In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a

**ratio** is a relationship between two numbers of the same kind (

*i.e.*, objects, persons, students, spoonfuls, units of whatever identical dimension), usually expressed as

*"a* to

*b"* or a:b, sometimes expressed arithmetically as a dimensionless

quotientIn mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

of the two which explicitly indicates how many times the first number contains the second (not necessarily an

integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

).

In layman's terms a ratio represents, simply, for every amount of one thing, how much there is of another thing. For example, suppose I have 10 pairs of socks for every pair of shoes then the ratio of shoes:socks would be 1:10 and the ratio of socks:shoes would be 10:1

## Notation and terminology

The ratio of numbers

*A* and

*B* can be expressed as:

- the ratio of
*A* to *B*
*A* is to *B*
*A*:*B*
- A rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

which is the quotient of *A* divided by *B*

The numbers

*A* and

*B* are sometimes called

*terms* with

*A* being the

*antecedent* and

*B* being the

*consequent*.

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

*B* and

*C* are called the

*means*. The equality of three or more proportions is called a continued proportion.

## History and etymology

It is impossible to trace the origin of the

*concept* of ratio, since the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society. However, it is possible to trace the origin of the word "ratio" to the

Ancient GreekAncient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...

λόγος (

logos' is an important term in philosophy, psychology, rhetoric and religion. Originally a word meaning "a ground", "a plea", "an opinion", "an expectation", "word," "speech," "account," "reason," it became a technical term in philosophy, beginning with Heraclitus ' is an important term in...

). Early translators rendered this into

LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

as

*ratio* ("reason"; as in the word "rational"). (A rational number may be expressed as the quotient of two integers.) A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. Medieval writers used the word

*proportio* ("proportion") to indicate ratio and

*proportionalitas* ("proportionality") for the equality of ratios.

Euclid collected the results appearing in the Elements from earlier sources. The

PythagoreansPythagoreanism was the system of esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism originated in the 5th century BCE and greatly influenced Platonism...

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) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to

EudoxusEudoxus or Eudoxos was the name of two ancient Greeks:* Eudoxus of Cnidus , Greek astronomer and mathematician.* Eudoxus of Cyzicus , Greek navigator....

. 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 to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seem 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. 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 ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

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 which "measures" it and conversely, a

*multiple* of a quantity is another quantity which 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) is that which, when multiplied by an integer greater than one, gives the quantity. Euclid does not define the term "measure" as used here but 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. Note that 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 to be 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 which exceeds the other. In modern notation, a ratio exists between quantities

*p* and

*q* if there exist integers

*m* and

*n* so that

*mp*>

*q* and

*nq*>

*m*. This condition is known as the

Archimedean propertyIn abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...

.

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 Euclid did not accept the existence of the quotients of incommensurables, so such a definition would have been meaningless to him. Thus, a more subtle definition is needed where quantities involved are not measured directly to one another. Though it may not be possible to assign a rational value 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 of

*p* to

*q* is less than, equal to, or greater than

*m*/

*n* when

*np* is less than, equal to, or greater than

*mq* respectively. Euclid's definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number. In modern notation this says that given quantities

*p*,

*q*,

*r* and

*s*, then

*p*:

*q*::

*r*:

*s* if for any positive integers

*m* and

*n*,

*np*<

*mq*,

*np*=

*mq*,

*np*>

*mq* according as

*nr*<

*ms*,

*nr*=

*ms*,

*nr*>

*ms* respectively. There is a remarkable similarity between this definition and the theory of

Dedekind cutIn mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....

s used in the modern definition of irrational numbers.

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*, then

*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 which have the property that the ratios of consecutive terms are equal are called

Geometric progressionIn mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a 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*. If

*p*,

*q* and

*r* are in proportion then

*q* is called a

*mean proportional* to (or the

geometric meanThe geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

of)

*p* and

*r*. Similarly, if

*p*,

*q*,

*r* and

*s* are in proportion then

*q* and

*r* are called two mean proportionals to

*p* and

*s*.

## Examples

The quantities being compared in a ratio might be physical quantities such as speed or length, or numbers of objects, or amounts of particular substances. A common example of the last case is the weight ratio of

water to cementThe water–cement ratio is the ratio of the weight of water to the weight of cement used in a concrete mix and has an important influence on the quality of concrete produced. A lower water-cement ratio leads to higher strength and durability, but may make the mix more difficult to place...

used in concrete, which is commonly stated as 1:4. This means that the weight of cement used is four times the weight of water used. It does not say anything about the total amounts of cement and water used, nor the amount of concrete being made. Equivalently 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..

