About

Alligation is a centuries-old arithmetic technique for determining the ratio in which two or more ingredients at different concentrations must be combined to yield a mixture at a specified target concentration D. A miscalculation in mixing proportions leads directly to dosing errors in compounding pharmacies, off-spec batches in chemical manufacturing, or failed quality control in food production. This calculator implements both alligation alternate (finding unknown proportions) and alligation medial (finding the resultant concentration from known quantities). It enforces the fundamental constraint that the desired concentration must lie between the lowest and highest component concentrations. Results include the exact ratio, absolute quantities for a given batch size, and a verification check confirming C_result = D within rounding tolerance.

The alligation alternate method works by computing cross-differences: each component receives parts equal to the absolute difference between the other component's concentration and the target. For multi-component systems, components are grouped above and below the target and paired systematically. This tool assumes ideal mixing (no volume contraction or expansion). For solutions where density changes significantly upon mixing (e.g., ethanol-water), apply density correction factors separately. Pro tip: in pharmacy compounding, always express concentrations in the same units (w/w, w/v, or v/v) before applying alligation.

Formulas

The alligation alternate method for two components with concentrations C₁ (higher) and C₂ (lower) to achieve desired concentration D:

Parts of Component 1 = |C₂ − D| = D − C₂

Parts of Component 2 = |C₁ − D| = C₁ − D

Ratio = Parts₁ : Parts₂ = (D − C₂) : (C₁ − D)

For alligation medial, the resultant concentration from known quantities:

C_avg = n∑i=1 C_i ⋅ Q_in∑i=1 Q_i

Quantity for a desired total batch T:

Q_i = Parts_in∑i=1 Parts_i × T

Where C_i = concentration of component i, Q_i = quantity of component i, D = desired target concentration, T = total batch size, n = number of components.

Reference Data

Application	Component A	Conc. A	Component B	Conc. B	Typical Target	Unit
Pharmacy: Hydrocortisone cream	Stock cream	10%	Base cream	0%	2.5%	w/w
Pharmacy: Dextrose IV	D50W	50%	D5W	5%	12.5%	w/v
Pharmacy: Alcohol dilution	Alcohol 95%	95%	Purified water	0%	70%	v/v
Chemistry: Acid dilution	Conc. HCl	37%	Water	0%	10%	w/w
Chemistry: NaCl saline	Hypertonic saline	23.4%	Normal saline	0.9%	3%	w/v
Food: Sugar syrup	Heavy syrup	65°Brix	Water	0°Brix	25°Brix	Brix
Food: Juice blending	Concentrate	42°Brix	Water	0°Brix	11.5°Brix	Brix
Beverage: Alcohol blending	Vodka 40%	40% ABV	Juice 0%	0% ABV	12% ABV	ABV
Agriculture: Fertilizer mix	Conc. NPK	20%N	Filler	5%N	12%N	w/w
Metallurgy: Gold alloy	24K gold	999‰	Base metal	0‰	750‰	fineness
Cosmetics: Essential oil	Pure oil	100%	Carrier oil	0%	2%	v/v
Paint: Color mixing (pigment)	Full-strength tint	100%	White base	0%	15%	w/w
Lab: Buffer preparation	Stock buffer 10×	10×	DI Water	0×	1×	fold
Fuel: Ethanol blend	Ethanol E100	100%	Gasoline E0	0%	10%	v/v
Dairy: Milk fat standardization	Cream	40%	Skim milk	0.1%	3.5%	fat %

Frequently Asked Questions

The alligation alternate method requires that the desired concentration D lies between the lowest and highest component concentrations. If D < C_min or D > C_max, no positive mixing ratio exists. You would need to source a component with a concentration beyond the current range. The calculator will flag this as an error.

No. Alligation assumes ideal mixing where total volume (or mass) equals the sum of individual volumes (or masses). For systems like ethanol-water, mixing 50 mL of each yields approximately 96.4 mL, not 100 mL. In such cases, work in mass units (w/w) rather than volume units (v/v), or apply a published volume correction factor after computing the ratio.

Components are split into two groups: those with concentration above D and those below D. Each higher-concentration component is paired with each lower-concentration component. For each pair, cross-differences are computed. The parts assigned to each component are then summed across all its pairings. This yields a valid ratio, though not necessarily the unique ratio - multiple valid solutions may exist for systems with more than two components.

Yes. Alligation applies to any system where a measurable property blends linearly by proportion. Common solid applications include mixing metal alloys by fineness (e.g., gold karat blending), combining fertilizer grades by nutrient percentage, and blending powdered pharmaceuticals. Use weight-based concentrations (w/w) for solids.

Alligation alternate solves the inverse problem: given component concentrations and a target, find the mixing ratio. Alligation medial solves the forward problem: given known quantities and concentrations, find the resulting mixture concentration. Medial uses a straightforward weighted average formula: C_avg = ∑C_i⋅Q_i∑Q_i.

Assign it a concentration of 0. This is standard practice. For example, diluting a 10% stock solution with water (0%) to 2%: parts of stock = 2 − 0 = 2, parts of water = 10 − 2 = 8, ratio 2:8 or 1:4.