Alligation: mix two strengths and check the mass balance

For a fictional 7% w/w target from 20% and 2% w/w starting preparations, high-strength parts are 7 - 2 = 5. Low-strength parts are 20 - 7 = 13. The ratio is therefore 5 parts high-strength to 13 parts low-strength, not the other way around.

There are 18 total parts. To calculate a 360 g final mixture, each part represents 20 g. You need 100 g of the 20% preparation and 260 g of the 2% preparation. Most of the mixture uses the weaker preparation because 7% is much closer to 2% than to 20%.

The strong preparation contributes 100 g × 0.20 = 20 g ingredient. The weak preparation contributes 260 g × 0.02 = 5.2 g. Together, 25.2 g ingredient in 360 g mixture is 7% w/w. That check catches a reversed ratio immediately.

You can solve the same problem without alligation. Let x be the grams of 20% preparation: 0.20x + 0.02(360 - x) = 0.07 × 360. Rearranging gives 0.18x = 18, so x = 100 g. Use the method you can set up reliably, not the one that looks shortest on paper.

A 25% target cannot be produced by mixing only 20% and 2% preparations of the same ingredient. Any weighted average must lie between the input strengths. A negative number of parts is not a practical solution; it tells you the target is outside the available range.

Alligation also assumes compatible concentration bases and a mixture model that conserves the ingredient. For w/w mixtures, use masses. For a volume-based question, use its stated volume assumptions. Do not mix a w/w percentage and a w/v percentage without the information required to reconcile them.

The example is an original calculation exercise, not an instruction to compound an actual medicine. Compatibility, homogeneous mixing, stability, ingredient identity and appropriate preparation procedures remain separate questions. The archived USP calculation proposal is linked for its mathematical explanation, not as a current standard of compounding practice.

For exam review, save the ratio, total parts and reverse check. If you keep reversing the preparations, compare the target with the low and high strengths before calculating. A target near the low strength should usually use more of the low-strength preparation. That qualitative check is fast and worth doing every time.

Try it before reading the answer

Write your setup, units and check first. These are original practice exercises, not recalled exam items.

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