In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7). The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive. A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient a/b. Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a proportion, for example, 6:8 is proportional to 3:4. Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. A more specific definition adopted in physical sciences (especially in metrology) for ratio is the dimensionless quotient between two physical quantities measured with the same unit. A quotient of two quantities that are measured with different units may be called a rate.
Notation and terminology The ratio of numbers A and B can be expressed as:
the ratio of A to B A:B A is to B (when followed by "as C is to D"; see below) a fraction with A as numerator and B as denominator that represents the quotient (i.e., A divided by B, or A B {\displaystyle {\tfrac {A}{B}}} ). This can be expressed as a simple or a decimal fraction, or as a percentage, etc. When a ratio is written in the form A:B, the two-dot character is sometimes the colon punctuation mark. In Unicode, this is U+003A : COLON, although Unicode also provides a dedicated ratio character, U+2236 ∶ RATIO. The numbers A and B are sometimes called terms of the ratio, with A being the antecedent and B being the consequent. A statement expressing the equality of two ratios A:B and C:D is called a proportion, written as A:B = C:D or A:B∷C:D. This latter form, when spoken or written in the English language, is often expressed as
(A is to B) as (C is to D). A, B, C and D are called the terms of the proportion. A and D are called its extremes, and B and C are called its means. The equality of three or more ratios, like A:B = C:D = E:F, is called a continued proportion. Ratios are sometimes used with three or even more terms. E.g., the proportion for the edge lengths of a "two by four" that is ten inches long is:
thickness : width : length = 2 : 4 : 10 {\displaystyle {\text{thickness : width : length }}=2:4:10}
A good concrete mix (in volume units) is sometimes quoted as:
cement : sand : gravel = 1 : 2 : 4. {\displaystyle {\text{cement : sand : gravel }}=1:2:4.}
The meaning of such a proportion of ratios with more than two terms is that the ratio of any two terms on the left-hand side is equal to the ratio of the corresponding two terms on the right-hand side.
History and etymology 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 ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios. Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers. The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables. The existence of multiple theories seems unnecessarily complex since ratios are, to a large extent, identified with quotients and their prospective values. However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.
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