Alligation Calculator — Mixing Two Solutions

Formula

How this is calculated

Alligation is a classical pharmacy and chemical arithmetic method for mixing two solutions of known concentrations to obtain a desired intermediate concentration. The "alligation alternate" approach arranges the three concentrations in a cross (or tic-tac-toe) pattern: the desired concentration C_D goes in the middle, the higher concentration C_H in the upper left and lower concentration C_L in the lower left. Subtracting diagonally gives the parts: C_D − C_L is the number of parts of the higher solution, and C_H − C_D is the number of parts of the lower solution. The total parts equal C_H − C_L.

To convert parts to actual volumes, divide each number of parts by the total parts to get the fraction, then multiply by the required total volume. For example, mixing 70% alcohol with 10% alcohol to make 30% alcohol: parts of 70% = 30 − 10 = 20; parts of 10% = 70 − 30 = 40; total parts = 60. To make 1000 ml: 1000 × (20/60) = 333 ml of 70% + 1000 × (40/60) = 667 ml of 10%. You can verify the result: (0.333 × 70%) + (0.667 × 10%) = 23.3% + 6.7% = 30%. ✓

Alligation applies to any two-component mixing problem where concentration is expressed as a percentage by volume, by weight, or by mole fraction — provided both solutions are ideal (no volume contraction on mixing) and the concentrations represent the same component. For non-ideal mixtures (e.g. concentrated ethanol with water, which contracts on mixing) the actual volume will differ slightly from the calculated value, and density adjustments are required.