Alligation Calculator
Solve pharmacy alligation problems: how much of a higher and a lower strength (or a diluent at 0%) to mix to make a desired percentage, with parts, mixing ratio and a check grid.
How it works
parts of higher strength = desired − lower · parts of lower strength = higher − desired · total parts = higher − lower · amount of each = total amount × its parts ÷ total parts
Alligation alternate is the pharmacy shortcut for mixing two strengths of the same ingredient to get one in between. Write the higher and lower strengths on the left, the desired strength in the middle, and subtract diagonally: the difference between the desired and the lower strength gives the parts of the higher strength, and the difference between the higher and the desired strength gives the parts of the lower one. A diluent such as water, white petrolatum or a cream base counts as 0%. It assumes the strengths are expressed the same way (all w/v, w/w or v/v) and that volumes are additive, which is close enough for dilute aqueous solutions but not exact for alcohol–water mixtures, which shrink on mixing.
Worked example
To make 500 mL of 20% alcohol from 70% alcohol and water (0%): parts of 70% = 20 − 0 = 20, parts of water = 70 − 20 = 50, total parts = 70. Use 500 × 20 ÷ 70 = 142.86 mL of 70% alcohol and 500 × 50 ÷ 70 = 357.14 mL of water, a mixing ratio of 1 : 2.50.
Sources: Wikipedia — Alligation · USP <795> Pharmaceutical Compounding — Nonsterile Preparations
Frequently asked questions
How does alligation work?
It is a weighted average solved backwards. The amount of each ingredient is inversely proportional to how far its strength is from the desired strength, so the stronger component is used in the proportion (desired − lower) and the weaker in the proportion (higher − desired).
How do I use alligation with a diluent?
Enter the diluent (water, petrolatum, cream base, normal saline as vehicle) as 0%. For example, 2.5% hydrocortisone ointment diluted with base to 1% uses 1 part of the 2.5% ointment to 1.5 parts of base.
What is the difference between alligation medial and alligation alternate?
Alligation medial finds the strength of a mixture you have already decided on (a weighted average of known amounts). Alligation alternate, which this calculator does, finds how much of each strength to mix to reach a strength you want.
Can the desired strength be outside the two strengths?
No. Mixing can only give a strength between the two you start with. To get something stronger you need a more concentrated starting material, or you must add pure drug (100%).
Does alligation work for ratio strengths?
Convert them first. A 1:1,000 solution is 0.1% w/v and 1:5,000 is 0.02%; the ratio strength calculator does this conversion.
Related calculators
Educational calculation for pharmacists, students and health professionals only — not medical advice or a prescription. Check every result against the product label, an approved compounding reference and local protocols, and have a second person verify any dose before it is given.