Alligation (alternate)

Determine the proportions of a stronger and a weaker preparation needed to produce a desired intermediate strength.

What is this for?

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.

How to use it

Enter the strengths of the two preparations you are mixing. If one is a plain diluent, enter 0 for the lower strength.
Enter the strength you need. It must lie between the two — alligation cannot produce a strength outside the bracketing pair.
Optionally enter the total quantity you want to prepare, and the parts will be converted into grams or millilitres.

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%.

Clinical pearls & pitfalls

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.
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.
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.

Assumptions & limitations

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.