Alligation Calculator
Our mixtures & solutions calculator computes alligation accurately. Enter measurements for results with formulas and error analysis.
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The alligation cross method calculates mixing ratios by finding the difference between the desired concentration and each source concentration. The difference from the lower concentration gives parts of the higher, and vice versa.
Last reviewed: December 2025
Worked Examples
Example 1: Pharmacy Compounding
Example 2: Saline Solution
Background & Theory
The Alligation Calculator applies the following established principles and formulas. Chemistry is the science of matter's composition, structure, properties, and transformations. At the heart of quantitative chemistry lies the mole concept. One mole of any substance contains exactly 6.022×10²³ entities (Avogadro's number, Nₐ), and the molar mass of an element or compound in grams per mole is numerically equal to its atomic or molecular mass in atomic mass units. This allows chemists to convert between measurable mass and the number of reacting particles. Stoichiometry uses balanced chemical equations to relate the amounts of reactants and products. A balanced equation conserves both mass and charge. Molarity, the most common concentration unit, is defined as M = n/V, where n is moles of solute and V is volume of solution in liters, giving units of mol/L. Acidity and basicity are quantified by the pH scale, defined as pH = −log₁₀[H⁺], where [H⁺] is the molar concentration of hydrogen ions. Pure water at 25°C has pH 7.00; acids have lower values and bases higher values. Each unit change represents a tenfold change in hydrogen ion concentration. Gas behavior is described by the ideal gas law PV = nRT, where P is pressure in pascals, V is volume in cubic meters, n is moles, R = 8.314 J/(mol·K), and T is temperature in kelvin. Special cases include Boyle's Law (P₁V₁ = P₂V₂ at constant temperature) and Charles's Law (V₁/T₁ = V₂/T₂ at constant pressure). Thermochemistry quantifies heat changes in reactions through enthalpy, H. Hess's Law states that the total enthalpy change for a reaction is the sum of enthalpy changes for any sequence of steps leading to the same overall reaction, making it possible to calculate enthalpies for reactions that cannot be measured directly. Electron configuration describes the distribution of electrons in atomic orbitals according to the Aufbau principle, Pauli exclusion principle, and Hund's rule. Periodic trends including atomic radius, ionization energy, and electronegativity arise systematically from electron configuration and nuclear charge, enabling chemists to predict and rationalize chemical behavior across the periodic table.
History
The history behind the Alligation Calculator traces back through the following developments. Chemistry's roots lie in alchemy, the medieval practice combining proto-scientific experimentation with mystical aims. Alchemists developed practical techniques including distillation, calcination, and the preparation of acids, building a body of empirical knowledge despite their theoretical misunderstandings. Modern chemistry is conventionally dated to Antoine Lavoisier (1743–1794), often called the father of modern chemistry. Lavoisier demonstrated the law of conservation of mass in 1789, showing that matter is neither created nor destroyed in chemical reactions. He identified oxygen's role in combustion, dismantling the phlogiston theory, and co-authored the first systematic chemical nomenclature, establishing the language still used today. John Dalton proposed the first modern atomic theory in 1803, asserting that all matter is composed of indivisible atoms, that atoms of the same element are identical in mass, and that compounds form from fixed ratios of different atoms. This provided a physical basis for Lavoisier's conservation law and Proust's law of definite proportions. Dmitri Mendeleev published his periodic table in 1869, arranging the 63 known elements by atomic mass and revealing repeating patterns of chemical behavior. He boldly left gaps for undiscovered elements and predicted their properties with remarkable accuracy, predictions confirmed by the subsequent discovery of gallium, scandium, and germanium. Ernest Rutherford's gold foil experiment in 1911 revealed the nuclear model of the atom: a tiny, dense, positively charged nucleus surrounded by electrons. Niels Bohr refined this in 1913 with a quantized model of electron orbits that explained the hydrogen emission spectrum. Quantum chemistry and molecular orbital theory, developed through the 1920s and 1930s, provided the full quantum mechanical description of chemical bonding. The latter 20th century saw the rise of computational chemistry, enabling molecular simulation at unprecedented scale. The green chemistry movement, articulated in the 12 Principles of Green Chemistry in 1998, reoriented the field toward sustainability, waste reduction, and benign chemical design, reflecting chemistry's growing awareness of its environmental responsibilities.
Frequently Asked Questions
Formula
Parts of higher = desired - lower | Parts of lower = higher - desired
The alligation cross method calculates mixing ratios by finding the difference between the desired concentration and each source concentration. The difference from the lower concentration gives parts of the higher, and vice versa.
Frequently Asked Questions
What is the alligation method in chemistry and pharmacy?
Alligation is a mathematical method used to determine the proportions in which two solutions of different concentrations must be mixed to obtain a desired intermediate concentration. It is widely used in pharmacy for compounding medications, in chemistry for preparing solutions, and in food science for blending ingredients. The method works by creating a simple cross-multiplication diagram where the desired concentration is placed in the center, and the differences from each source concentration give the mixing ratio.
How does the alligation cross method work?
The alligation cross (or tic-tac-toe method) arranges the higher concentration in the top-left and the lower concentration in the bottom-left, with the desired concentration in the center. You subtract diagonally: desired minus lower gives parts of the higher concentration, and higher minus desired gives parts of the lower concentration. The resulting numbers form the mixing ratio. For example, mixing 70% and 30% solutions to get 50% yields a ratio of (50-30):(70-50) = 20:20 = 1:1.
When should I use alligation versus the dilution equation?
Use alligation when mixing two solutions of different known concentrations to achieve a target concentration. Use the simple dilution equation (C1V1 = C2V2) when diluting a single concentrated solution with pure solvent (effectively 0% concentration). Alligation is more general because it handles mixing any two concentrations, while the dilution equation is a special case of alligation where the lower concentration is zero. Both methods are fundamental tools in pharmaceutical compounding and laboratory preparation.
Can alligation be used for more than two components?
Standard alligation (alligation alternate) handles two components at a time. For three or more components, you can use alligation medial to find the resulting concentration when mixing known quantities, or apply alligation alternate iteratively by grouping components. In practice, multi-component mixing problems are often solved using systems of linear equations rather than the cross method. However, the two-component alligation remains the most commonly used version in pharmacy and chemistry education.
Can I use Alligation Calculator on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
How do I verify Alligation Calculator's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy