Alligation Calculator

Comprehensive Guide to Alligation in Pharmaceutical and Chemical Calculations

In pharmaceutical compounding, chemical manufacturing, and clinical nursing mathematics, professionals regularly need to prepare mixtures of precise concentration from stock ingredients of differing strengths. When a required commercial concentration is unavailable on hospital shelves, compounding pharmacists turn to an arithmetic method known as Alligation.

Originating from the Latin alligare, meaning "to bind or tie together," alligation provides an intuitive algebraic shortcut for solving simultaneous equations involving weighted mixtures. Whether mixing high-strength alcohol solutions with purified water, blending hydrocortisone ointments of different potencies, or preparing hyperalimentation parenteral infusions in clinical settings across the United States, Canada, and the United Kingdom, mastering alligation is essential for patient safety and dosage accuracy.

The Two Core Alligation Methods: Alternate vs. Medial

Alligation is divided into two distinct computational procedures based on the starting information and objective:

1. Alligation Alternate (Finding Proportional Parts)

Alligation alternate is used when you know the concentrations of two or more starting components and must determine what proportion of each is needed to achieve an intermediate target concentration. It uses the familiar "Tic-Tac-Toe" or cross-grid layout:

Once total parts are established, each component's volume or mass is found by multiplying its fractional part by the overall target quantity. To confirm dilution ratios before compounding, you can also cross-reference steps using our Solution Dilution Calculator.

2. Alligation Medial (Finding Final Mixture Strength)

Alligation medial is applied in reverse: when given known quantities of two or more solutions of known strengths, it computes the resulting weighted average percentage strength of the combined mixture:

When working with basic volumetric solutions where pure solvents are added to liquid solutes, practitioners often check their values against our Dilution Factor Calculator.

Step-by-Step Practical Example: Compounding Ethyl Alcohol

Suppose a hospital pharmacist receives a prescription for 500 mL of 35% (v/v) Ethyl Alcohol. The pharmacy stockroom only carries 70% Ethyl Alcohol and sterile water (0% alcohol). How much 70% alcohol and how much water must be blended?

1. Set up the grid: Higher strength = 70%, Lower strength = 0%, Target = 35%.
2. Subtract diagonally:
   • Parts of 70% alcohol = |35 - 0| = 35 parts.
   • Parts of water = |70 - 35| = 35 parts.
3. Sum the parts: 35 parts + 35 parts = 70 total parts.
4. Calculate volume for each component:
   • Volume of 70% Alcohol = (35 / 70) × 500 mL = 250 mL.
   • Volume of Water = (35 / 70) × 500 mL = 250 mL.

Mixing equal parts of 70% stock and pure diluent yields the target 35% solution. When standardizing molar solutions in analytical chemistry, consider validating your base formula with our Molarity Calculator.

Ointments, Creams, and Solid Reagent Compounding

Alligation applies equally to solid and semisolid mixtures measured by weight (w/w). For example, preparing an intermediate concentration of zinc oxide ointment by combining 20% ointment with pure petrolatum diluent (0% active zinc oxide) follows the identical alligation cross framework.

Similarly, if combining 100 g of 2% hydrocortisone cream with 200 g of 0.5% cream, Alligation Medial gives:

For broader chemical proportions, compounding technicians can also verify dry powder and solvent ratios using our Mixing Ratio Calculator or compute elemental ratios with our Mass Percent Calculator. If you need to express concentrations in terms of equivalent chemical reactivity rather than raw percentage, reference our Normality Calculator.

Common Pitfalls in Compounding Calculations

• Inverting the Target Strength: The desired concentration must always fall strictly between the higher and lower stock strengths. If the desired strength is higher than all available stocks, alligation cannot be performed without adding pure active drug (100%).
• Treating Pure Diluent as Non-Zero: Pure water, simple syrup, or white petrolatum contains 0% active drug. When using pure solvents, the lower strength is entered as 0%.
• Ignoring Volume Contraction: Mixing concentrated alcohol with water causes molecular contraction due to hydrogen bonding. For precise preparation, add the calculated volume of alcohol first, then bring to the final volume graduation with water in a volumetric flask.