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In mathematics, a ratio is a relationship between two numbers of the same kind[1] (i.e., objects, persons, students, spoonfuls, units of whatever identical dimension), usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two,[2] which explicitly indicates how many times the first number contains the second.[3]


Notation and terminology

The ratio of numbers A and B can be expressed as:[4]

  • the ratio of A to B
  • A is to B
  • A:B

The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.

The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D. this latter form, when spoken or written in the English language, is often expressed as

Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion.[5]

History and etymology

It would be impossible to trace the origin of the concept of ratio since the ideas from which it developed would have been familiar to preliterate cultures. For example the idea of one village being twice as large as another or a distance being half that of another are so basic that they would have been understood in prehistoric society.[6] However, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio, meaning "reason" (as in "rational"). (A rational number may be expressed as the quotient of two integers.) A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.[7] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.[8]

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