Alligation Calculator
Find the mixing ratio and exact component volumes needed to combine two concentrations into a target concentration.
Mixture inputs
Must be greater than the target concentration.
Must be less than the target concentration.
Must lie strictly between the higher and lower values.
Use mL, L, gal, or another unit consistently.
Live result
Mix four parts of the higher concentration with three parts of the lower concentration.
Mixture breakdown
| Component | Concentration | Ratio parts | Share | Volume |
|---|---|---|---|---|
| Higher solution | 12 | 4 | 57.14% | 350 |
| Lower solution | 5 | 3 | 42.86% | 262.5 |
| Final mixture | 9 | 7 | 100.00% | 612.5 |
The verification uses a concentration-weighted average. Concentration units must match each other, and all volume figures remain in the unit you entered.
How to use this alligation calculator
What this calculator does
This calculator determines the proportion of two solutions, mixtures, or preparations with different concentrations that must be combined to obtain an intermediate target concentration. It reports the classic alligation ratio and, when you provide a final volume, converts that ratio into the exact amount of each component. It also checks the result with a weighted-average calculation. The tool assumes the two starting materials are compatible, their concentration values use the same basis, and volumes are additive. It does not assess chemical compatibility, density contraction, clinical suitability, manufacturing tolerances, or regulatory requirements.
When to use it
Use alligation when blending a stronger and weaker stock solution, combining two ointments or creams of different strengths, preparing a laboratory mixture at an intermediate concentration, or planning a batch from two feed streams. For pharmaceutical or clinical work, treat the result as arithmetic support only and follow the applicable formulation, compounding, labeling, and checking procedures.
How to calculate
- The calculator opens with a complete demonstration: concentrations 12 and 5, target 9, and final volume 612.5. The example workbook is immediately available from Download Excel.
- Replace the four sample values with your own. Enter concentrations as plain decimal numbers using a period as the decimal separator. The three concentration entries must share one unit or basis, such as percent, mol/L, mg/mL, or w/v.
- Read the alligation ratio first, then use the component volumes and mixture breakdown to measure the batch. The volume unit is whatever consistent unit you chose for the final volume.
- Download the current calculation as a validated XLSX workbook. Reset clears the demonstration and results; the export button remains unavailable until a complete valid set is entered again.
Input guide
Higher concentration is required and must be a finite positive number greater than the target; 12 is a realistic example. Raising it generally reduces the high-strength share needed for a fixed lower and target value. Do not enter a percent sign or mix concentration bases. Lower concentration is required and must be nonnegative and below the target; the sample is 5. Lowering it changes the balance toward the higher solution. Required concentration is required and must fall strictly between the two starting values; the sample is 9. A target equal to either endpoint is not a two-component alligation problem. Required concentration volume is required and must be greater than zero; 612.5 is the demonstration value. It scales both component volumes but does not change the ratio.
Output guide
Alligation ratio (higher : lower) is the exact relative-parts relationship before optional simplification for display. Higher concentration volume and Lower concentration volume are the component amounts in your chosen volume unit. Required concentration volume repeats the final batch size as a cross-check. Verified concentration is the weighted-average concentration calculated from the displayed component volumes; it should match the target within rounding. The summary pills show the ratio and each component's percentage share. In the table, Concentration identifies each stream, Ratio parts gives proportional parts, Share gives percent of the final batch, and Volume gives the measured amount.
Worked example
For a higher concentration of 12, a lower concentration of 5, and a target of 9, the high-solution parts are 9 – 5 = 4 and the low-solution parts are 12 – 9 = 3. The ratio is therefore 4:3. The higher share is 4/7 = 57.142857%, and the lower share is 3/7 = 42.857143%. Applied to 612.5 volume units, these shares give 350 units of the higher solution and 262.5 units of the lower solution. The weighted concentration is (12 × 350 + 5 × 262.5) / 612.5 = 9, matching the first-open result exactly.
Learn more
The underlying arithmetic is a weighted mean; the NIST explanation of the weighted mean provides the mathematical context. For measurement conventions, see the NIST guidance on SI volume units. In regulated pharmacy settings, consult the FDA overview of compounding laws and policies alongside local requirements.
How the alligation formula works
Alligation converts the distance between each starting concentration and the target into opposite-side ratio parts. The number of higher-strength parts equals the target minus the lower concentration. The number of lower-strength parts equals the higher concentration minus the target. This cross-difference arrangement works because the excess concentration contributed by the stronger component must balance the deficit contributed by the weaker component.
Lower parts = higher – target
Higher volume = final volume × higher parts ÷ total parts
Lower volume = final volume × lower parts ÷ total parts
Interpretation, assumptions, and common mistakes
The ratio is independent of batch size. Doubling the required volume doubles both component volumes but leaves the ratio and shares unchanged. Moving the target toward the higher concentration increases the share of the higher-strength solution; moving it toward the lower concentration increases the share of the lower-strength solution. A target outside the interval cannot be reached by mixing only those two starting materials. The most common errors are reversing the ratio order, using concentrations expressed on different bases, typing a target equal to an endpoint, and assuming that measured volumes always add perfectly in systems where mixing causes contraction or expansion.
For laboratory work, use equipment with suitable resolution and account for significant figures, temperature, density, purity, and uncertainty. For products intended for people or animals, independent checking and applicable professional standards are essential. This calculator performs a deterministic mass-balance identity; it does not replace a validated formulation or process specification.