Ratio

A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another.

Definitions and notation

Ratios are typically unitless, as they relate quantities of the same dimension. A rate is a special kind of ratio in which the two quantities being compared are of different units. The units of a rate are the units of the first quantity "per" unit of the second — for example, a rate of speed or velocity can be expressed in "miles per hour".

Fractions and percentages are both specific applications of ratios. Fractions relate the part (the numerator) to the whole (the denominator) while percentages indicate parts per 100.

Ratio can be written as two numbers separated by a colon (<tt>:</tt>) which is read as the word "to". For example, a ratio of 2:3 ("two to three") means that the whole is made up of 2 parts of one thing and 3 parts of another — thus, the whole contains five parts in all. To be specific, if a basket contains 2 apples and 3 oranges, then the ratio of apples to oranges is 2:3. If another 2 apples and 3 oranges are added to the basket, then it will contain 4 apples and 6 oranges, resulting in a ratio of 4:6, which is equivalent to a ratio of 2:3 (thus ratios "reduce" like regular fractions). In this case, 2/5 or 40% of the fruit are apples and 3/5 or 60% are oranges in the basket.

Note that in the previous example the proportion of apples in the basket is 2/5 ("two of five" fruits, "two out of five" fruits, "two fifths" of the fruits, or 40% of the fruits). Thus a proportion compares part to whole instead of part to part.

Throughout the physical sciences, ratios of physical quantities are treated as real numbers. For example, the ratio of metres to 1 metre (say, the ratio of the circumference of a certain circle to its radius) is the real number . That is, m/1m = . Accordingly, the classical definition of measurement is the estimation of a ratio between a quantity and a unit of the same kind of quantity. (See also the article on commensurability in mathematics.)

In algebra, two quantities having a constant ratio are in a special kind of linear relationship called proportionality.

Ratios and fractions

The relationship between ratios and fractions is often a source of confusion. The source is the erroneous belief that a ratio is simply a fraction that has been turned on its side. For example, 1/2 is a ratio of 1:1 and not 1:2. A more accurate description of the relationship between ratios and fractions is that a ratio is a comparison of two fractions. Uncovering the fractions implied in a ratio is accomplished by adding the two (or more) parts of the ratio to form a common denominator for the parts.

For example:

If you have three apples for every four oranges then you have a 3:4 ratio
If you want to determine what fraction of the total fruit will be apples or oranges then you add the parts of the ratio to determine the total fruit, in this case: 3 + 4 = 7
The total fruit becomes the common denominator and the parts of the ratio become the numerators, in this case: 3/7 of the fruit is apples and 4/7 is oranges
The fractions implied in a ratio will always total one whole (or 100% of the fruit), in this case: 3/7 + 4/7 = 7/7 = 1

This also works for ratios where there are more than two quantities being compared:

4 : 6 : 5 : 20 : 56 : 2
4 + 6 + 5 + 20 + 56 + 2 = 93
4/93 and 6/93 and 5/93 and 20/93 and 56/93 and 2/93
4/93 + 6/93 + 5/93 + 20/93 + 56/93 + 2/93 = 93/93 = 1

Any two or more fractions that total one whole can also be converted into a ratio by simply finding the common denominator and then removing it:

5/8 and 6/16
5/8 and 3/8
5:3

More examples

The ratio of heights of the Eiffel Tower (300 m) and the Great Pyramid of Giza (139 m) is 300:139, so one structure is more than twice the height of the other (more precisely, 2.16 times).
The ratio of the mass of Jupiter to the mass of the Earth is approximately 318:1.
If two axles are connected by gear wheels, the number of times one axle turns for each turn of the other is known as the gear ratio, one familiar example of which is the number of turns of the pedals of a bicycle compared with number of turns of the rear wheel.
The ratio of hydrogen atoms to oxygen in water (H<sub>2</sub>O) is 2:1.
Most movie theater screens have an aspect ratio of 16:9, which means that the screen is 16/9 as wide as it is high.
In probability, the ratio of the probability of something happening to the probability of it not happening is called the odds of the thing happening.
In music, the interval of a perfect fifth is formed by two pitches, or frequencies, at a ratio of 6:4, with the higher note being 1.5 times the frequency of the lower.

The actual descriptive text portion of the article above (and its Contents menu) are licensed under the GNU Free Documentation License then in effect published by the Free Software Foundation. The text article uses material from http://en.wikipedia.org/wiki/Ratio (retrieval date: 2007-05-27) and is provided "as-is" and without warranty. Permission is granted to copy, distribute and/or modify only the actual descriptive text portion of the article above (and its Contents Box) under the GNU Free Documentation License. A copy of the license is available at: http://en.wikipedia.org/wiki/Wikipedia:Text_of_the_GNU_Free_Documentation_License. Any advertisements (or active links not embedded in the actual article) and the layout or selection thereof are excluded from such license.
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