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. Alligation alternate provides answers when an algebraic solution (e.g., using simultaneous equations) is not possible (e.g., you have three variables but only two equations). Note that in this class of problem, there may be multiple feasible answers.
Two further variations on Alligation occur : Alligation Partial and Alligation Total (see John King's Arithmetic Book 1795 which includes worked examples.) The technique is not used in schools although it is used still in pharmacies for quick calculation of quantities.