A ratio is a comparison of two quantities expressed as a quotient, showing how many times one contains the other.
Ratios can be written as a:b, a/b, or "a to b." Equivalent ratios are found by multiplying or dividing both terms. Ratios are used in scaling, maps (scale ratios), recipes, mixtures, speed, and financial ratios. Proportions state that two ratios are equal (a/b = c/d).
Ratios were used by ancient Egyptians and Babylonians for construction and trade. The Pythagoreans (500 BCE) studied ratios extensively, discovering the mathematical basis of musical harmony through whole-number ratios.