Inmathematics, aratio (/ˈreɪ.ʃ(i.)oʊ/) shows how many times onenumber 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 bepositive.
A ratio may be specified either by giving both constituting numbers, written as "a tob" or "a:b", or by giving just the value of theirquotienta/b.[1][2][3] Equal quotients correspond to equal ratios.A statement expressing the equality of two ratios is called aproportion.
Consequently, a ratio may be considered as an ordered pair of numbers, afraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero)natural numbers, arerational numbers, and may sometimes be natural numbers.
The ratio of numbersA andB can be expressed as:[6]
the ratio ofA toB
A:B
A is toB (when followed by "asC is toD"; see below)
afraction withA as numerator andB as denominator that represents the quotient (i.e.,A divided byB, or). This can be expressed as a simple or a decimal fraction, or as a percentage, etc.[7]
When a ratio is written in the formA:B, the two-dot character is sometimes thecolon punctuation mark.[8] InUnicode, this isU+003A:COLON, although Unicode also provides a dedicated ratio character,U+2236∶RATIO.[9]
The numbersA andB are sometimes calledterms of the ratio, withA being theantecedent andB being theconsequent.[10]
A statement expressing the equality of two ratiosA:B andC:D is called aproportion,[11] written asA:B =C:D orA:B∷C:D. This latter form, when spoken or written in the English language, is often expressed as
(A is toB) as (C is toD).
A,B,C andD are called the terms of the proportion.A andD are called itsextremes, andB andC are called itsmeans. The equality of three or more ratios, likeA:B =C:D =E:F, is called acontinued proportion.[12]
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
(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
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.
It is possible to trace the origin of the word "ratio" to theancient Greekλόγος (logos). Early translators rendered this intoLatin asratio ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning.[14] Medieval writers used the wordproportio ("proportion") to indicate ratio andproportionalitas ("proportionality") for the equality of ratios.[15]
Euclid collected the results appearing in the Elements from earlier sources. ThePythagoreans developed a theory of ratio and proportion as applied to numbers.[16] 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 toirrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due toEudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.[17]
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.[18]
Book V ofEuclid's Elements has 18 definitions, all of which relate to ratios.[19] 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 apart of a quantity is another quantity that "measures" it and conversely, amultiple 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 (meaningaliquot 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 quantitymeasures 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.[20] Euclid defines a ratio as between two quantitiesof 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 quantitiesp andq, if there exist integersm andn such thatmp>q andnq>p. This condition is known as theArchimedes 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 quantitiesp,q,r ands,p:q∷r:s if and only if, for any positive integersm andn,np < mq,np = mq, ornp > mq according asnr < ms,nr = ms, ornr > ms, respectively.[21] This definition has affinities withDedekind cuts as, withn andq both positive,np stands tomq asp/q stands to the rational numberm/n (dividing both terms bynq).[22]
Definition 6 says that quantities that have the same ratio areproportional orin 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 quantitiesp,q,r ands,p:q > r:s if there are positive integersm andn so thatnp > mq andnr ≤ ms.
As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three termsp,q andr to be in proportion whenp:q∷q:r. This is extended to four termsp,q,r ands asp:q∷q:r∷r:s, and so on. Sequences that have the property that the ratios of consecutive terms are equal are calledgeometric progressions. Definitions 9 and 10 apply this, saying that ifp,q andr are in proportion thenp:r is theduplicate ratio ofp:q and ifp,q,r ands are in proportion thenp:s is thetriplicate ratio ofp:q.
In general, a comparison of the quantities of a two-entity ratio can be expressed as afraction derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is 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 that of the third entity.
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, or to express them in parts per hundred (percent).
If a mixture contains substancesA,B,C andD in the ratio 5:9:4:2, then there are 5 parts ofA for every9 parts ofB, 4 parts ofC, and 2 parts ofD. As5 + 9 + 4 + 2 = 20, the total mixture contains 5/20 ofA (5 parts out of 20), 9/20 ofB, 4/20 ofC, and 2/20 ofD. If we divide all numbers by the total and multiply by 100, we have converted topercentages: 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,, or 40% of the whole is apples and, 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, oldertelevisions have a 4:3aspect ratio, 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.
Ratios can bereduced (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 insimplest form or lowest terms.
Sometimes it is useful to write a ratio in the form 1:x orx:1, wherex 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 ormultiplier.
Ratios may also be established betweenincommensurable quantities (quantities whose ratio, as value of a fraction, amounts to anirrational number). The earliest discovered example, found by thePythagoreans, is the ratio of the length of the diagonald to the length of a sides of asquare, which is thesquare root of 2, formally Another example is the ratio of acircle's circumference to its diameter, which is calledπ, and is not just anirrational number, but atranscendental number.
Also well known is thegolden ratio of two (mostly) lengthsa andb, which is defined by the proportion
or, equivalently
Taking the ratios as fractions and as having the valuex, yields the equation
or
which has the positive, irrational solutionThus at least one ofa andb 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 consecutiveFibonacci numbers: 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, thesilver ratio ofa andb is defined by the proportion
corresponding to
This equation has the positive, irrational solution so again at least one of the two quantitiesa andb in the silver ratio must be irrational.
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.
Ratios may beunitless, as in the case they relate quantities in units of the samedimension, even if theirunits of measurement are initially different.For example, the ratioone minute : 40 seconds can be reduced by changing the first value to 60 seconds, so the ratio becomes60 seconds : 40 seconds. 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 asrates (sometimes also as ratios).[23][24]In 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.
The locations of points relative to a triangle withverticesA,B, andC and sidesAB,BC, andCA are often expressed in extended ratio form astriangular coordinates.
Inbarycentric coordinates, a point with coordinatesα, 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 atA andB beingα :β, the ratio of the weights atB andC beingβ :γ, and therefore the ratio of weights atA andC beingα :γ.
Intrilinear coordinates, a point with coordinatesx :y :z hasperpendicular distances to sideBC (across from vertexA) and sideCA (across from vertexB) in the ratiox :y, distances to sideCA and sideAB (across fromC) in the ratioy :z, and therefore distances to sidesBC andAB in the ratiox :z.
Since all information is expressed in terms of ratios (the individual numbers denoted byα, β, γ, x, y, andz have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle.
^Decimal fractions are frequently used in technological areas where ratio comparisons are important, such as aspect ratios (imaging), compression ratios (engines or data storage), etc.
^"ASCII Punctuation"(PDF).The Unicode Standard, Version 15.0. Unicode, Inc. 2022. Retrieved2022-11-26.[003A is] also used to denote division or scale; for that mathematical use 2236 ∶ is preferred.
^"Ratio as a Rate. The first type [of ratio] defined byFreudenthal, 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"[1]
