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In mathematics, a ratio is a relationship between two numbers of the same kind[1] (e.g., objects,persons, students, spoonfuls, units of whatever identical dimension), usually expressed as "a tob" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two[2] thatexplicitly indicates how many times the first number contains the second (not necessarily aninteger).[3]In layman's terms a ratio represents, for every amount of one thing, how much there is of anotherthing. For example, supposing one has 8 oranges and 6 lemons in a bowl of fruit, the ratio oforanges to lemons would be 4:3 (which is equivalent to 8:6) while the ratio of lemons to orangeswould be 3:4. Additionally, the ratio of oranges to the total amount of fruit is 4:7 (equivalent to8:14). The 4:7 ratio can be further converted to a fraction of 4/7 to represent how much of the fruitis oranges.Contents [hide]1 Notation and terminology2 History and etymology2.1 Euclid's definitions3 Examples3.1 Fraction4 Number of terms4.1 Percentage ratio5 Proportions

6 Reduction6.1 Dilution ratio7 Odds8 Different units9 See also10 References11 Further reading12 External linksNotation and terminology[edit]

The ratio of numbers A and B can be expressed as:[4]the ratio of A to B

A is to B

A:BA fraction (rational number) that is the quotient of A divided by B\tfrac{A}{B}The numbers A and B are sometimes called terms with A being the antecedent and B being theconsequent.[citation needed]The proportion expressing the equality of the ratios A:B and C:D is written 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.Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and Band C are called the means. The equality of three or more proportions is called a continuedproportion.[5]Ratios are sometimes used with three or more terms. The dimensions of a two by four that is teninches long are 2:4:10.

History and etymology[edit]

Look up ratio in Wiktionary, the free dictionary.It is impossible to trace the origin of the concept of ratio, because the ideas from which itdeveloped would have been familiar to preliterate cultures. For example, the idea of one villagebeing twice as large as another is so basic that it would have been understood in prehistoricsociety.[6] However, it is possible to trace the origin of the word "ratio" to the Ancient Greek (logos). Early translators rendered this into Latin as ratio ("reason"; as in theword "rational"). (A rational number may be expressed as the quotient of two

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integers.) A more modern interpretation of Euclid's meaning is more akin tocomputation or reckoning.[7] Medieval writers used the word proportio ("proportion")to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.[8]Euclid collected the results appearing in the Elements from earlier sources. ThePythagoreans developed a theory of ratio and proportion as applied to numbers.[9]

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, asthe Pythagoreans also discovered, incommensurable ratios (corresponding toirrational numbers) exist. The discovery of a theory of ratios that does not assumecommensurability is probably due to Eudoxus. The exposition of the theory ofproportions that appears in Book VII of The Elements reflects the earlier theory ofratios of commensurables.[10]

The existence of multiple theories seems unnecessarily complex to modernsensibility since ratios are, to a large extent, identified with quotients. This is acomparatively recent development however, as can be seen from the fact thatmodern geometry textbooks still use distinct terminology and notation for ratios andquotients. The reasons for this are twofold. First, there was the previously mentionedreluctance to accept irrational numbers as true numbers. Second, the lack of a widelyused symbolism to replace the already established terminology of ratios delayed thefull acceptance of fractions as alternative until the 16th century.[11]Euclid's definitions[edit]Book V of Euclid's Elements has 18 definitions, all of which relate to ratios.[12] Inaddition, Euclid uses ideas that were in such common usage that he did not includedefinitions for them. The first two definitions say that a part of a quantity is anotherquantity that "measures" it and conversely, a multiple of a quantity is anotherquantity that it measures. In modern terminology, this means that a multiple of aquantity is that quantity multiplied by an integer greater than oneand a part of aquantity (meaning aliquot part) is a part that, when multiplied by an integer greaterthan one, gives the quantity.Euclid does not define the term "measure" as used here, However, one may infer thatif a quantity is taken as a unit of measurement, and a second quantity is given as anintegral number of these units, then the first quantity measures the second. Notethat these definitions are repeated, nearly word for word, as definitions 3 and 5 inbook VII.Definition 3 describes what a ratio is in a general way. It is not rigorous in amathematical sense and some have ascribed it to Euclid's editors rather than Euclidhimself.[13] Euclid defines a ratio as between two quantities of the same type, so bythis definition the ratios of two lengths or of two areas are defined, but not the ratioof a length and an area. Definition 4 makes this more rigorous. It states that a ratio oftwo quantities exists when there is a multiple of each that exceeds the other. Inmodern notation, a ratio exists between quantities p and q if there exist integers mand n so that mp>q and nq>m. This condition is known as the Archimedean property.Definition 5 is the most complex and difficult. It defines what it means for two ratiosto be equal. Today, this can be done by simply stating that ratios are equal when thequotients of the terms are equal, but Euclid did not accept the existence of thequotients of incommensurables, so such a definition would have been meaningless to

