How to Use Alligation for Pharmacy Compounding
The alligation alternate method for mixing two preparations of known strength to a required strength in between, with a full worked example.
The problem alligation solves
Alligation alternate solves the mixing problem: given two preparations of known strength, in what ratio must they be combined to give a required strength in between? The answer is read diagonally — the parts of the strong preparation equal the difference between the desired and weak strengths, and vice versa.
Setting up the diagonal
Place the desired strength between the two available strengths. Subtracting diagonally gives the proportion of each: the parts of the higher-strength preparation equal the difference between the desired strength and the lower one, and the parts of the lower-strength preparation equal the difference between the higher strength and the desired one.
Worked example
You stock 20% and 5% benzoyl peroxide gels. Prepare 240 g of an 8% gel.
Answer: 3 parts of 20% to 12 parts of 5%, a total of 15 parts. For 240 g: (3/15) × 240 = 48 g of 20% and (12/15) × 240 = 192 g of 5%.
Special cases
- A pure diluent — petrolatum, water, lactose — is 0%. Treating it as anything else is a frequent setup error.
- A pure active ingredient is 100%. Adding pure drug to strengthen a preparation is an alligation problem with the high strength set to 100.
Checking your answer
Always verify: multiply each quantity by its strength, add them, and check the total equals the desired strength times the total quantity. The result panel does this for you on the "check" line.
Common setup errors
- When the desired strength is very close to one of the two available strengths, the calculated quantity of the other becomes small — check that it exceeds your balance's minimum weighable quantity.
- Assumes the two preparations are miscible and that the strengths are expressed in the same way (both w/w, or both w/v). Mixing a w/w with a w/v strength gives a wrong answer.
When alligation does not apply
- Handles two components only. Three-component problems require alligation medial or a system of simultaneous equations.
- Assumes the two preparations are miscible and that the strengths are expressed in the same way (both w/w, or both w/v). Mixing a w/w with a w/v strength gives a wrong answer.
- Assumes additive quantities. For semi-solids this is reliable; for alcohol-water systems it is not.
References
- United States Pharmacopeia. General Chapter <795> Pharmaceutical Compounding — Nonsterile Preparations.
- Allen LV. The Art, Science, and Technology of Pharmaceutical Compounding. American Pharmacists Association.