what is the mathematical definition of the ratio of one quantity to another?

Relationship between 2 numbers of the same kind

For non-dimensionless ratios, see Rates.

"is to" redirects here. For the grammatical construction, come across am to.

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, so the ratio of oranges to lemons is viii to six (that is, viii:six, which is equivalent to the ratio iv:three). Similarly, the ratio of lemons to oranges is 6:viii (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 whatsoever kind, such as counts of people or objects, or such as measurements of lengths, weights, fourth dimension, etc. In nearly contexts, both numbers are restricted to be positive.

A ratio may be specified either by giving both constituting numbers, written every bit "a to b" or "a:b", or past giving just the value of their quotient a / b .^{[1] [2] [3]} Equal quotients correspond to equal ratios.

Consequently, a ratio may exist considered every bit an ordered pair of numbers, a fraction with the outset number in the numerator and the second in the denominator, or as the value denoted past this fraction. Ratios of counts, given by (non-aught) natural numbers, are rational numbers, and may sometimes be natural numbers. When 2 quantities are measured with the same unit, as is frequently the instance, their ratio is a dimensionless number. A caliber of ii quantities that are measured with different units is chosen a rate.^[4]

Notation and terminology [edit]

The ratio of numbers A and B tin be expressed as:^[5]

• the ratio of A to B
• A:B
• A is to B (when followed by "as C is to D "; see below)
• a fraction with A every bit numerator and B as denominator that represents the caliber (i.eastward., A divided past B, or ${\displaystyle {\tfrac {A}{B}}}$ ). This tin be expressed equally a simple or a decimal fraction, or every bit a per centum, etc.^[6]

A colon (:) is oft used in place of the ratio symbol, Unicode U+2236 (:).

The numbers A and B are sometimes called terms of the ratio, with A being the antecedent and B existence the consequent.^[7]

A statement expressing the equality of 2 ratios A:B and C:D is chosen a proportion,^[eight] written as A:B = C:D or A:B∷C:D. This latter form, when spoken or written in the English linguistic communication, is often expressed as

(A is to B) equally (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 ways. The equality of three or more ratios, like A:B = C:D = Eastward:F, is chosen a continued proportion.^[nine]

Ratios are sometimes used with three or even more terms, east.g., the proportion for the edge lengths of a "2 by iv" that is ten inches long is therefore

${\displaystyle {\text{thickness : width : length }}=2:iv:x;}$

(unplaned measurements; the start two numbers are reduced slightly when the wood is planed smooth)

a skilful physical mix (in volume units) is sometimes quoted every bit

${\displaystyle {\text{cement : sand : gravel }}=1:2:4.}$ ^[10]

For a (rather dry) mixture of iv/1 parts in volume of cement to h2o, it could exist said that the ratio of cement to h2o is 4:1, that there is four times as much cement every bit water, or that there is a quarter (1/iv) as much water equally cement.

The pregnant of such a proportion of ratios with more than 2 terms is that the ratio of any 2 terms on the left-mitt side is equal to the ratio of the respective two terms on the right-mitt side.

History and etymology [edit]

It is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin equally ratio ("reason"; as in the give-and-take "rational"). A more modern interpretation^{[ compared to? ]} of Euclid's meaning is more akin to computation or reckoning.^[eleven] Medieval writers used the discussion proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.^[12]

Euclid collected the results actualization in the Elements from earlier sources. The Pythagoreans adult a theory of ratio and proportion as practical to numbers.^[13] The Pythagoreans' formulation of number included simply what would today be called rational numbers, casting uncertainty on the validity of the theory in geometry where, every bit the Pythagoreans also discovered, incommensurable ratios (respective to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book Seven of The Elements reflects the earlier theory of ratios of commensurables.^[14]

The beingness 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, equally tin can be seen from the fact that modern geometry textbooks all the same utilise distinct terminology and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to have irrational numbers as true numbers, and 2nd, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the total acceptance of fractions equally culling until the 16th century.^[fifteen]

Euclid's definitions [edit]

Book 5 of Euclid's Elements has eighteen definitions, all of which relate to ratios.^[16] In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first 2 definitions say that a part of a quantity is some other quantity that "measures" information technology and conversely, a multiple of a quantity is another quantity that information technology measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a function of a quantity (meaning aliquot function) is a function 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 every bit a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the 2nd. These definitions are repeated, well-nigh word for give-and-take, as definitions 3 and v in book VII.

Definition 3 describes what a ratio is in a general fashion. It is not rigorous in a mathematical sense and some have ascribed information technology to Euclid's editors rather than Euclid himself.^[17] Euclid defines a ratio equally between two quantities of the same type, so past this definition the ratios of two lengths or of 2 areas are divers, but not the ratio of a length and an surface area. Definition 4 makes this more rigorous. Information technology states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In mod annotation, a ratio exists between quantities p and q, if there exist integers m and n such that mp>q and nq>p. This condition is known as the Archimedes property.

