a ratio. Suppose you have two numbers A and B. You can express the ratio between them as:

- A:B
- A to B
- A/B

When reading the ratio aloud, you always say "A to B." The term for A is the antecedent, and the term for B is the consequent.

As an example, consider a grade school class which has 32 students, 17 of whom are girls and 15 of which are boys. The ratio of girls to boys can be written as 17:15, 17 to 15 or 17/15, while the ratio of boys to girls is 15:17, 15 to 17 or 15/17. The classroom has 32 students, so the ratio of girls to the total number of students is 17:32, and the ratio of boys to the total number of students is 15:32.

When comparing part of a whole to the whole, you can convert the ratio to a percentage by expressing it in fractional form, dividing the antecedent by the consequent and multiplying by 100. In our example, we find that the class is 17/32 x 100 = 53% female and 15/32 x 100 = 47% male. In terms of percentages, the ratio of girls to boys is 53:47, and the ratio of boys to girls is 47:53.

## Scaling a Ratio

You scale a ratio by multiplying both the antecedent and consequent by the same number. In the above example, we scaled the ratio by multiplying by 100 to give us percentages, which are often more useful than raw numbers. Cooks often need to scale ratios to adapt recipes for different numbers of people.

For example, a recipe intended to feed 4 people calls for 2 cups of soup mix to be added to 6 cups of water. The ratio of soup mix to water is therefore 2:6. If a cook wants to make this soup for 12 people, he or she needs to multiply each term by 3, because 12 divided by 4 = 3. The ratio then becomes 6:18. The cook needs to add 6 cups of soup mix to 12 cups of water.

## Simplifying a Ratio

When a ratio compares two large numbers, it's often useful to simplify it by dividing the antecedent and consequent by a common factor. For example, you can simplify the ratio 128:512 by dividing each term by 128. This produces the more convenient ratio 1:4.

To illustrate, consider a referendum on a proposition to ban assault weapons. Ten thousand people voted at a certain polling station, and when the results were tallied, it turned out that 4,800 people voted for the proposition, 3,200 voted against it and 2,000 were undecided. The ratio of those for the proposition to those against it was 4,800:3,200. Simplify this by dividing each term by 1,600 to find that the ratio of those for the proposition to those against it was 3:2. On the other hand, the ratio of those who had an opinion on the proposition to those who didn't was 8,000:2,000. or 4:1 after dividing each term by 2,000.

When reporting voting results, news media often convert the ratios to percentages. In this case, the percentage of those for the proposition was 4,800/10,000 = 48/100 =0.48 x 100 = 48%. The percentage of voters against the proposition was 3,200/10,000 = 32/100 = 0.32 x 100 = 32%, and the percentage of voters who were undecided was 2,000/10,000 = 20/100 = 0.2 x 100 = 20%.