Alligation Calculator

Alligation is a practical mathematical shortcut derived from simultaneous equations, used primarily in pharmacy and chemistry. The method relies on finding the proportional parts of a higher concentration solution and a lower concentration solution needed to reach a target required concentration. The core formula determines the relative parts of each solution by calculating the absolute difference between the desired concentration and the starting concentrations. Specifically, the part of the higher concentration solution is proportional to the difference between the required concentration and the lower concentration (|Required - Lower|), while the part of the lower concentration solution is proportional to the difference between the higher concentration and the required concentration (|Higher - Required|). These ratios can then be scaled up or down to find exact required volumes for any batch size.

Imagine you need to prepare 120 milliliters of an 8 Molar solution using a stock high concentration of 13 M and a stock low concentration of 4 M. First, find the parts for the higher concentration by subtracting the lower concentration from the required concentration: 8 - 4 = 4 parts. Next, find the parts for the lower concentration by subtracting the required concentration from the higher concentration: 13 - 8 = 5 parts. The total number of parts is 4 + 5 = 9 parts. To find the exact volume of the 13 M solution needed, multiply the total required volume (120 mL) by the fraction of high parts: (4 / 9) * 120 mL = 53.33 mL. For the low concentration solution, calculate (5 / 9) * 120 mL = 66.67 mL. Mixing 53.33 mL of the 13 M solution with 66.67 mL of the 4 M solution yields precisely 120 mL of your target 8 M solution.

Always ensure that all concentration units match before beginning your calculation; for instance, do not mix percentages with molarity or volume-per-volume units unless properly converted. Double-check that your desired target concentration lies strictly between your high and low stock concentrations, as outside values are mathematically impossible to blend using standard two-component alligation. When measuring precise liquid volumes in a laboratory setting, remember to account for volumetric contraction or expansion that can occasionally occur when mixing concentrated acids or polar solvents.