Older

televisionTelevision is a telecommunication medium for transmitting and receiving moving images that can be monochrome or colored, with accompanying sound...

s have a 4:3 "

aspect ratioThe aspect ratio of a shape is the ratio of its longer dimension to its shorter dimension. It may be applied to two characteristic dimensions of a three-dimensional shape, such as the ratio of the longest and shortest axis, or for symmetrical objects that are described by just two measurements,...

", which means that the width is 4/3 of the height; modern widescreen TVs have a 16:9 aspect ratio.

### Fraction

If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, whereas the fraction of oranges to total fruit is 2/5.

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 fraction of concentrate is 1/5 and the fraction of water is 4/5.

## Number of terms

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, the amount/size/volume/number of the first quantity will be

that of the second quantity. This pattern also works with ratios with more than two terms. However, a ratio 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 only compare two numbers. If the ratio deals with objects or amounts of objects, this is often expressed as "for every two parts of the first quantity there are three parts of the second quantity".

### Percentage ratio

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 denominatorIn mathematics, the lowest common denominator or least common denominator is the least common multiple of the denominators of a set of vulgar fractions...

, 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 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, this is converted to percentages: 25% A, 45% B, 20% C, and 10% D (equivalent to writing the ratio as 25:45:20:10).

## Proportions

If the two or more ratio quantities encompass all of the quantities in a particular situation, for example two apples and three oranges in a fruit basket containing no other types of fruit, it could be said that "the whole" contains five parts, made up of two parts apples and three parts oranges. In this case,

, or 40% of the whole are apples and

, or 60% of the whole are oranges. This comparison of a specific quantity to "the whole" is sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.

## Reduction

Note that ratios can be

reducedIn mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible is called "reducing a fraction"...

(as fractions are) by dividing each quantity by the common factors of all the quantities. This is often called "cancelling." As for fractions, the simplest form is considered to be that in which the numbers in the ratio are the smallest possible integers.

Thus, the ratio 40:60 may be considered equivalent in meaning to the ratio 2:3 within contexts 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 formAn irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...

or lowest terms.

Sometimes it is useful to write a ratio in the form 1:

*n* or

*n*:1 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, 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 written without the 1 and the colon, though, mathematically, this makes it a

factorIn mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

or

multiplierThe term multiplier may refer to:In electrical engineering:* Binary multiplier, a digital circuit to perform rapid multiplication of two numbers in binary representation* Analog multiplier, a device that multiplies two analog signals...

.

### Dilution ratio

Ratios are often used for simple dilutions applied in chemistry and biology. A simple dilution is one in which a unit volume of a liquid material of interest is combined with an appropriate volume of a solvent liquid to achieve the desired concentration. The dilution factor is the total number of unit volumes in which your material will be dissolved. The diluted material must then be thoroughly mixed to achieve the true dilution. For example, a 1:5 dilution (verbalize as "1 to 5" dilution) entails combining 1 unit volume of solute (the material to be diluted) + 4 unit volumes (approximately) of the solvent to give 5 units of the total volume. (Some solutions and mixtures take up slightly less volume than their components.)

The dilution factor is frequently expressed using exponents: 1:5 would be 5e−1 (5

^{−1} i.e. one-fifth:one); 1:100 would be 10e−2 (10

^{−2} i.e. one hundredth:one), and so on.

There is often confusion between dilution ratio (1:n meaning 1 part solute to n parts solvent) and dilution factor (1:n+1) where the second number (n+1) represents the total volume of solute + solvent. In scientific and serial dilutions, the given ratio (or factor) often means the ratio to the final volume, not to just the solvent. The factors then 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 is normally given as a plain ratio of solvent to solute.

## 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.

## Different units

Ratios are

unitlessIn dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...

when they relate quantities which have units of the same

dimensionIn physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...

.

For example, the ratio 1 minute : 40 seconds can be reduced by changing the first value to 60 seconds. Once the units are the same, they can be omitted, and the ratio can 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 every 100mL of solution. This cannot easily be converted to a pure ratio because of density considerations, and the second figure is the

*total* amount, not the volume of

solventA solvent is a liquid, solid, or gas that dissolves another solid, liquid, or gaseous solute, resulting in a solution that is soluble in a certain volume of solvent at a specified temperature...

## See also

- Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

- Aspect ratio
The aspect ratio of a shape is the ratio of its longer dimension to its shorter dimension. It may be applied to two characteristic dimensions of a three-dimensional shape, such as the ratio of the longest and shortest axis, or for symmetrical objects that are described by just two measurements,...

- Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

- Rule of three (mathematics)
- Price/performance ratio
In economics and engineering, the price/performance ratio refers to a product's ability to deliver performance, of any sort, for its price. Generally speaking, products with a higher price/performance ratio are more desirable, excluding other factors....

## Further reading