Alligation

CALCULATIONS

Alligation is a calculation method used in pharmacy to solve problems involving mixtures and the preparation of various pharmaceutical forms, such as solutions, creams, gels, and ointments, that contain the same substance at different concentrations.

This document is designed to give you the tools you need to:

• Calculate the final concentration of a mixture involving products at different concentrations;
• Determine the amount needed of ingredients with different concentrations to prepare a finished product at a specific concentration.

Medial alligation is used to calculate the final concentration of a mixture made from products at different concentrations.

Alternate alligation is used to determine the quantities of ingredients needed to obtain a finished product at a specific concentration.

Medial Alligation

Let’s start with medial alligation, which allows us to find the final concentration obtained after mixing products with different concentrations.

Let’s consider a case where we need to mix the contents of three bottles of alcohol. Suppose the first bottle contains 500 mL of alcohol at 40 % (v/v), the second 200 mL at 60 % (v/v), and the third 50 mL at 70 % (v/v).

Next, we calculate the final volume of the mixture by simply adding the volumes of the three bottles, giving us a total of 750 mL.

The next step is to determine the final concentration obtained after mixing these 750 mL.

The volume of each component is multiplied by its respective concentration to calculate its part in the final mixture. Then, these parts are added together to determine the total number of parts, which helps show the proportional contribution of each component to the composition of the finished product.

Then, to obtain the final concentration, we divide the total number of parts by the final volume:

This calculation tells us that the final concentration of the mixture is approximately 47.3 % (v/v).

Medial alligation in summary

Although the 50 mL at 70 % represents only a small portion of the total volume, it plays a significant role in determining the final concentration. Its impact is greater than that of the same amount of a less concentrated substance. This is because the 50 mL at 70 % represents 3500 parts of the final concentration, whereas, for example, 50 mL at 40 % would represent only 2000 parts. In other words, even if the volume is the same, the higher concentration contributes more to the final concentration. Therefore, in this calculation, the concentration of each component is just as important as its volume. In short, the 50 mL at 70 % has a greater influence on the final concentration than 50 mL of a substance at a lower concentration because it adds a larger amount of active substance to the mixture, increasing the final concentration more strongly than the same amount of a less concentrated substance.

PHARMACY ASSISTANT St Clements

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PHARMACIST Timmins

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Alternate Alligation

Let’s now look at alternate alligation, a particularly useful calculation in pharmacy when we need to create a product at a specific concentration that is not directly available.

This method allows us to combine ingredients with different concentrations in the right proportions to reach the desired concentration in the finished product.

Imagine we have two jars of cream with the same active ingredient but at different concentrations: one at 20 % (w/w) and the other at 5 % (w/w). Our goal is to prepare 250 g of cream at 12 % (w/w).

To solve this problem, we use a cross calculation. We start by writing the available concentrations on the left, regardless of whether they are placed at the top or bottom. Then, we place the desired concentration in the center and connect the numbers to form an X. We subtract along the lines of the X. The differences obtained represent the parts of each concentration in the final mixture.

Cross calculation

The sum of the parts corresponds to the total number of parts that make up the preparation; 15 parts = 250 g of cream at 12 % (w/w).

The number of parts found must then be associated with the concentration of the ingredient located opposite it on the same line:

• The cream at 20 % (w/w) will make up 7 parts out of 15;
• The cream at 5 % (w/w) will make up 8 parts out of 15.

We then use the rule of three to calculate the required quantity of each cream.

Therefore, to obtain 250 g of cream at 12 % (w/w), we need to mix: 116.7 g of cream at 20 % (w/w) + 133.3 g of cream at 5 % (w/w)

Validation Using Medial Alligation

Use medial alligation to validate the results obtained with alternate alligation:

Practice | Multiple Choice

PHARMACIST Thunder Bay

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PHARMACIST Surrey

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