This alligation calculator tells you exactly how much of a high-strength and a low-strength preparation to mix to hit a target concentration — the pharmacist’s alligation cross, computed with exact volumes for your batch size.

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The Pharmacist's Cross, Five Centuries Old

Alligation is the arithmetic of blending: given two preparations of the same substance at different strengths, in what proportion do they mix to land exactly on a target strength? The medieval method — still taught in every pharmacy program — draws a cross: target in the middle, the two strengths at the left corners, and differences taken diagonally. Parts of high = target − low; parts of low = high − target. The defaults run the canonical case: 70% and 20% alcohol to make 40% → parts of 70% = 40 − 20 = 20, parts of 20% = 70 − 40 = 30, a 2:3 ratio — for 500 mL, 200 mL of the 70% plus 300 mL of the 20%, and the mixture self-verifies: (200×70 + 300×20)/500 = 40.0%. Why the diagonal differences work is weighted-average algebra in disguise — the target is the parts-weighted mean of the two strengths, and solving the mean equation for the ratio yields exactly the cross's differences (each strength is weighted by its distance from the target on the OPPOSITE side: the closer a stock sits to the target, the more of it you use — intuitive in hindsight, magical the first time the cross just works).

Scope and honest caveats. Alligation assumes the mixed quantities (volumes or masses) ADD — true by definition for mass, approximately true for volumes, with alcohol-water the famous exception: ethanol and water contract on mixing (50 mL + 50 mL gives ≈ 96 mL, not 100), so pharmacopoeias prepare alcoholic strengths to a final VOLUME ("qs to volume") rather than by summing parts, and precise work in percent w/w sidesteps the issue entirely. Keep units consistent — both strengths in the same basis (% w/w, % v/v, mg/mL, proof — for spirits, work in proof or % ABV uniformly) — because the cross blends whatever quantity the percentages measure. The method extends gracefully: diluent is simply a 0% stock (diluting 95% alcohol to 70% is alligation with low = 0), pure drug is 100%, and the same cross prices blended teas, alloys and coffee in the older "alligation medial" tradition — the technique predates algebra notation and survived because two subtractions replace a system of equations. For more than two stocks the cross generalizes to systems of mixtures, at which point linear equations (or this site's other mixture tools) take over — two-component blending is alligation's eternal home turf.

Worked Examples (Verified)

  • Defaults: 70% + 20% → 40%, 500 total → 200 of 70% + 300 of 20% (ratio 2:3).
  • Dilution as alligation: 95% + water (0%) → 70%, 1,000 total → parts 70 : 25 → 736.8 of 95% + 263.2 water.
  • Fortifying: 10% + 90% → 30% → parts 20 : 60 = 1:3.
  • Close target: 50% + 40% → 48% → parts 8 : 2 = 4:1 — the near stock dominates, as it should.

Alligation Discipline

  • The target MUST lie between the two strengths — no mixture of 20% and 70% can yield 80%; the cross returns negative parts as its warning (the calculator blocks it).
  • Same basis for both strengths (% w/w with % w/w, % v/v with % v/v) — the cross blends the units you feed it.
  • Alcohol-water volumes contract on mixing: prepare to final volume for pharmacopoeial accuracy.
  • Parts first, then scale: the ratio is size-independent; multiply parts by (total ÷ parts sum) for any batch.
  • Always run the verification mean — (Σ amount×strength) ÷ total must return the target exactly.

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Frequently Asked Questions

How does alligation work?

Write the target strength in the center, the higher strength top-left, the lower bottom-left, and subtract diagonally: parts of the HIGH strength = target − low; parts of the LOW strength = high − target. The differences cross because each stock is used in proportion to the OTHER stock's distance from the target. Scale the parts to your batch size (amount = total × parts ÷ total parts). This calculator performs the cross, the scaling, and an automatic weighted-mean verification of the result.

What is the difference between alligation and dilution?

Dilution is alligation's special case where the low "strength" is pure diluent at 0%: mixing 95% alcohol with water to reach 70% is an alligation with parts 70 : 25. Conversely, C₁V₁ = C₂V₂ (the dilution formula) solves directly for the stock volume when one side is diluent. Alligation earns its separate name when BOTH sides carry active ingredient — blending two real preparations — where the dilution formula alone doesn't apply and the cross (or the weighted-mean equation behind it) is the tool.

Can alligation mix more than two strengths?

Classical alligation is a two-stock method. With three or more stocks there are usually infinitely many blends hitting the target (the problem becomes underdetermined), and pharmacists either fix one stock's amount and alligate the remaining pair, or solve the mixture as linear equations with an added criterion (cheapest blend, use up old stock). Some textbook "alligation alternate" chains handle multi-stock cases when stocks pair off above and below the target — but for genuine multi-component formulation, linear programming on cost is the professional tool.

Why does the mixture strength formula use parts from the opposite corner?

Because of how weighted means balance. The mixture strength is (parts_high × high + parts_low × low) ÷ total parts. Setting that equal to the target and rearranging gives parts_high × (high − target) = parts_low × (target − low) — so the parts are proportional to the opposite differences: parts_high ∝ (target − low), parts_low ∝ (high − target). Intuition: the stock closer to the target can be used more freely, while the far stock acts as the strong lever used sparingly — each one's share is set by the other's distance. The algebra is the cross; the cross is the algebra.

What percentages can I use with alligation?

Any consistent concentration measure: % w/w, % v/v, % w/v, mg/mL, ppm, proof, even unit prices for blended goods (the historical "alligation medial" priced mixed teas and grains). Both inputs and the target must share the SAME measure and basis — blending a % w/w stock with a % v/v stock without density conversions is the silent error. For alcohol, remember proof = 2 × % ABV in the US system; work the whole problem in one scale.

How accurate is alligation with alcohol and water?

The RATIO math is exact; the physical volumes are not — ethanol-water mixtures contract (up to ~4% near mid-strengths) and release heat, so summing the two measured volumes slightly overshoots the final volume and the delivered strength drifts from nominal. Pharmacopoeial practice: mix the computed amounts, then bring to the final volume with diluent ("quantity sufficient to make"), or work entirely in % w/w where mass is conserved exactly. For typical beverage and lab purposes the calculator's volumes are the accepted answer; for compendial products, qs-to-volume is the standard finish.

What if my target is outside the two strengths?

Physically impossible by mixing alone — a blend always lands between its ingredients (it's a weighted average, and averages live inside the range). The cross signals this with negative parts. Your options: add a third, stronger (or weaker) component and alligate in stages; concentrate one stock first (evaporate, or fortify with pure solute = the 100% case); or dilute the whole target concept and remake at an achievable strength. The calculator refuses out-of-range targets rather than printing plausible-looking nonsense.

Is this alligation calculator free?

Yes — free, instant, no sign-up. Enter two strengths, a target and a batch size for exact amounts, the parts ratio and a built-in verification. Educational only — verify compounded preparations against your pharmacopoeia and protocols.