Ratio
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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, a ratio indicates 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 In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ...
Equal quotients correspond to equal ratios. Consequently, a ratio may be considered as an ordered pair of numbers, a
fraction A fraction (from Latin ', "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 ...
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 File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s, are
rational number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, and may sometimes be natural numbers. When two quantities are measured with the same unit, as is often the case, their ratio is a
dimensionless number In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric curre ...
. A quotient of two quantities that are measured with ''different'' units is 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 Latin ', "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 ...
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. A
colon Colon commonly refers to: * Colon (punctuation) (:), a punctuation mark * Major part of large intestine, the final section of the digestive system Colon may also refer to: Places * Colon, Michigan, US * Colon, Nebraska, US * Kowloon, Hong Kong, s ...

colon
(:) is often used in place of the ratio symbol,
Unicode Unicode, formally the Unicode Standard, is an information technology standard Standard may refer to: Flags * Colours, standards and guidons * Standard (flag), a type of flag used for personal identification Norm, convention or requireme ...

Unicode
U+2236 (∶). The numbers ''A'' and ''B'' are sometimes called ''terms of the ratio'', with ''A'' being the ''
antecedent An antecedent is a preceding event, condition, cause, phrase, or word. More specifically, it may refer to: * Antecedent (behavioral psychology), the stimulus that occurs before a trained behavior * Antecedent (genealogy), antonym of descendant, gen ...
'' and ''B'' being the ''
consequent A consequent is the second half of a hypothetical proposition In linguistics and logic, a proposition is the meaning of a declarative sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which i ...
''. 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 used in and the from around 1500 BC to 300 BC. It is often roughly divided into the following periods: (), Dark Ages (), the period (), and the period (). Ancient Greek was the language of an ...
(''
logos ''Logos'' (, ; grc, , lógos; from , , ) is a term in , , , and derived from a Greek word variously meaning "ground", "plea", "opinion", "expectation", "word", "speech", "account", "reason", "proportion", and "discourse".Henry George Liddell ...

logos
''). Early translators rendered this into
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became the dominant la ...

Latin
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 the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...
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 which 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 (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from ...
. 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'' ( grc, Στοιχεῖον ''Stoikheîon'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
) 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 In abstract algebra and analysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics ( ...
. 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 Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citize ...
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''≤''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 4 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

geometric progression
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 Latin ', "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 ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, or to express them in parts per hundred (
percent In mathematics, a percentage (from Latin ''per centum'' "by a hundred") is a number or ratio expressed as a fraction (mathematics), fraction of 100. It is often Denotation, denoted using the percent sign, "%", although the abbreviations "pct.", ...

percent
). 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 5+9+4+2=20, 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
: 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, sometimes shortened to TV or telly, is a telecommunication Media (communication), medium used for transmitting moving images in grayscale, black-and-white or in color, and in two or 3D television, three dimensions and sound. The ...

television
s have a 4∶3 ''
aspect ratio The aspect ratio of a geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμό ...
'', 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 FACTOR (the Foundation to Assist Canadian Talent on Records) is a private non-profit organization "dedicated to providing assistance toward the growth and development of the Music of Canada, Canadian music industry". FACTOR was founded in 1982 by r ...

factor
or
multiplierMultiplier may refer to: Mathematics * Multiplier (coefficient), the number of multiples being computed in multiplication, also known as a coefficient in algebra * Lagrange multiplier, a scalar variable used in mathematics to solve an optimisati ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
). The earliest discovered example, found by the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...
, is the ratio of the length of the diagonal to the length of a side of a
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...

square
, which is the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Re ...
, formally a:d = 1:\sqrt. Another example is the ratio of a
circle A circle is a consisting of all in a that are at a given distance from a given point, the ; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is . The distance between any po ...

circle
's circumference to its diameter, which is called

, and is not just an algebraically irrational number, but a transcendental irrational. Also well known is the
golden ratio In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

golden ratio
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 numbers, commonly denoted , form a sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it cont ...

Fibonacci number
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...
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 In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the International ...
, as in the case they relate quantities in units of the same
dimension In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...
, even if their
units of measurement A unit of measurement is a definite magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics ...
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 ratios, also known as
rates Rate or rates may refer to: Finance * Rates (tax) Rates are a type of property tax system in the United Kingdom, and in places with systems deriving from the British one, the proceeds of which are used to fund local government. Some other co ...
.''"''Ratio as a Rate''. The first type f ratiodefined by , 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 coordinatesImage:Trilinear coordinates.svg, 300px In geometry, the trilinear coordinates ''x:y:z'' of a point relative to a given triangle describe the relative directed distances from the three extended side, sidelines of the triangle. Trilinear coordinates ...

trilinear coordinates
, a point with coordinates ''x'':''y'':''z'' has
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propertie ...

perpendicular
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

*
Dilution ratio In chemistry and biology, the dilution ratio is the ratio of solute to solvent. It is often used for ''simple dilutions,'' one in which a unit volume of a liquid material of interest is combined with an appropriate volume of a solvent liquid to ac ...
* Displacement–length ratio *
Dimensionless quantity In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the International ...
*
Financial ratio A financial ratio or accounting ratio is a relative magnitude of two selected numerical values taken from an enterprise's financial statement Financial statements (or financial reports) are formal records of the financial activities and positio ...
* Fold change *
Interval (music) In music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "Elements of music, rudiments", that are ...

Interval (music)
*
Odds ratio An odds ratio (OR) is a statistic A statistic (singular) or sample statistic is any quantity computed from values in a Sample (statistics), sample that is used for a statistical purpose. Statistical purposes include estimating a population parame ...
*
Parts-per notation In science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), organizes knowledge in the form of Testability, testable explanations and predictio ...
*
Price–performance ratio In economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...
*
Proportionality (mathematics) In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
*
Ratio distribution A ratio distribution (also known as a quotient distribution) is a probability distribution In probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presenta ...
* Ratio estimator *
Rate (mathematics) In mathematics, a rate is the ratio between two related quantity, quantities in different units. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed system ...
*
Rate ratioIn epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined populations. It is a cornerstone of public health, and shapes pol ...
*
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 be ...
*
Rule of three (mathematics) In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two Fraction (mathematics), fractions or rational fraction, rational expressions, one can cross-multiply to simplify the equation or determine th ...
*
Scale (map) File:Maßstabsleiste.png, A bar scale with the nominal scale , expressed as both "1cm = 6km" and "1:600 000" (equivalent, because 6km = 600 000cm) The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground ...
*
Scale (ratio) The scale ratio of a Physical model, model represents the Proportionality (mathematics), 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 sca ...
*
Sex ratio The sex ratio is the ratio In mathematics, a ratio indicates 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 ...
*
Superparticular ratio In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons i ...
*
Slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a ...

Slope


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