ratio

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The ratio of width to height of typical computer displays.

A ratio is an expression that compares quantities relative to each other. The most common examples involve two quantities, but any number of quantities can be compared. Ratios are represented mathematically by separating each quantity with a colon – for example, the ratio 2:3, which is read as the ratio "two to three". The quantities separated by colons are sometimes called terms.

History and etymology

Look up ratio in Wiktionary, the free dictionary.

The word ratio comes from the Ancient Greek λόγος (logos) appearing in Book V of Euclid's Elements. Early translators rendered this into Latin as ratio, meaning "reason". However 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.^[2]

Euclid collected the results appearing the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.^[3] 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. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.^[4]

Examples

The quantities being compared in a ratio might be physical quantities such as speed or temperature, or may simply refer to amounts of particular objects. A common example of the latter 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 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.

Ratios are often used for simple dilutions applied in 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 diluent (the material to be diluted) + 4 unit volumes of the solvent medium (hence, 1 + 4 = 5 = dilution factor). The dilution factor is frequently expressed using exponents: 1:5 would be 5e-1; 1:100 would be 10e-2, and so on.

Special notice should be made to ratios (usually) comparing one item to another item, as opposed to the whole. If there are 2 oranges and 3 apples, the ratio of oranges to apples is shown as 2:3 or 2/3, however the fraction of oranges to total fruit is 2/5.

Number of terms

In general, a ratio of 2:3 means that the amount of the first quantity is (two thirds) of the amount of the second quantity. This pattern 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. 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".

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 reduced as fractions are, by dividing each quantity by the common factors of all the quantities. Thus the ratio 4:6 may be considered equivalent in meaning to the ratio 2:3 within contexts concerned only with relative quantities.

Different units

Ratios are unit-less when they relate quantities which have the same units. When the two quantities being compared are of different types, the units are the first quantity "per" unit of the second – for example, a speed or velocity can be expressed in "miles per hour". A ratio for which the second unit is a measure of time is called a rate.

Ratios are used frequently throughout the physical sciences, and in many cases a ratio is thought of as a single value. For example, the ratio 60 metres to 1 second, or 60:1 is written as 60 m/s, or 60 ms⁻¹, "60 metres per second" and is thought of as a measurement of velocity. In this case, the measurement is actually a ratio between two quantities with different units.

Direct proportion

In algebra, two variable quantities having a constant ratio are in a special kind of relationship called direct proportion.

References

1. ^ Penny Cyclopedia, p. 307
2. ^ Smith, p. 478
3. ^ Heath, p. 112
4. ^ Heath, p. 113

• "Ratio" The Penny Cyclopædia vol. 19, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London pp. 307ff
• The thirteen books of Euclid's Elements, vol 2. trans. Sir Thomas Little Heath (1908). Cambridge Univ. Press. pp. 112ff. http://books.google.com/books?id=lxkPAAAAIAAJ&pg=RA3-PA112. 
• D.E. Smith, History of Mathematics, vol 2 Dover (1958) pp. 477ff

External links

• Online Etymology Dictionary[[