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:

grams per liter.

Alligation alternate

Suppose you like 1_ps_ milk, but you have only 3_ps_ whole milk and ½_ps_ low fat milk. How much of each should you mix together to make an 8 ounce cup of 1_ps_ 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_ps_ from the desired 1_ps_, 2_ps_, is assigned to the low fat milk, and the difference of ½_ps_ from the desired 1_ps_, ½_ps_, is assigned alternately to the whole milk. The total amount, 8 ounces, is then divided by the sum to yield 16/5, and the amounts of the two ingredients are

ounces whole milk and ounces low fat milk.

This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain.

External links

• Alligation Alternate and the Composition of Medicines: Arithmetic and Medicine in Early Modern England
• Robinson's Progressive Practical Arithmetic

Source: adapted by the editor from Wikipedia, the free encyclopedia; from the article "Alligation". Image Credit.

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