Ratio

A ratio is a unitless quantity denoting an amount or magnitude of one quantity relative to another. Fractions and percentages are both specific applications of ratios. Fractions relate the part (the numerator) to the whole (the denominator) while percentages always indicate parts per hundred.

A ratio can be written as two numbers separated by a colon (:) 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 both cases, there are 2/3 as many apples as oranges in the basket, or 3/2 as many oranges as apples.

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 $2\pi$ metres to 1 metre (say, the ratio of the circumference of a certain circle to its radius) is the real number $2\pi$ . That is, $2\pi$ m/1m = $2\pi$ . 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.)

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".

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

More examples

A new grey color of paint is made by mixing 3 parts of black paint with 5 parts of white. The ratio of black to white is therefore 3:5 and the ratio of white to black 5:3. The black paint constitutes 3/8 (37.5%) of the grey paint and the white paint 5/8 (62.5%). Such a mixture of grey paint would be slightly lighter than a 2:3 mixture of black to white, since the latter is 2/5 (40%) black paint and only 3/5 (60%) white. Note that these ratios can be compared directly as regular fractions (3/5 is less than 2/3), but this method may obscure the true meaning of the ratios, as explained above.
If a school has a 20:1 student-teacher ratio, there are twenty times as many students as teachers.
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 two times 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 (H₂O) 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.

Ratio analysis

More colloquially, a ratio is a value calculated by dividing one number by another. Five divided by two gives a "ratio" of 2.5. (More accurately, this gives a ratio of 2.5:1, but this shortcut disregards the latter half of the expression in favor of simpler notation.)

In the business world it is typical to use ratios to analyze financial statements. For example, the current ratio assesses liquidity, or time required for some asset to be converted to cash. The current ratio looks at current assets relative to current liabilities.

One indicator, or ratio, for strength or stability of revenue in government is own source revenues (property taxes, for example) divided by total revenues (property tax and outside grants). In some respects, a high ratio suggests safety and stability. Grants or intergovernmental revenues can be taken away and heavy reliance on these outside sources, which would produce a low ratio, can spell trouble for a state or local government.