him. Thus, a more subtle definition is needed where quantities involved are notmeasured directly to one another. Though it may not be possible to assign a rationalvalue to a ratio, it is possible to compare a ratio with a rational number. Specifically,given two quantities, p and q, and a rational number m/n we can say that the ratio ofp to q is less than, equal to, or greater than m/n when np is less than, equal to, orgreater than mq respectively. Euclid's definition of equality can be stated as that tworatios are equal when they behave identically with respect to being less than, equalto, or greater than any rational number. In modern notation this says that givenquantities p, q, r and s, then p:q::r:s if for any positive integers m and n, np

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np=mq, np>mq according as nrms respectively. There is aremarkable similarity between this definition and the theory of Dedekind cuts used inthe modern definition of irrational numbers.[14]Definition 6 says that quantities that have the same ratio are proportional or inproportion. Euclid uses the Greek (analogon), this has the same root as and is related to the English word "analog".

Definition 7 defines what it means for one ratio to be less than or greater thananother and is based on the ideas present in definition 5. In modern notation it saysthat given quantities p, q, r and s, then p:q>r:s if there are positive integers m and nso that np>mq and nrms.As with definition 3, definition 8 is regarded by some as being a later insertion byEuclid'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 thathave the property that the ratios of consecutive terms are equal are called Geometricprogressions. Definitions 9 and 10 apply this, saying that if p, q and r are inproportion then p:r is the duplicate ratio of p:q and if p, q, r and s are in proportionthen p:s is the triplicate ratio of p:q. If p, q and r are in proportion then q is called amean proportional to (or the geometric mean of) p and r. Similarly, if p, q, r and s arein proportion then q and r are called two mean proportionals to p and s.Examples[edit]

The quantities being compared in a ratio might be physical quantities such as speedor length, or numbers of objects, or amounts of particular substances. A commonexample of the last case is the weight ratio of water to cement used in concrete,which is commonly stated as 1:4. This means that the weight of cement used is fourtimes the weight of water used. It does not say anything about the total amounts ofcement and water used, nor the amount of concrete being made. Equivalently itcould be said that the ratio of cement to water is 4:1, that there is 4 times as muchcement as water, or that there is a quarter (1/4) as much water as cement..Older televisions have a 4:3 aspect ratio, which means that the width is 4/3 of theheight; modern widescreen TVs have a 16:9 aspect ratio.Fraction[edit]Main article: Fraction (mathematics)If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and theratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also beexpressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of thepieces of fruit are oranges. If orange juice concentrate is to be diluted with water inthe ratio 1:4, then one part of concentrate is mixed with four parts of water, givingfive 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 ratiosand fractions, it is important to be clear what is being compared to what, andbeginners often make mistakes for this reason.Number of terms[edit]

In general, when comparing the quantities of a two-quantity ratio, this can be

expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, theamount/size/volume/number of the first quantity is \tfrac{2}{3} that of the secondquantity. This pattern also works with ratios with more than two terms. However, aratio with more than two terms cannot be completely converted into a single fraction;a single fraction represents only one part of the ratio since a fraction can onlycompare two numbers. If the ratio deals with objects or amounts of objects, this isoften expressed as "for every two parts of the first quantity there are three parts ofthe second quantity".Percentage ratio[edit]If we multiply all quantities involved in a ratio by the same number, the ratio remains

