ratios

A ratio is a relationship between two quantities that expresses how many times the second quantity fits into the first, or how the two compare in size. It can be written as a:b or as a fraction a/b. When the quantities share the same units, the ratio is often considered dimensionless. Ratios are typically reduced to simplest terms by dividing both parts by their greatest common divisor.

Two related concepts accompany ratios: proportion and rate. A proportion is an equality between two ratios, such as a:b = c:d, which implies ad = bc by cross-multiplication. A rate is a specific kind of ratio that compares a quantity to a unit of another quantity, for example speed (miles per hour) or density (people per square kilometer).

Common types include part-to-part ratios, which compare two parts of a whole (for example, 3:5), and part-to-whole ratios, which compare a part to the entire quantity (such as 2:10 representing a portion of the whole). Ratios can be scaled by multiplying numerator and denominator by the same positive number, yielding an equivalent ratio. A ratio with zero in the denominator is undefined.

Applications span everyday and scientific contexts. In cooking, ratios help balance ingredients; in maps and models, ratios express scale. In finance, ratios summarize relationships between metrics (for example, debt-to-equity, price-earnings). In statistics, ratios and related measures compare probabilities or frequencies, aiding interpretation and decision making.