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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a ratio () shows how many times one
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
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 In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
Equal quotients correspond to equal ratios. A statement expressing the equality of two ratios is called a ''proportion''. Consequently, a ratio may be considered as an ordered pair of numbers, a
fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
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 number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, are
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s, and may sometimes be natural numbers. A more specific definition adopted in
physical sciences Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together is called the "physical sciences". Definition ...
(especially in
metrology Metrology is the scientific study of measurement. It establishes a common understanding of Unit of measurement, units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to stan ...
) for ''ratio'' is the
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
quotient between two
physical quantities A physical quantity (or simply quantity) is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a '' numerical value'' and a '' ...
measured with the same unit. A quotient of two quantities that are measured with 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 A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
with ''A'' as numerator and ''B'' as denominator that represents the quotient (i.e., ''A'' divided by ''B, or'' \tfrac). 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 Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...
, this is , although Unicode also provides a dedicated ratio character, . 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 therefore :\text = 2:4:10 :(unplaned measurements; the first two numbers are reduced slightly when the wood is planed smooth); a good concrete mix (in volume units) is sometimes quoted as :\text = 1:2:4. For a (rather dry) mixture of 4/1 parts in volume of cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement. 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 Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...
(''
logos ''Logos'' (, ; ) is a term used in Western philosophy, psychology and rhetoric, as well as religion (notably Logos (Christianity), Christianity); among its connotations is that of a rationality, rational form of discourse that relies on inducti ...
''). Early translators rendered this into
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
as ' ("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 ' ("proportion") to indicate ratio and ' ("proportionality") for the equality of ratios. Euclid collected the results appearing in the Elements from earlier sources. The
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...
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 In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to
Eudoxus of Cnidus Eudoxus of Cnidus (; , ''Eúdoxos ho Knídios''; ) was an Ancient Greece, ancient Greek Ancient Greek astronomy, astronomer, Greek mathematics, mathematician, doctor, and lawmaker. He was a student of Archytas and Plato. All of his original work ...
. 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.


Euclid's definitions

Book V of
Euclid's Elements The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the w ...
has 18 definitions, all of which relate to ratios. In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a ''part'' of a quantity is another quantity that "measures" it and conversely, a ''multiple'' of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning aliquot part) is a part that, when multiplied by an integer greater than one, gives the quantity. Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity ''measures'' the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII. Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself. Euclid defines a ratio as between two quantities ''of the same type'', so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities ''p'' and ''q'', if there exist integers ''m'' and ''n'' such that ''mp''>''q'' and ''nq''>''p''. This condition is known as the Archimedes property. Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's definition of equality is that given quantities ''p'', ''q'', ''r'' and ''s'', ''p'':''q''∷''r'':''s'' if and only if, for any positive integers ''m'' and ''n'', ''np'' < ''mq'', ''np'' = ''mq'', or ''np'' > ''mq'' according as ''nr'' < ''ms'', ''nr'' = ''ms'', or ''nr'' > ''ms'', respectively. This definition has affinities with Dedekind cuts as, with ''n'' and ''q'' both positive, ''np'' stands to ''mq'' as stands to the rational number (dividing both terms by ''nq''). Definition 6 says that quantities that have the same ratio are ''proportional'' or ''in proportion''. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog". Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities ''p'', ''q'', ''r'' and ''s'', ''p'':''q'' > ''r'':''s'' if there are positive integers ''m'' and ''n'' so that ''np'' > ''mq'' and ''nr'' â‰¤ ''ms''. As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms ''p'', ''q'' and ''r'' to be in proportion when ''p'':''q''∷''q'':''r''. This is extended to four terms ''p'', ''q'', ''r'' and ''s'' as ''p'':''q''∷''q'':''r''∷''r'':''s'', and so on. Sequences that have the property that the ratios of consecutive terms are equal are called
geometric progression A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the s ...
s. Definitions 9 and 10 apply this, saying that if ''p'', ''q'' and ''r'' are in proportion then ''p'':''r'' is the ''duplicate ratio'' of ''p'':''q'' and if ''p'', ''q'', ''r'' and ''s'' are in proportion then ''p'':''s'' is the ''triplicate ratio'' of ''p'':''q''.


Number of terms and use of fractions

In general, a comparison of the quantities of a two-entity ratio can be expressed as a
fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is \tfrac that of the second entity. If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while the amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason. Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities. A separate fraction can be used to compare the quantities of any two of the entities covered by the ratio: for example, from a ratio of 2:3:7 we can infer that the quantity of the second entity is \tfrac that of the third entity.


