proportions

Proportions are equations that express the equality of two ratios. The basic form is a/b = c/d, where b and d are nonzero. This means that the two fractions represent the same value, and the quantities involved are in proportion to one another. Cross-multiplication provides a practical check: ad = bc. This relationship is commonly used to solve for an unknown term when three of the four quantities are known.

Proportions are closely related to direct proportionality: if a/b = k, then a = k·b, so increasing one quantity scales the other by the same factor. In geometry, proportions underpin the concept of similarity, where corresponding sides of similar figures are in proportion, allowing measurements to be inferred from a scale factor. Proportions also arise in scaling drawings, models, and maps.

It is important to distinguish a proportion from a ratio. A ratio compares two quantities, while a proportion asserts that two ratios are equal. In all cases, units should be consistent, and denominators must be nonzero.

In statistics, the term proportion refers to a part of a whole, expressed as p = x/n, where x is a count of successes and n is the total population or sample. Proportions can be conveyed as fractions, decimals, or percentages.

Common applications include recipe scaling, adapting quantities while preserving taste, and adjusting measurements in engineering or design to maintain the same ratios.