In

/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.

"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.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, 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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s, are rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

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 cur ...

. 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) *afraction
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'' $\backslash tfrac$). 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 ...

(:) is often used in place of the ratio symbol, Unicode
Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

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
:$\backslash text\; =\; 2:4:10;$
:(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
:$\backslash text\; =\; 1:2:4.$
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 theAncient Greek
Ancient Greek includes the forms of the Greek language
Greek ( el, label=Modern Greek
Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...

(''logos
''Logos'' (, ; grc, λόγος
''Logos'' (, ; grc, λόγος
''Logos'' (, ; grc, λόγος, lógos; from , , ) is a term in Western philosophy
Western philosophy refers to the philosophy, philosophical thought and work of the W ...

''). 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 in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) 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 ofEuclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, ...

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 ...

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 afraction
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 $\backslash tfrac$ 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 $\backslash tfrac$ 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 thelowest common denominator
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple, lowest common multiple of the denominators of a set of fraction (mathematics), fractions. It simplifies adding, subtracting, ...

, or to express them in parts per hundred (percent
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

).
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 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 ...

: 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, $\backslash tfrac$, or 40% of the whole is apples and $\backslash tfrac$, 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
Telecommunication is the transmission of information by various types of technologies over wire
A wire is a single usually cylindrical
A cylinder (from Gre ...

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, one of the oldest branches of mathematics. It is concerned with properties of space that are ...

'', 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 afactor
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 ...

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 ...

.
Irrational ratios

Ratios may also be established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to anirrational 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
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

, which is the square root of 2
The square root of 2 (approximately 1.4142) is a positive real 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 ...

, formally $a:d\; =\; 1:\backslash sqrt.$ Another example is the ratio of a circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

'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 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 ...

of two (mostly) lengths and , which is defined by the proportion
: $a:b\; =\; (a+b):a\; \backslash quad$ or, equivalently $\backslash quad\; a:b\; =\; (1+b/a):1.$
Taking the ratios as fractions and $a:b$ as having the value , yields the equation
:$x=1+\backslash tfrac\; 1x\; \backslash quad$ or $\backslash quad\; x^2-x-1\; =\; 0,$
which has the positive, irrational solution $x=\backslash tfrac=\backslash tfrac.$
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 , a sequence is an enumerated collection of in which repetitions are allowed and matters. Like a , it contains (also called ''elements'', or ''terms''). The numbe ...

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
:$a:b\; =\; (2a+b):a\; \backslash quad\; (=\; (2+b/a):1),$ corresponding to $x^2-2x-1\; =\; 0.$
This equation has the positive, irrational solution $x\; =\; \backslash tfrac=1+\backslash sqrt,$ 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 beunitless
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 physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

, 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 (mathematic ...

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 ...

.''"''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''. Intrilinear 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 ...

, a point with coordinates ''x'':''y'':''z'' has perpendicular
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

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
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 cur ...

*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
Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. Genera ...

*Odds ratio
An odds ratio (OR) is a statistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population
Population ty ...

*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 , a rate is the between two related 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 systematically (i.e., is an ), then the n ...

*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, ...

*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'', ...

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