The alligation calculator above works the two questions the alligation method answers. Given two preparations of known percentage strength and a target strength that lies between them, it returns the parts and the actual quantities of each that blend to the target. Run in reverse, it takes two quantities you have already decided on and returns the strength of the blend they produce. Both are pure mass-balance arithmetic, and both are shown with the intermediate parts so the working is visible rather than hidden inside a single number.
Arb Digital publishes free calculators for the steps where the method matters as much as the answer. Alligation is a case in point: it is a piece of proportional reasoning several centuries old, and the reason it survives is that its crossed layout makes a weighted-average problem solvable on paper in seconds. This page keeps the parts visible for exactly that reason. It is a teaching and method tool, and it works in percentage strengths only.
What This Alligation Calculator Does
In alligation alternate mode you enter a stronger strength, a weaker strength, a target strength and the total quantity you need. The page returns how many parts of each go into the mix, the actual quantity of each in your chosen unit, the proportion of the blend that comes from the stronger preparation, and a verification of the strength the blend actually reaches. That last figure is a check, not a repeat of your input: it recomputes the strength from the two quantities the tool just produced, so a mismatch would be visible.
In alligation medial mode you enter the two strengths and the two quantities, and the page returns the resulting strength. This is the direction you use when the quantities are fixed for a practical reason — you have a part-used container, or you are working to a container size — and you want to know what strength that combination lands on rather than what quantities a target demands.
The unit selector is a label only. Alligation is dimensionless: the same parts apply whether you are working in grams, millilitres, kilograms or litres, provided both preparations are measured on the same basis and the strengths are expressed on that same basis too. That proviso does more work than it looks like it does, and there is a section on it below.
How to Use It
- Pick the direction. Alternate if you know the target and need quantities; medial if you know the quantities and need the strength.
- Enter the two known strengths as percentages, higher one first. A pure diluent is entered as 0 percent, and an undiluted concentrate as 100.
- Enter the target in alternate mode. If it sits outside the two strengths the page will say so rather than return a nonsensical negative quantity.
- Enter the total quantity you need and choose its unit, or in medial mode enter the two quantities you are actually combining.
- Read the two quantities and check the verified strength. If the verification does not equal your target to within rounding, one of the inputs is on a different basis from the others.
The Formula and How It Is Calculated
Alligation alternate is a rearranged weighted average. Write the stronger strength as H, the weaker as L and the target as T. The parts of the stronger preparation are T − L, and the parts of the weaker are H − T. The crossing is what gives the method its name and its traditional layout: each strength is paired with the difference on the opposite side of the target. Total parts are H − L, so the quantity of the stronger from a total Q is Q(T − L)/(H − L), and the quantity of the weaker is Q(H − T)/(H − L).
Working the default: with H = 70 percent, L = 20 percent and T = 40 percent, the parts of the stronger are 40 − 20 = 20 and the parts of the weaker are 70 − 40 = 30, a ratio of 2 to 3. For a total of 500 g that is 500 × 20/50 = 200 g of the stronger and 500 × 30/50 = 300 g of the weaker. Checking by mass balance: 200 × 0.70 = 140 units of active plus 300 × 0.20 = 60 units gives 200 units in 500 g, which is 40 percent. The verification line reproduces exactly that check.
Alligation medial runs the same balance forwards: the blend strength is (QHH + QLL)/(QH + QL). It is a quantity-weighted mean of the two strengths and nothing more. The percentage conventions all of this rests on — by mass, by volume, and mass in volume — are set out in the reference on units of concentration.
Alternate and Medial Answer Different Questions
The two directions are not simply the same calculation run backwards with the same reliability, and it is worth understanding why. Alligation alternate has exactly one solution for a given target: the parts ratio is fixed by the three strengths, and the total quantity merely scales it. There is no room for choice, which is what makes it useful when a target is specified and must be met.
Alligation medial has no target at all. It reports where an arbitrary combination lands. That makes it the right tool when the quantities are constrained by something outside the arithmetic, and it also makes it the honest way to check a blend that was made up by eye or by container size rather than by calculation. Running medial on what was actually combined, rather than on what was intended, is how a discrepancy gets caught.
There is a third case the traditional method handles that this page deliberately does not: alligation with more than two components, where several stocks are paired off against the target in combinations. That problem is underdetermined — there are usually infinitely many valid answers — and a calculator that picked one of them without saying which pairing it had chosen would be misleading. Two components have one answer, so that is what this page computes.
Why the Target Must Sit Between the Two Strengths
If the target is above the stronger preparation, one of the parts comes out negative, and a negative quantity of a preparation is not a thing you can weigh. The arithmetic does not fail; it produces an answer that means nothing. This page detects the condition and says so in words rather than printing the negative number, because the negative number looks plausible enough to be acted on.
The underlying reason is that blending is an averaging operation. A weighted mean of two values always lies between them, at the endpoints when one weight is zero and strictly between otherwise. No combination of a 70 percent preparation and a 20 percent preparation reaches 75 percent, in the same way that no average of two exam marks exceeds the higher of them. To go above the stronger preparation you have to add more of the active component or remove solvent, which is concentration rather than blending, and a different calculation entirely.
