# Alligation

:dablink|Alligation should not be mistaken for allegation.

Alligation is an old and practical method of solving arithmetic problems related to mixtures of ingredients. There are two types of alligation: alligation medial, used to find the quantity of a mixture given the quantities of its ingredients, and alligation alternate, used to find the amount of each ingredient needed to make a mixture of a given quantity. Alligation medial is merely a matter of finding a weighted mean. Alligation alternate is more complicated and involves organizing the ingredients into high and low pairs which are then traded off.

Examples

Alligation medial

Suppose you make a cocktail drink combination out of 1/2 Coke, 1/4 Sprite, and 1/4 orange soda. The Coke has 120 grams of sugar per liter, the Sprite has 100 grams of sugar per liter, and the orange soda has 150 grams of sugar per liter. How much sugar does the drink have? This is an example of alligation medial because you want to find the amount of sugar in the mixture given the amounts of sugar in its ingredients. The solution is just to find the weighted average by composition:

:$\left\{1over2\right\} imes 120 + \left\{1over4\right\} imes 100 + \left\{1over4\right\} imes 150 = 122.5$ grams per liter.

Alligation alternate

Suppose you like 1% milk, but you have only 3% whole milk and ½% low fat milk. How much of each should you mix together to make an 8 ounce cup of 1% milk? This is an example of alligation alternate because you want to find the amount of two ingredients to mix together to form a mixture with a given amount of fat. Since there are only two ingredients, there is only one possible way to form a pair. The difference of 3% from the desired 1%, 2%, is assigned to the low fat milk, and the difference of ½% from the desired 1%, ½%, is assigned alternately to the whole milk. The total amount, 8 ounces, is then divided by the sum $2 + \left\{1over2\right\} = \left\{5over2\right\}$ to yield 16/5, and the amounts of the two ingredients are

:$\left\{16over5\right\} imes\left\{1over2\right\} = \left\{8over5\right\}$ ounces whole milk and $\left\{16over5\right\} imes2 = \left\{32over5\right\}$ ounces low fat milk.

* [http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1172291 Alligation Alternate and the Composition of Medicines: Arithmetic and Medicine in Early Modern England]
* [http://books.google.com/books?id=vaYXAAAAIAAJ&pg=PA307&dq=Alligation&num=30&as_brr=1#PPA308,M1 Robinson's Progressive Practical Arithmetic]

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### Look at other dictionaries:

• Alligation — Al li*ga tion, n. [L. alligatio.] 1. The act of tying together or attaching by some bond, or the state of being attached. [R.] [1913 Webster] 2. (Arith.) A rule relating to the solution of questions concerning the compounding or mixing of… …   The Collaborative International Dictionary of English

• Alligation — (v. lat.), Beimischung, so v.w. Alliage. Daher Alligationsrechnung (Alligationsregel), Rechnungsregel, welche lehrt aus dem Werthe zweier zu mischenden Gegenstände u. dem Werthe der Mischung die Anzahl der, von jedem Gegenstande zu nehmenden… …   Pierer's Universal-Lexikon

• Alligation — (lat.), s. Legierungen …   Meyers Großes Konversations-Lexikon

• Alligation — (lat.), Beimischung …   Kleines Konversations-Lexikon

• Alligation — L alligation est un type de calcul pouvant entre autres être utilisé en pharmacie. Il existe plusieurs type de calculs pharmaceutiques dont l alligation médiale et l alligation alterne. Alligation médicale L alligation médicale permet de calculer …   Wikipédia en Français

• alligation — ˌalə̇ˈgāshən noun ( s) Etymology: Latin alligation , alligatio tying, band, from alligatus (past participle of alligare to tie to, from ad + ligare to tie) + ion , io ion more at ligature 1. : the action of attaching or the state of being… …   Useful english dictionary

• alligation — A rule of mixtures whereby 1) the cost of a mixture may be determined, given the proportions and prices of the several ingredients; or 2) in pharmacy, the relative amounts of solutions of different percentages which must be taken to form a… …   Medical dictionary

• Alligation — Al|li|ga|ti|on, die; , en [lat. alligatio = das Anbinden] (Fachspr.): Mischung [von Metallen]; Zusatz …   Universal-Lexikon

• Alligation — Al|li|ga|ti|on die; , ... en <aus lat. alligatio »das Anbinden, Umwicklung« zu alligare, vgl. ↑alligieren> Mischung (meist von Metallen); Zusatz …   Das große Fremdwörterbuch

• alligation — n. act of tying together or attaching by some bond …   English contemporary dictionary