alligation

Alligation is a method used in pharmacy, chemistry, and related fields to determine the proportions of ingredients with different strengths that must be mixed to obtain a desired final concentration. The technique relies on comparing the available strengths to the target concentration and using the inverse differences to set the relative amounts of each source. It is commonly presented in two forms: alligation medial and alligation alternate. Alligation medial is typically used for simple mixings involving two ingredients to reach a specified concentration, while alligation alternate is employed when more than two sources are involved, using a lattice or grid and cross-differences to derive the required proportions.

In practice, the method involves listing the strengths of the available ingredients and the target strength, then calculating the differences between each strength and the target. The amounts of each ingredient are taken in proportion to these differences, with higher-strength components paired against larger deviations and lower-strength components paired against smaller deviations. This yields a simple ratio that, when applied to the total quantity to be prepared, specifies how much of each ingredient to use.

Example: To prepare 100 units of a solution at 40% from a 60% solution and a 20% solution, compute differences: 60 − 40 = 20 and 40 − 20 = 20. The required amounts are in the ratio 20:20, i.e., 1:1. Therefore, mix 50 units of the 60% solution with 50 units of the 20% solution to achieve 100 units at 40%.

Applications include pharmaceutical compounding, formulation development, and any process requiring precise concentration adjustments. Limitations include assumptions of linear additive behavior, no chemical interaction, and accurate knowledge of each source’s concentration.