chemistry

Alligation Calculator

Alligation is a foundational technique in pharmacy, chemistry, and medicine for determining the exact proportions of two solutions needed to produce a target concentration. Rather than diluting a single solution with water, alligation blends two stock solutions of known strengths to achieve a precise intermediate concentration. This method is critical in pharmaceutical compounding, where accuracy directly impacts patient safety and product efficacy.

Last updated: May 3, 2026

Creators Hanna Pamuła, PhD

Reviewers Davide Borchia and Dominik Czernia

2,462 people find this calculator helpful

Understanding Alligation

Alligation is a mathematical procedure for determining the ratio in which two solutions of differing concentrations must be combined to create a solution of an intermediate concentration. Unlike dilution—which weakens a concentrated solution by adding solvent—alligation mixes two existing solutions without introducing extra diluent.

The method is especially valuable in pharmaceutical contexts, where you often have access to two standard preparations and need to prepare an intermediate strength. For instance, if a pharmacy stocks 25% and 5% hydrocortisone creams, alligation provides the exact proportion to blend them into a 15% cream.

The core principle rests on a balance equation: the difference between the higher concentration and the target equals the parts of the lower solution, while the difference between the target and the lower concentration equals the parts of the higher solution.

Alligation Ratio and Volume Formulas

The alligation method uses the following relationships, where H represents the higher concentration, L the lower concentration, and R the required (target) concentration:

Ratio of higher solution = R − L

Ratio of lower solution = H − R

Final ratio = (R − L) : (H − R)

Volume of higher = [(R − L) / (H − L)] × V_required

Volume of lower = [(H − R) / (H − L)] × V_required

H — Higher concentration of the stock solution
L — Lower concentration of the stock solution
R — Required (target) concentration of the final mixture
Vrequired — Desired total volume of the final solution

Alligation in Pharmaceutical Practice

Pharmaceutical compounding demands precision because even small deviations in concentration can affect therapeutic outcomes. Alligation simplifies the task of preparing intermediate-strength preparations when only two standard concentrations are available.

Consider a practical example: a compounding pharmacist needs to prepare 500 mL of a 12% antibiotic suspension. Stock solutions available are 18% and 6%. Using alligation:

Ratio calculation: (12 − 6) : (18 − 12) = 6 : 6 = 1 : 1
Equal parts are needed, so 250 mL of each stock solution
Mixing produces exactly 500 mL at 12% concentration

This approach eliminates guesswork and ensures compliance with pharmaceutical standards. The method scales to any volume, making it indispensable for batch production and individual patient doses alike.

Alligation versus Dilution

Although both methods adjust solution concentration, alligation and dilution are fundamentally different processes.

Dilution weakens a solution by adding solvent (usually water). A pharmacist starting with 40% dextrose and adding water can produce 20% dextrose, but only weaker concentrations are possible.

Alligation blends two existing solutions without adding solvent, allowing creation of intermediate concentrations. Mixing 40% and 10% solutions can yield 25%, 15%, or any concentration between the two extremes.

For pharmacy and industry, alligation is preferred when you must match a target strength precisely and have multiple stock solutions available. Dilution is simpler when you only have one concentrate and wish to weaken it.

Common Pitfalls and Practical Tips

Avoid these frequent mistakes when applying alligation in the laboratory or pharmacy.

Reversing the ratio — The most common error is calculating which concentration contributes which parts. Remember: the higher concentration's contribution equals (Required − Lower), not the reverse. Double-check by ensuring the two calculated parts sum correctly.
Neglecting volume rounding — When the alligation ratio doesn't divide evenly into your total volume, you'll get decimal milliliters. Always round to the nearest 0.1 mL (or your balance's precision) and verify the final volume adds up by re-measuring the combined mixture.
Confusing percentage with molarity — Alligation works identically for percentage concentration (%), molarity (M), or any other unit—as long as you use the same unit throughout. Mixing a 20% solution with a 0.5 M solution using alligation will give nonsensical results.
Overlooking solution incompatibility — Mathematically sound alligation may produce an unusable mixture if the two solutions are chemically incompatible (e.g., mixing aqueous and oil-based formulations). Always verify chemical compatibility before blending.

Frequently Asked Questions

What ratio of 20% and 8% solutions produces 14%?

Using alligation, the ratio is (14 − 8) : (20 − 14) = 6 : 6, or 1 : 1. Equal volumes of the two solutions must be mixed. If you need 200 mL final volume, combine 100 mL of the 20% solution with 100 mL of the 8% solution to obtain 200 mL at 14% concentration.

How do I calculate the actual volumes needed for a total batch?

Once you know the alligation ratio, divide the total desired volume by the sum of ratio parts. For a 1 : 1 ratio and 500 mL total, divide 500 by 2, yielding 250 mL of each. For a 3 : 2 ratio and 500 mL total, divide by 5 (3 + 2), then multiply: 500 ÷ 5 × 3 = 300 mL (higher) and 500 ÷ 5 × 2 = 200 mL (lower).

Can alligation be used for solids or only liquids?

Alligation applies to any mixture in which concentration is measurable—solutions, suspensions, emulsions, or even solids. A pharmacy might use alligation to blend two powders of different strengths (e.g., 30% and 5% active ingredient) in the correct proportion. The mathematical principle is identical; only the unit of measurement and physical handling differ.

Why can't I just mix the two solutions 50-50?

Mixing 50-50 works only if you want the numerical average of the two concentrations, and only when the two starting concentrations are equidistant from the target. If you need 14% from 20% and 8%, a 50-50 mix gives (20 + 8) ÷ 2 = 14%—which happens to work here. But mixing 25% and 5% 50-50 yields 15%, not every intermediate concentration. Alligation calculates the exact ratio for any target.

What happens if my target concentration is outside the range of the two stock solutions?

Alligation cannot produce a concentration outside the range bounded by the two stock solutions. If your stocks are 10% and 30%, you cannot make 40% or 5% by mixing them alone. You would instead need dilution (to go lower) or access to a higher-concentration stock. Always verify that your target falls between the two available concentrations before applying alligation.

How does alligation differ from a weighted average?

Alligation and weighted average look similar but serve different purposes. A weighted average combines values with assigned weights (e.g., exam scores with different credit hours). Alligation calculates the ratio of volumes (or masses) needed such that the concentration of the final mixture reaches a target. The alligation ratio is the weight, and it's determined by the concentration differences, not assigned beforehand.

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