Two boundary cases are legitimate and the page handles them without complaint. A target equal to the weaker strength gives zero parts of the stronger, meaning the answer is to use the weaker preparation alone. A target equal to the stronger strength gives zero parts of the weaker. Both are correct, and both are worth seeing rather than being treated as errors, because they tell you the blend was unnecessary.
Percentage Strength Is Not a Single Thing
The most consequential assumption on this page is that both strengths are on the same basis, and percentage strength has at least three common bases. Percent weight in weight is grams of active per hundred grams of preparation. Percent weight in volume is grams per hundred millilitres. Percent volume in volume is millilitres per hundred millilitres. They are numerically different for the same preparation, and mixing bases inside one alligation produces an answer that is wrong without looking wrong.
The trap is that the difference is small in dilute aqueous preparations, because a millilitre of dilute aqueous solution weighs close to a gram. It stops being small as soon as density departs from one, which it does for anything containing a significant proportion of a solvent other than water, for anything viscous, and for anything concentrated. Blending two preparations of different density on a volume basis also carries a second problem: volumes are not strictly additive, so the finished volume can be slightly less than the sum of the parts.
The safe practice is to convert everything to one basis before the alligation rather than after it. Where a density is available, the concentration converter handles moving between mass and volume bases, and the percentage calculator covers the plain proportional arithmetic. If you need molar rather than percentage terms, the molarity calculator is the bridge.
Where Alligation Stops and Dilution Begins
Alligation and dilution overlap at exactly one point and are otherwise different jobs. The solution dilution calculator takes one stock and adds pure diluent to reach a lower concentration, working from the relationship C₁V₁ = C₂V₂. This page blends two preparations that both contain active material. When the weaker preparation is entered as 0 percent, alligation reduces exactly to that dilution problem, and the two tools agree.
Beyond that point they diverge. Dilution cannot use up a part-strength preparation you already have on the shelf; alligation is precisely how you do. Alligation cannot handle a chain of successive reductions across several orders of magnitude, where accumulated pipetting error makes a single large step unreliable; that is what the serial dilution calculator exists for. And where a preparation is described as a ratio strength such as one in a thousand rather than as a percentage, the dilution ratio calculator converts the notation before the alligation begins.
Compounding a medicine is a regulated activity performed by a licensed pharmacist working to a verified formula, under the standards for compounding quality published by the United States Pharmacopeia. This page computes the proportional arithmetic that underlies the alligation method and nothing beyond it.
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Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Pairing each strength with its own difference — the parts cross over, so the stronger preparation takes the difference between the target and the weaker strength.
- Mixing percentage bases — a weight-in-weight strength and a weight-in-volume strength cannot be alligated against each other until one has been converted.
- Setting a target outside the two strengths — blending is an average, so it can never exceed the stronger component or fall below the weaker.
- Treating parts as quantities — parts are a ratio and mean nothing until they are scaled by the total quantity you actually need.
- Assuming volumes add exactly — combining two liquids of different composition can give slightly less than the arithmetic sum, which matters when working to a fixed final volume.
Related Free Tools From Arb Digital
For diluting a single stock with pure diluent, use the solution dilution calculator, and for a chain of reductions the serial dilution calculator. The dilution ratio calculator converts ratio strengths into percentages before you start, the ratio calculator simplifies and scales the parts, and the weighted average calculator handles the general case that alligation medial is a two-item example of. For molar rather than percentage work, start at the molarity calculator. The full free online tools hub lists everything else.
Frequently Asked Questions
It is a proportional technique for finding how much of a stronger and a weaker preparation to combine to reach a target strength that lies between them. The parts of each are the crossed differences between the target and the opposite strength.
Alternate starts from a target strength and returns the quantities needed. Medial starts from two known quantities and returns the strength the blend reaches. Alternate has one answer; medial simply reports where a chosen combination lands.
Because the amount of the stronger preparation is set by how far the target sits above the weaker one. The larger that gap, the more of the stronger preparation is needed, so the stronger takes the target minus the weaker strength as its parts.
No. Blending produces a weighted average of the two strengths, and an average always falls between them. Reaching a higher strength requires adding more active component or removing solvent, which is a concentration step rather than a blend.
Either, as long as both preparations and both strengths are on the same basis throughout. The parts themselves are dimensionless, so the unit only enters when the parts are scaled to a total quantity.
Enter it as 0 percent. Alligation then reduces exactly to a simple dilution, and the result agrees with the dilution calculation from a stock concentration and a final volume.
The traditional method can, by pairing stocks against the target, but the problem then has many valid answers rather than one. This page handles two components, where the answer is unique.
This calculator is provided for education and general reference. It computes the proportional arithmetic of the alligation method only, and is not pharmaceutical, compounding, dosing or clinical guidance. Compounding a medicine is carried out by a licensed pharmacist working to a verified formula under the applicable standards.
