Calculate the required proportions of two different concentrations to achieve a specific target concentration.
Last updated: March 2026
Alligation is a mathematical method used to calculate the proportions of two substances with different concentrations that must be mixed to obtain a third substance of a desired intermediate concentration. It is widely used in pharmacy, chemistry, brewing, and food manufacturing for precision blending.
The "Alligation Alternate" method (also known as the Tic-Tac-Toe method) provides a visual way to calculate these ratios without complex algebraic equations. It tells you how many "parts" of each concentration you need to combine. This method has been used for centuries and remains the fastest way to solve mixture problems by hand.
The alligation method is a visual shortcut derived from a weighted average equation. When mixing two concentrations A and B to get target C, the amount each contributes must balance:
(A × VA) + (B × VB) = C × (VA + VB)
Rearranging this equation shows that the ratio of volumes is inversely proportional to the difference from target. The cross method captures this: the "high" solution contributes in proportion to how far the "low" is from target—and vice versa. This diagonal subtraction eliminates algebraic manipulation while preserving mathematical correctness.
Mix 70% alcohol and 20% alcohol to get 500ml of 40% alcohol:
High Volume: (20/50) × 500 = 200 ml
Low Volume: (30/50) × 500 = 300 ml
Final Answer: Mix 200 ml of 70% with 300 ml of 20%.
The standard cross method is for two. For three or more, you must pair them up (one higher than target, one lower) and solve in stages, or use a system of linear equations.
Yes. Alligation works for any concentration by weight or volume, provided the units are consistent (e.g., all w/w or all v/v).
Simply use 0% as your Low Concentration. This is essentially the same as the standard dilution formula (C1V1=C2V2).
While computers handle most calculations now, Alligation remains a core skill taught to pharmacists for manual compounding and as a quick mental check for accuracy.
You cannot mix two concentrations to create a target outside their range—it violates mass balance. If your target is higher than both inputs, you need a more concentrated source.
If both inputs have the same concentration, mixing them produces that same concentration—no calculation needed. The alligation method requires two different starting values.
For standard percentage concentrations (w/w or v/v), values over 100% are physically impossible. Use decimal ratios or different units if working with supersaturated solutions.
Parts give you the ratio independent of total volume, making the method universal. You can scale the result to any batch size by multiplying parts by your desired total.
Yes - the cross method IS the quick approach. But the mental shortcut: to move halfway between two concentrations, you need roughly equal parts. To move closer to the high concentration, use more high and less low.
Plug your volumes back into the weighted average formula: (High percent x High Vol + Low percent x Low Vol) / Total Vol should equal your target percent. If it doesn't, recalculate the cross method.