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valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduceterms to the lowest common denominator, or to express them in parts per hundred(percent).If a mixture contains substances A, B, C & D in the ratio 5:9:4:2 then there are 5 partsof A for every 9 parts of B, 4 parts of C and 2 parts of D. As 5+9+4+2=20, the totalmixture 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, this is converted topercentages: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as25:45:20:10).Proportions[edit]

If the two or more ratio quantities encompass all of the quantities in a particularsituation, for example two apples and three oranges in a fruit basket containing noother types of fruit, it could be said that "the whole" contains five parts, made up oftwo parts apples and three parts oranges. In this case, \tfrac{2}{5}, or 40% of thewhole are apples and \tfrac{3}{5}, or 60% of the whole are oranges. Thiscomparison of a specific quantity to "the whole" is sometimes called a proportion.Proportions are sometimes expressed as percentages as demonstrated above.Reduction[edit]

Main article: Reduction (mathematics)Note that ratios can be reduced (as fractions are) by dividing each quantity by thecommon factors of all the quantities. This is often called "cancelling." As for fractions,the simplest form is considered that in which the numbers in the ratio are thesmallest possible integers.

Thus, the ratio 40:60 may be considered equivalent in meaning to the ratio 2:3 withincontexts concerned only with relative quantities.Mathematically, we write: "40:60" = "2:3" (dividing both quantities by 20).Grammatically, we would say, "40 to 60 equals 2 to 3."An alternative representation is: "40:60::2:3"Grammatically, we would say, "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:n or n:1 to enable comparisons ofdifferent ratios.For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4)Alternatively, 4:5 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 writtenwithout the 1 and the colon, though, mathematically, this makes it a factor ormultiplier.Dilution ratio[edit]Ratios are often used for simple dilutions applied in chemistry and biology. A simpledilution is one in which a unit volume of a liquid material of interest is combined withan appropriate volume of a solvent liquid to achieve the desired concentration. Thedilution factor is the total number of unit volumes in which your material is dissolved.

The diluted material must then be thoroughly mixed to achieve the true dilution. Forexample, a 1:5 dilution (verbalize as "1 to 5" dilution) entails combining 1 unitvolume of solute (the material to be diluted) + 4 unit volumes (approximately) of thesolvent to give 5 units of the total volume. (Some solutions and mixtures take upslightly less volume than their components.)

The dilution factor is frequently expressed using exponents: 1:5 would be 5e1 (51i.e. one-fifth:one); 1:100 would be 10e2 (102 i.e. one hundredth:one), and so on.

There is often confusion between dilution ratio (1:n meaning 1 part solute to n partssolvent) and dilution factor (1:n+1) where the second number (n+1) represents thetotal volume of solute + solvent. In scientific and serial dilutions, the given ratio (or

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factor) often means the ratio to the final volume, not to just the solvent. The factorsthen can easily be multiplied to give an overall dilution factor.In other areas of science such as pharmacy, and in non-scientific usage, a dilution isnormally given as a plain ratio of solvent to solute.Odds[edit]

Main article: OddsOdds (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 everythree chances that it will happen. On the other hand, the probability of success is30%. In every ten trials, there are three wins and seven losses.Different units[edit]

Ratios are unitless when they relate quantities in units of the same dimension.For example, the ratio 1 minute : 40 seconds can be reduced by changing the firstvalue to 60 seconds. Once the units are the same, they can be omitted, and the ratiocan be reduced to 3:2.In chemistry, mass concentration "ratios" are usually expressed as w/v percentages,and are really proportions.

For example, a concentration of 3% w/v usually means 3g of substance in every100mL of solution. This cannot easily be converted to a pure ratio because of densityconsiderations, and the second figure is the total amount, not the volume of solvent.See also[edit]

Aspect ratioFraction (mathematics)Golden ratioInterval (music)Parts-per notationPrice/performance ratioProportionality (mathematics)Ratio distribution

Ratio estimatorRule of three (mathematics)Slope