Alligation is an arithmetic method for figuring out how to mix two different concentrations of the same substance to get a desired concentration in between. If you have a strong solution and a weak solution and need something in the middle, alligation tells you exactly how much of each to combine. It’s most commonly used in pharmacy compounding, where technicians and pharmacists mix medications at precise strengths, but the technique applies anywhere you need to blend two versions of the same thing at different concentrations.
How Alligation Works
The core idea is simple: when you mix a higher-concentration solution with a lower-concentration one, the result will always fall somewhere between the two. Alligation gives you the ratio of how much of each you need to hit your target. The method uses subtraction rather than algebra, which is why it has remained popular for centuries as a shortcut for mixture problems.
Here’s the process. Say you have a 10% solution and a 4% solution, and you need to make an 8% solution. You subtract the target concentration from the higher one: 10 minus 8 equals 2. Then you subtract the lower concentration from the target: 8 minus 4 equals 4. Those two numbers, flipped across from their original solutions, become your ratio. You need 4 parts of the 10% solution for every 2 parts of the 4% solution, giving you a ratio of 2:1 (higher to lower).
The “parts” that alligation produces aren’t specific units yet. They’re proportions. To convert them into actual volumes, you divide your total desired quantity by the total number of parts, then multiply. For example, if you need 1,000 mL of a 5% solution made from 10% and 2% stocks, the alligation gives you 3 parts of the 10% and 5 parts of the 2%. That’s 8 total parts. So you’d use 1,000 times 3/8, which is 375 mL of the 10% solution, and 625 mL of the 2% solution.
Two Types: Medial and Alternate
The term “alligation” actually covers two related techniques. Alligation alternate is the one described above and by far the more commonly used. It works backward from a desired concentration to find the mixing ratio. This is the version pharmacy students learn and the one that appears on certification exams.
Alligation medial works in the opposite direction. If you already know the quantities and concentrations you’re mixing, alligation medial calculates the final concentration of the combined mixture. It’s a straightforward weighted average, less of a problem-solving tool and more of a verification step.
Why Pharmacies Still Use It
You can solve any alligation problem with algebra, using concentration-times-volume equations. But algebra requires you to set up the right formula before you start, and for mixing problems with multiple components, that setup can get complicated quickly. Alligation skips the equation entirely. You subtract, cross-multiply, and read the answer. For someone compounding prescriptions under time pressure, that speed matters.
Alligation is a tested skill on both the NAPLEX (for pharmacists) and the PTCE (for pharmacy technicians) in the United States. Pharmacy programs teach it as a core competency alongside other dosage calculations. Even with calculators and software widely available, the method remains part of the curriculum because it builds intuition about how concentrations behave when mixed, not just the ability to punch numbers into a formula.
A Surprisingly Old Technique
Alligation has roots stretching back to the proportional reasoning found in Euclid’s Elements. The first major English-language treatment appeared in 1543, when Robert Record’s arithmetic textbook included a chapter called “The Rule of Alligation” and noted it “hath great use in composition of medicines.” By the 1650s, at least three separate English arithmetic books had chapters specifically on using alligation for compounding medicines. The method has been a standard tool in pharmaceutical education for nearly 500 years.
Beyond the Pharmacy
While pharmacy is the field most associated with alligation, the same math applies to any blending problem. Mixing two grades of metal alloy, combining paints of different pigment density, adjusting the alcohol content of a beverage, or blending fuels at different octane ratings all follow the same logic. Any time you have two quantities of the same substance at different strengths and need a specific strength in between, alligation gives you the ratio in a few seconds of subtraction.