Proportions and percentage ratios

If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduce terms to the
lowest common denominator In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The l ...
, or to express them in parts per hundred (
percent In mathematics, a percentage () is a number or ratio expressed as a fraction of 100. It is often denoted using the ''percent sign'' (%), although the abbreviations ''pct.'', ''pct'', and sometimes ''pc'' are also used. A percentage is a dime ...
). If a mixture contains substances ''A'', ''B'', ''C'' and ''D'' in the ratio 5:9:4:2, then there are 5 parts of ''A'' for every ''9'' parts of ''B'', 4 parts of ''C'', and 2 parts of ''D''. As the total mixture contains 5/20 of ''A'' (5 parts out of 20), 9/20 of ''B'', 4/20 of ''C'', and 2/20 of ''D''. If we divide all numbers by the total and multiply by 100, we have converted to percentages: 25% ''A'', 45% ''B'', 20% ''C'', and 10% ''D'' (equivalent to writing the ratio as 25:45:20:10). If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, \tfrac, or 40% of the whole is apples and \tfrac, or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion. If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older
television Television (TV) is a telecommunication medium for transmitting moving images and sound. Additionally, the term can refer to a physical television set rather than the medium of transmission. Television is a mass medium for advertising, ...
s have a 4:3 ''
aspect ratio The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...
'', which means that the width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places). More recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. One of the popular widescreen movie formats is 2.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparison. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider image. Such a comparison works only when values being compared are consistent, like always expressing width in relation to height.


Reduction

Ratios can be reduced (as fractions are) by dividing each quantity by the common factors of all the quantities. As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers. Thus, the ratio 40:60 is equivalent in meaning to the ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3. The verbal equivalent is "40 is to 60 as 2 is to 3." A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in simplest form or lowest terms. Sometimes it is useful to write a ratio in the form 1:''x'' or ''x'':1, where ''x'' is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4) Alternatively, it can be written as 0.8:1 (dividing both sides by 5). Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the ratio symbol (:), though, mathematically, this makes it a factor or multiplier.


Irrational ratios

Ratios may also be established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to an
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
). The earliest discovered example, found by the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...
, is the ratio of the length of the diagonal to the length of a side of a
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, which is the
square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
, formally a:d = 1:\sqrt. Another example is the ratio of a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
's circumference to its diameter, which is called , and is not just an
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
, but a
transcendental number In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
. Also well known is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
of two (mostly) lengths and , which is defined by the proportion : a:b = (a+b):a \quad or, equivalently \quad a:b = (1+b/a):1. Taking the ratios as fractions and a:b as having the value , yields the equation :x=1+\tfrac 1x \quad or \quad x^2-x-1 = 0, which has the positive, irrational solution x=\tfrac=\tfrac. Thus at least one of ''a'' and ''b'' has to be irrational for them to be in the golden ratio. An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive
Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
s: even though all these ratios are ratios of two integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio. Similarly, the
silver ratio In mathematics, the silver ratio is a geometrical aspect ratio, proportion with exact value the positive polynomial root, solution of the equation The name ''silver ratio'' results from analogy with the golden ratio, the positive solution of ...
of and is defined by the proportion :a:b = (2a+b):a \quad (= (2+b/a):1), corresponding to x^2-2x-1 = 0. This equation has the positive, irrational solution x = \tfrac=1+\sqrt, so again at least one of the two quantities ''a'' and ''b'' in the silver ratio must be irrational.


Odds

''Odds'' (as in gambling) are expressed as a ratio. For example, odds of "7 to 3 against" (7:3) mean that there are seven chances that the event will not happen to every three chances that it will happen. The probability of success is 30%. In every ten trials, there are expected to be three wins and seven losses.


Units

Ratios may be unitless, as in the case they relate quantities in units of the same
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
, even if their
units of measurement A unit of measurement, or unit of measure, is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other qua ...
are initially different. For example, the ratio can be reduced by changing the first value to 60 seconds, so the ratio becomes . Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2. On the other hand, there are non-dimensionless quotients, also known as ''rates'' (sometimes also as ratios).''"''Ratio as a Rate''. The first type f ratiodefined by Freudenthal, above, is known as rate, and illustrates a comparison between two variables with difference units. (...) A ratio of this sort produces a unique, new concept with its own entity, and this new concept is usually not considered a ratio, per se, but a rate or density."'', "Ratio and Proportion: Research and Teaching in Mathematics Teachers

/ref> In chemistry, Mass concentration (chemistry), mass concentration ratios are usually expressed as weight/volume fractions. For example, a concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, as in weight/weight or volume/volume fractions.