Definition v is the most circuitous and difficult. It defines what information technology 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, just such a definition would have been meaningless to Euclid. In modern note, Euclid's definition of equality is that given quantities p, q, r and southward, p:q∷r :s if and merely 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.^[18] This definition has affinities with Dedekind cuts every bit, with due north and q both positive, np stands to mq as p / q stands to the rational number chiliad / north (dividing both terms past nq).^[xix]

Definition half-dozen says that quantities that accept 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 i ratio to be less than or greater than another and is based on the ideas nowadays in definition 5. In modern annotation it says that given quantities p, q, r and s, p:q>r:s if there are positive integers m and n and then that np>mq and nr≤ms.

As with definition 3, definition eight is regarded past some as being a later on 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 iv 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 chosen geometric progressions. Definitions 9 and 10 use this, saying that if p, q and r are in proportion so p:r is the indistinguishable ratio of p:q and if p, q, r and s are in proportion then p:due south is the triplicate ratio of p:q.

Number of terms and employ of fractions [edit]

In general, a comparing of the quantities of a two-entity ratio can exist expressed equally a fraction derived from the ratio. For example, in a ratio of two:3, the amount, size, book, or quantity of the first entity is ${\displaystyle {\tfrac {2}{3}}}$ that of the 2nd entity.

If there are 2 oranges and three apples, the ratio of oranges to apples is 2:iii, and the ratio of oranges to the total number of pieces of fruit is ii:v. These ratios can also exist expressed in fraction form: there are 2/iii as many oranges every bit apples, and ii/5 of the pieces of fruit are oranges. If orangish juice concentrate is to be diluted with water in the ratio 1:4, and so one part of concentrate is mixed with iv parts of h2o, giving five parts total; the amount of orange juice concentrate is 1/four the amount of water, while the amount of orangish juice concentrate is 1/5 of the full liquid. In both ratios and fractions, information technology is important to be articulate what is being compared to what, and beginners often brand mistakes for this reason.

Fractions can too be inferred from ratios with more 2 entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction tin only compare ii 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 two:3:vii we can infer that the quantity of the second entity is ${\displaystyle {\tfrac {three}{7}}}$ that of the third entity.

Proportions and percentage ratios [edit]

If nosotros multiply all quantities involved in a ratio past the same number, the ratio remains valid. For example, a ratio of 3:two is the same every bit 12:8. It is usual either to reduce terms to the lowest mutual denominator, or to limited them in parts per hundred (percentage).

If a mixture contains substances A, B, C and D in the ratio 5:9:four:2 and so there are 5 parts of A for every ix parts of B, 4 parts of C and two parts of D. As v+9+4+2=20, the full mixture contains 5/xx of A (5 parts out of 20), 9/20 of B, 4/20 of C, and 2/20 of D. If we divide all numbers by the total and multiply by 100, we have converted to percentages: 25% A, 45% B, 20% C, and ten% 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 iii oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, ${\displaystyle {\tfrac {2}{v}}}$ , or forty% of the whole is apples and ${\displaystyle {\tfrac {3}{5}}}$ , or sixty% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion.

If the ratio consists of only 2 values, it can be represented equally a fraction, in detail as a decimal fraction. For example, older televisions accept a 4:3 attribute ratio, which ways that the width is 4/3 of the height (this can also exist expressed as ane.33:1 or just 1.33 rounded to ii decimal places). More than recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. 1 of the popular widescreen film formats is two.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparing. When comparing one.33, 1.78 and 2.35, it is obvious which format offers wider image. Such a comparing works just when values existence compared are consistent, similar always expressing width in relation to height.

Reduction [edit]

Ratios can be reduced (equally fractions are) past dividing each quantity past the mutual 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:iii, the latter beingness obtained from the one-time by dividing both quantities past xx. Mathematically, we write twoscore:60 = 2:three, or equivalently 40:lx∷2:3. The exact 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 grade ane:x or x:1, where x is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:5 tin exist written as 1:i.25 (dividing both sides past 4) Alternatively, it can exist written as 0.8:ane (dividing both sides by v).

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 information technology a factor or multiplier.

Irrational ratios [edit]

Ratios may also exist established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to an irrational number). The earliest discovered example, found by the Pythagoreans, is the ratio of the length of the diagonal d to the length of a side s of a foursquare, which is the square root of 2, formally ${\displaystyle a:d=1:{\sqrt {two}}.}$ Another example is the ratio of a circle's circumference to its bore, which is called π, and is non just an algebraically irrational number, merely a transcendental irrational.

