Alligation is a centuries-old shortcut — still taught in every pharmacy compounding course — for figuring out how much of two differently-concentrated solutions to combine to hit a target strength. This calculator automates the classic alligation-cross method so you can check compounding math, scale a recipe, or verify a homework problem instantly.
How the Alligation Calculator works
The alligation cross arranges the two source concentrations (C1 and C2) diagonally around the target (Ct), then cross-subtracts: the parts needed of C1 equal the distance from C2 down to the target, and the parts needed of C2 equal the distance from the target up to C1. The two differences form the mixing ratio, which the calculator reduces to lowest terms. This is algebraically equivalent to solving the weighted-average equation C1·x + C2·(1−x) = Ct for the mixing fraction x, but the cross method is faster to do by hand and is the version taught in pharmacy and chemistry courses.
Inputs and what they mean
Concentration 1 and Concentration 2 are the strengths of the two solutions you already have — percent, mg/mL, molarity, or any consistent unit works, as long as both inputs use the same unit. The target concentration is the strength you want to end up with, and it must land strictly between the two source values; alligation cannot produce a target outside that range, since mixing two solutions can only average their strengths. The optional total volume scales the reduced ratio into real, usable amounts on the Volumes tab — leave it blank if you only need the ratio.
Limits and edge cases
If the target concentration equals one of the two source concentrations, the ratio degenerates (all of one solution, none of the other) — the calculator flags this as an invalid combination rather than a ratio. If C1 and C2 are equal, no ratio can hit a different target, so that combination is rejected too. Alligation assumes the two components mix linearly by volume, which holds for most aqueous solutions and simple dilutions but can break down for concentrated acids, alcohols, or other mixtures with non-additive volumes — for those, consult a compounding reference or pharmacist rather than relying on the linear approximation.