Triangular coordinates

The locations of points relative to a triangle with vertices ''A'', ''B'', and ''C'' and sides ''AB'', ''BC'', and ''CA'' are often expressed in extended ratio form as ''triangular coordinates''. In barycentric coordinates, a point with coordinates ''α, β, γ'' is the point upon which a weightless sheet of metal in the shape and size of the triangle would exactly balance if weights were put on the vertices, with the ratio of the weights at ''A'' and ''B'' being ''α'' : ''β'', the ratio of the weights at ''B'' and ''C'' being ''β'' : ''γ'', and therefore the ratio of weights at ''A'' and ''C'' being ''α'' : ''γ''. In
trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
, a point with coordinates ''x'':''y'':''z'' has
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟠...
distances to side ''BC'' (across from vertex ''A'') and side ''CA'' (across from vertex ''B'') in the ratio ''x'':''y'', distances to side ''CA'' and side ''AB'' (across from ''C'') in the ratio ''y'':''z'', and therefore distances to sides ''BC'' and ''AB'' in the ratio ''x'':''z''. Since all information is expressed in terms of ratios (the individual numbers denoted by ''α, β, γ, x, y,'' and ''z'' have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle.


See also

*
Cross ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defin ...
* Dilution ratio * Displacement–length ratio *
Dimensionless quantity Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
*
Financial ratio A financial ratio or accounting ratio states the relative magnitude of two selected numerical values taken from an enterprise's financial statements. Often used in accounting, there are many standard ratios used to try to evaluate the overall fin ...
*
Fold change Fold change is a measure describing how much a quantity changes between an original and a subsequent measurement. It is defined as the ratio between the two quantities; for quantities ''A'' and ''B'' the fold change of ''B'' with respect to ''A' ...
*
Interval (music) In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or har ...
*
Odds ratio An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B ...
*
Parts-per notation In science and engineering, the parts-per notation is a set of pseudo-units to describe the small values of miscellaneous dimensionless quantities, e.g. mole fraction or mass fraction. Since these fractions are quantity-per-quantity meas ...
*
Price–performance ratio In economics, engineering, business management and marketing the price–performance ratio is often written as cost–performance, cost–benefit or capability/price (C/P), refers to a product's ability to deliver performance, of any sort, for i ...
*
Proportionality (mathematics) In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called ''coefficient of proportionality'' (or ''proportionality ...
*
Ratio distribution A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually statistically independent, independ ...
*
Ratio estimator The ratio estimator is a statistical estimator for the ratio of means of two random variables. Ratio estimates are biased and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical and symm ...
*
Rate (mathematics) In mathematics, a rate is the quotient of two quantities, often represented as a fraction. If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed s ...
* Ratio (Twitter) *
Rate ratio In epidemiology, a rate ratio, sometimes called an incidence density ratio or incidence rate ratio, is a relative difference measure used to compare the incidence rates of events occurring at any given point in time. It is defined as: : \text = \f ...
*
Relative risk The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio, relative risk measures the association bet ...
*
Rule of three (mathematics) In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is a ...
*
Scale (map) The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation ...
*
Scale (ratio) The scale ratio of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a ...
*
Sex ratio A sex ratio is the ratio of males to females in a population. As explained by Fisher's principle, for evolutionary reasons this is typically about 1:1 in species which reproduce sexually. However, many species deviate from an even sex ratio, ei ...
* Superparticular ratio *
Slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...


References


Further reading


"Ratio" ''The Penny Cyclopædia'' vol. 19
The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
"Proportion" ''New International Encyclopedia, Vol. 19'' 2nd ed. (1916) Dodd Mead & Co. pp270-271"Ratio and Proportion" ''Fundamentals of practical mathematics'', George Wentworth, David Eugene Smith, Herbert Druery Harper (1922) Ginn and Co. pp. 55ff
* *D.E. Smith, ''History of Mathematics, vol 2'' Ginn and Company (1925) pp. 477ff. Reprinted 1958 by Dover Publications.


External links

{{Fractions and ratios Elementary mathematics Algebra Quotients