proporii

Proporii, or proportions, are mathematical relationships that express the equality of two ratios. They show how quantities relate to one another when they change by the same factor, a principle central to measurements, comparisons, and modeling.

A proportion is an equation of the form a/b = c/d, where b and d are nonzero. When two ratios are equal, their cross products are equal: ad = bc. This property provides a practical method for solving missing terms in a proportion or for checking whether two fractions are equivalent.

Related ideas include ratios, unit rates, and direct variation. A ratio compares two quantities, while a unit rate expresses the amount per one unit. In direct variation, as one quantity increases, the other does so in a constant proportion, while inverse variation involves reciprocal relationships.

In geometry, proportions underpin similarity: similar figures have corresponding sides in proportional lengths and equal corresponding angles. The scale factor between figures multiplies lengths; areas change by the square of the scale factor.

In statistics and everyday use, proportions describe parts of a whole, such as a percentage or fraction of a population. Proportional reasoning is a common tool in problem solving, design, recipe adjustment, and data interpretation.