Also well known is the golden ratio of two (by and large) lengths a and b, which is divers past the proportion

${\displaystyle a:b=(a+b):a\quad }$ or, equivalently ${\displaystyle \quad a:b=(i+b/a):one.}$

Taking the ratios equally fractions and ${\displaystyle a:b}$ as having the value x, yields the equation

${\displaystyle x=1+{\tfrac {1}{x}}\quad }$ or ${\displaystyle \quad x^{2}-ten-ane=0,}$

which has the positive, irrational solution ${\displaystyle ten={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{two}}.}$ Thus at to the lowest degree 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 2 consecutive Fibonacci 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, the silver ratio of a and b is defined by the proportion

${\displaystyle a:b=(2a+b):a\quad (=(2+b/a):one),}$ respective to ${\displaystyle x^{2}-2x-ane=0.}$

This equation has the positive, irrational solution ${\displaystyle x={\tfrac {a}{b}}=i+{\sqrt {2}},}$ and then over again at least one of the two quantities a and b in the silver ratio must be irrational.

Odds [edit]

Main article: 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 result will not happen to every three chances that information technology will happen. The probability of success is thirty%. In every ten trials, there are expected to exist three wins and seven losses.

Units [edit]

Ratios may be unitless, every bit in the instance they relate quantities in units of the same dimension, fifty-fifty if their units of measurement are initially dissimilar. For example, the ratio 1 minute : xl seconds can be reduced by changing the first value to lx seconds, so the ratio becomes 60 seconds : 40 seconds. Once the units are the aforementioned, they can exist omitted, and the ratio can be reduced to 3:2.

On the other hand, there are non-dimensionless ratios, likewise known as rates.^{[xx] [21]} In chemical science, mass concentration ratios are usually expressed as weight/volume fractions. For example, a concentration of three% w/v commonly means three 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 [edit]

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 grade as triangular coordinates.

In barycentric coordinates, a point with coordinates α, β, γ is the betoken upon which a weightless sail of metallic in the shape and size of the triangle would exactly remainder if weights were put on the vertices, with the ratio of the weights at A and B being α : β, the ratio of the weights at B and C being β : γ, and therefore the ratio of weights at A and C being α : γ.

In trilinear coordinates, a point with coordinates x :y :z has 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, ten, y, and z have no significant by themselves), a triangle assay using barycentric or trilinear coordinates applies regardless of the size of the triangle.

Meet likewise [edit]

• Dilution ratio
• Deportation–length ratio
• Dimensionless quantity
• Financial ratio
• Fold alter
• Interval (music)
• Odds ratio
• Parts-per notation
• Cost–performance ratio
• Proportionality (mathematics)
• Ratio distribution
• Ratio estimator
• Rate (mathematics)
• Rate ratio
• Relative risk
• Rule of three (mathematics)
• Scale (map)
• Scale (ratio)
• Sex activity ratio
• Superparticular ratio
• Slope

References [edit]

1. ^ New International Encyclopedia
2. ^ "Ratios". www.mathsisfun.com . Retrieved 2020-08-22 .
3. ^ Stapel, Elizabeth. "Ratios". Purplemath . Retrieved 2020-08-22 .
4. ^ "The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)", "The Mathematics Dictionary" [ane]
5. ^ New International Encyclopedia
6. ^ Decimal fractions are frequently used in technological areas where ratio comparisons are important, such equally aspect ratios (imaging), compression ratios (engines or data storage), etc.
7. ^ from the Encyclopædia Britannica
8. ^ Heath, p. 126
9. ^ New International Encyclopedia
10. ^ Belle Grouping physical mixing hints
11. ^ Penny Cyclopædia, p. 307
12. ^ Smith, p. 478
13. ^ Heath, p. 112
14. ^ Heath, p. 113
15. ^ Smith, p. 480
16. ^ Heath, reference for section
17. ^ "Geometry, Euclidean" Encyclopædia Britannica Eleventh Edition p682.
18. ^ Heath p.114
19. ^ Heath p. 125
20. ^ "'Velocity' can be defined as the ratio... 'Population density' is the ratio... 'Gasoline consumption' is mensurate every bit the ratio...", "Ratio and Proportion: Research and Teaching in Mathematics Teachers" [2]
21. ^ "Ratio as a Rate. The first type [of ratio] defined by Freudenthal, in a higher place, 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: Inquiry and Teaching in Mathematics Teachers" [3]

Further reading [edit]

• "Ratio" The Penny Cyclopædia vol. xix, The Society for the Diffusion of Useful Noesis (1841) Charles Knight and Co., London pp. 307ff
• "Proportion" New International Encyclopedia, Vol. xix 2d 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
• The 13 books of Euclid'due south Elements, vol 2. trans. Sir Thomas Petty Heath (1908). Cambridge Univ. Printing. 1908. pp. 112ff. {{cite book}}: CS1 maint: others (link)
• D.E. Smith, History of Mathematics, vol ii Ginn and Company (1925) pp. 477ff. Reprinted 1958 by Dover Publications.

External links [edit]

Await up ratio in Wiktionary, the free dictionary.

nicolllogesterme.blogspot.com

Source: https://en.wikipedia.org/wiki/Ratio

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