When alligation does not work

Alligation is a shortcut for finding the relative amounts of two ingredients with different strengths that will produce a desired intermediate strength. Its setup assumes the target is between the two starting strengths. If that condition fails, the cross-difference method cannot yield a meaningful positive mixture.

The range test comes first

Suppose you have 10% and 40% solutions. A mixture can be 20% or 30%, because those targets fall between 10% and 40%. It cannot be 5% or 50% by mixing only those two solutions: a weighted average cannot be lower than the lowest ingredient or higher than the highest ingredient. Use different starting materials or another process.

Example where it works

To make a 20% mixture from 10% and 40% ingredients, subtract diagonally: 40 − 20 = 20 parts of the 10% ingredient; 20 − 10 = 10 parts of the 40% ingredient. The ratio is 20:10, or 2:1. For 90 mL total, divide into three parts: 60 mL of the 10% ingredient and 30 mL of the 40% ingredient. Check: (0.10 × 60 + 0.40 × 30) ÷ 90 = 0.20, or 20%.

When to use another setup

Alligation does not choose the units

The ratio gives relative parts, not automatically milliliters, grams, or doses. To find actual amounts, add the ratio parts and scale them to the requested total. A 2:1 ratio with a 90 mL total means one part is 30 mL. If the problem then asks for a dose, a concentration conversion may still be needed.

Common errors

• Using alligation when the target is outside the ingredient range.
• Reversing the diagonal differences and assigning the wrong amount to each ingredient.
• Reporting parts as if they were final measurement units.
• Using a mixture shortcut without accounting for added solvent or other constraints.
• Forgetting to verify the answer with a weighted average.
• Mixing percentages, ratios, or units that do not share the same basis.

A reasonableness check catches most errors: the stronger ingredient should contribute fewer parts when the target is closer to the weaker ingredient, and the final strength must remain between the starting strengths. The PTCE outline includes alligations among calculation skills. For real medication preparation or dosing, follow the prescription and pharmacy procedure and refer uncertainties to the pharmacist.

Alligation is a shortcut for finding proportions when combining two materials of known concentrations to produce a target concentration that lies between them, under the problem’s assumptions. It is not a general method for every dilution or compounding calculation. If the target is outside the range of the starting strengths, a simple two-source blend cannot reach it without another component or process. If the ingredients are incompatible or the final volume does not behave additively, the arithmetic alone cannot establish a valid preparation.

Check whether the inputs describe compatible quantities

Before using alligation, confirm that concentrations use the same basis: for example, percent by weight with percent by weight, or mg/mL with mg/mL. Do not combine unlike units without converting first. Make sure each strength refers to the same active ingredient or compatible preparation and that the prescribed target uses that same basis. A percentage, ratio strength, and mass-per-volume concentration may look comparable but represent different units. Convert and label every quantity before calculating.

Know when a direct equation is clearer

For simple dilution with a stock solution and diluent, C1V1=C2V2 is often the direct method. For reconstitution, displacement volume, density differences, overfill, or non-linear potency, follow the product or compounding procedure instead of forcing an alligation setup. Alligation does not determine compatibility, stability, beyond-use date, sterility, or legal authorization to compound. A correct numerical ratio can still produce an unsafe product if the formulation or workflow is not approved.

Worked example: target outside the strengths

Suppose two available liquids are 10 mg/mL and 20 mg/mL, but the requested target is 25 mg/mL. A mixture of those two solutions cannot reach 25 mg/mL; blending them only produces a value between 10 and 20 mg/mL. The right response is not to reverse the alligation diagonals. Stop and ask the pharmacist whether a different stock, concentration, or preparation is intended. By contrast, a 15 mg/mL target lies between the two inputs and may be mathematically solvable if all other conditions are satisfied.

Use a final reasonableness check

After calculation, verify that proportions are positive, the target lies between input strengths, component quantities sum to the requested amount under stated assumptions, and the weighted concentration equals the target. Then independently confirm the product can be mixed and the prescription authorizes it. If the problem provides incompatible units or missing information, identify the gap rather than inventing an assumption.

A mathematically possible ratio may still be inappropriate if the ingredients are not compatible, the final preparation is unstable, or the product is not authorized for compounding. Alligation only allocates proportions; it does not account for pH, solubility, degradation, preservative concentration, container compatibility, sterility, or beyond-use dating. Those decisions require an approved formulation and pharmacist oversight.

If a calculation mixes percent strength with mg/mL, convert each concentration to a common basis before setting up the ratio. If the instructions provide a final volume after mixing or a displacement factor, include that information rather than assume volumes are additive. Show the ratio and units, then verify the result using an independent method when required. If any assumption is missing, stop and ask rather than inventing a value to make the arithmetic work.

Final workflow check

Do not use the shortcut to justify a preparation,Alligation is a way to solve a proportion, not a compounding authorization. The ingredients, formulation, route, equipment, and beyond-use date still need an approved basis. If a patient-specific preparation is required, use the pharmacy’s master formulation record and documented process. A technician may perform an assigned calculation, but the pharmacist is responsible for the professional decision to compound and verify the final preparation.