In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.^[1] For example, if a bowl of fruit contains eight oranges and six lemons, 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:bquotient a/b,^[2] since the product of the quotient and the second number yields the first, as required by the above definition.

Consequently, a ratio may be considered as an ordered pair of numbers, as a fraction with the first number in the numerator and the second as denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, 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. A quotient of two quantities that are measured with different units is called a rate.^[3]

The ratio of numbers A and B can be expressed as:^[2]

The numbers A and B are sometimes called terms of the ratio with A being the antecedent and B being the consequent.^[6]

A statement expressing the equality of two ratios A∶ and C∶Dis called a proportion written as A∶ = C∶Dor A∶B*::C∶D. This latter form, when spoken or written in the English language, is often expressed as

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∶ = C∶D*=* E∶F*, is called a* continued proportion ^[2]

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

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 LHS makes up a standard proportion with the corresponding two terms on the RHS.

The corresponding terms are called the homologues in the proportion.

It is impossible to trace the origin of the concept of ratio because 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.^[9] However, it is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος (logos). Early translators rendered this into Latin 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.^[1] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios.^[11]

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

The Pythagoreans developed a theory of ratio and proportion as applied to numbers.^[12] 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 Eudoxus 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.^[13]

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 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.

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.^[14]

Book V of Euclid's Elements has 18 definitions, all of which relate to ratios.^[15] 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) 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.^[16]Encyclop%C3%A6dia%20Britannica%20Ele]]uclid defines a ratio as between two quantitiesn 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 qf there exist integers mand nso that mp>** and nq>p. This condition is known as the Archimedes 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 Euclid did not accept the existence of the quotients of incommensurate, 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 qand a rational number m/ we can say that the ratio of pto qis less than, equal to, or greater than m*/nwhen* npis less than, equal to, or greater than mqrespectively. 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*,* rand s*, then* p*:q::r:sif for any positive integers* mand n*,* np*<mq,* np*=mq,* np*>mqaccording as* nr*<ms,* nr*=ms,* nr*>ms respectively. There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers.^[17]

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 sthen p:>r:sif there are positive integers mand nso that np*>mqand* 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 rbe in proportion when p:::q:r. This is extended to 4 terms p*,* q*,* rand sas p*:q::q:r::r:sgeometric progressions. Definitions 9 and 10 apply this, saying that if p,* qand rare in proportion then p*:ris the* duplicate ratioof p*:qand if* p*,* q*,* rand sare in proportion then p*:sis the* triplicate ratioof p*:q. If* p*,* qand rare in proportion then qis called a mean proportionalgeometric mean of) pand r*. Similarly, if* p*,* q*,* rand sare in proportion then qand rare called two mean proportionals to pand s*.

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.

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 denominator, or to express them in parts per hundred (percent).

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

If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction.

For example, older televisions have a 4:3 aspect ratio

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:xor* 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 colon, though, mathematically, this makes it a factor or multiplier.

Also well known is the golden ratio of two (mostly) lengths a and b, which is defined by the proportion

Similarly, the silver ratio of a and b is defined by the proportion

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 be unitless, as in the case they relate quantities in units of the same dimension, even if their units of measurement are initially different. 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.

On the other hand, there are non-dimensionless ratios, also known as rates.^[18][19] 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 with vertices A, B, and C and sides AB, BC, and CA are often expressed in extended ratio form as triangular coordinates.

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.