Ratio

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For the use of ratio as a human capacity, see reason.

In number and more generally in algebra, a ratio is the linear relationship between two quantities. It is expressed as two numbers separated by a colon (pronounced "to"). The ratio of 2:3 means 2 parts of one to 3 parts of the other. This means 2 fifths of the mixture is of the one and 3 fifths is of the other since there are 5 parts in total. A rate is a special kind of ratio where the two quantities being compared are of a different unit.

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Examples

  • A new grey colour of paint is made up by mixing 2 parts black to 8 parts of white. This ratio is 2:8 or 1:4 in its simplest form. Two tenths or 1 fifth of the grey paint is made up of black paint. If a larger quantity of mixture was required this ratio must be kept the same for the same grey colour to be produced.

This concept produces problems like "Which is a darker grey? A 2:5 black:white mixture or a 3:7 mixture?"

  • If a school has a twenty-to-one student-teacher ratio, that means that there are twenty times as many students as teachers.
  • The ratio of heights of the Eiffel Tower (300 m) and the Great Pyramid (137 m) is 300:137, so one structure is more than twice the height of the other (or more accurately, 2.2 times).
  • The ratio of the mass of Jupiter to the mass of the Earth is approximately 317.8:1.
  • The musical interval of a perfect fifth, the pitch ratio 3:2, consists of two pitches, one approximately 1.5 times the frequency of another.
  • 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. The best example being the number of turns of the pedals of a bicycle compared with number of turns of the bicycle's rear wheel.
  • The ratio of hydrogen atoms to oxygen in water is 2:1, or two parts to one.

Note the use of words such as "times", "parts", "number", etc. Because two objects are being compared using the same measure, ratios are unitless; the units cancel out of the ratio. For example, the ingredients in a recipe that required 500 grams and 300 grams of each, would be in the ratio of 5:3, with no units.

Note also the difference between ratios and vulgar fractions. For example, if there are three raspberry candies and five blackcurrant candies, then the ratio of raspberry candies to blackcurrant candies is 3:5.

This indicates that there are three fifths as many raspberry candies as blackcurrant candies. This compares the two groups of different candies as separate entities. The problem is really about a single group of candies, some of which are different. To say there are "three fifths as many raspberry candies as blackcurrant candies" is not a very lucid way of looking at the problem. A better way follows.

Three eighths of the candies are raspberry which is less than half. Five eighths are blackcurrant.

However the fraction of all the candies that are raspberry is three out of a total of all eight candies or 3/(3+5) = 3/8. Thus the chances of a randomly selected candy being raspberry are three in eight.

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See also

de:Quotient et:Jagatis es:Cociente he:יחס (בין מספרים) ja:比 lt:Santykis sv:Kvot vi:Tỷ lệ

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