CALCULATIONS

by RXpharmaLAB.com

ALLIGATION
December 2023
Alligation is a calculation
method used in pharmacy for
solving problems in mixing and
preparing various pharmaceutical
forms, such as solutions, creams,
gels, and ointments, which contain
the same substance at different
concentrations.

This document provides the tools to:

calculate the final concentration of a mixture

involving products at different concentrations;
determine the required amount of ingredients with
distinct concentrations to produce a final product at a
specific concentration.

Alligation MEDIAL allows for the calculation of the final

concentration of a mixture of products with different concentrations.
Alligation ALTERNATE allows you to determine the quantities of ingredients necessary to
obtain a final product at a specific concentration.
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Let's begin with alligation, which allows us to determine the final concentration resulting from mixing
products of different concentrations.

Consider a scenario where we need to mix the contents of three alcohol bottles. For example, the first
bottle contains 500 mL of alcohol at 40% (v/v), the second has 200 mL at 60% (v/v), and the third
50 mL at 70% (v/v).

We start by calculating the final volume of the mixture. Simply adding the volumes of the three
bottles gives us a total of 750 mL.

500 mL + 200 mL + 50 mL = 750 mL 1

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

For information related to percentages (w/w, v/v, w/v),

feel free to consult the document titled CALCULATIONS – PERCENTAGES,
also available at RXpharmaLAB.com.
The volume of each component is multiplied by its respective concentration to
calculate its part in the final mixture. These parts are added to find the total
number of parts, which shows the proportional contribution of each
component to the composition of the final product.

2
200 mL à 60% 200 x 60 = 12 000 p
500 mL à 40% 500 x 40 = 20 000 p
+ Medial
50 mL à 70% 50 x 70 = 3 500 p
+ Alligation
35 500 p
in summary
2
200 mL x 60 % = 12 000 p
Then, to obtain the final concentration, + +
500 mL x 40 % = 20 000 p
we divide the total parts by the final volume: + +
50 mL x 70 % = 3 500 p

35 500 p / 750 mL = 47,33333 3 1 750 mL ÷ 35 500 p

This calculation indicates that the final ≈ 47,3% 3

concentration of the mixture is approximately
47.3% (v/v).

Although the 50 mL at 70% represents

Each component
only a small portion of the total volume,
influences the final it plays a significant role in determining
concentration of the the final concentration. Its impact is
mixture based on its more pronounced than that of an equal
volume and amount of a less concentrated
concentration. substance. This is because the 50 mL at
70% accounts for 3,500 parts of the final
concentration, while, for example, 50 mL
at 40% would have represented only
6,67 % of the total volume 2,000 parts. Indeed, even though the

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3500 parts of the volume is the same, the higher
final concentration concentration implies a more substantial
contribution to the final concentration.
50 mL at 70% Thus, in the calculation, the
26,67 % of the total volume concentration of each component is as
important as its volume. In summary,
200 mL at 60% 12 000 parts of the
the 50 mL at 70% has a greater influence
final concentration
on the final concentration than 50 mL of
a substance at a lower concentration, as
500 mL at 40% 66,67 % of the total volume
it contributes a larger amount of active
substance to the mixture, thereby
20 000 parts of the increasing the final concentration more
final concentration significantly than the same amount of a
less concentrated substance.
Let's now address alligation alternate, a particularly useful calculation in
pharmacy when faced with the task of creating a product at a specific
concentration that is not directly available. This method allows for the effective
combination of ingredients at different concentrations to achieve the desired
concentration in the final 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).
20 % 7 parts
To solve this problem, we apply a cross-multiplication -
calculation. We start by noting the available concentrations on
the left, regardless of their position. Then, we place the desired 12 %
ù

concentration in the center and connect the figures to form -

an 'X'. We proceed with subtractions following the lines of the
'X'. The differences obtained represent the 'shares' of each 5% 8 parts
concentration in the final mixture.
* Note that the absolute value must be used in this calculation. The difference between 20% and 12% = 8 parts
The absolute value of a number represents its value regardless The difference between 5% and 12% = *7 parts
of its sign, ensuring it is always positive. This is important in
alternate alligation to accurately reflect the physical quantities The absolute value* must be used.
of the ingredients to be used.
The sum of these parts = the total number of parts
that make up the preparation; 15 parts = 250 g of
20 % 7 parts cream at 12 % (w/w).

The numbers of parts thus found must be associated with the

12 % + concentration of the ingredient located on the opposite end of the
ù same line:
the 20% (w/w) cream, will occupy 7 out of 15 shares,
5% 8 parts the 5% (w/w) cream, will occupy 8 out of 15 shares.
Let's now use the rule of three to calculate the necessary
15 parts quantities of each cream:

250 g of 12 % cream = 15 parts 250 g of 12 % cream = 15 parts

X g of 20 % cream = 7 parts X g of 5 % cream = 8 parts

X g of 20 % cr. = 250 g of 12 % cr. X 7 parts X g of 5 % cr. = 250 g of 12 % cr. X 8 parts

15 parts 15 parts

X g of 20 % cr. ≈ 116,7 g X g of 5 % cr. ≈ 133,3 g

Therefore, to obtain 250 g of cream at 12% (w/w), we need to mix:

116.7 g of 20% (w/w) cream + 133.3 g of 5% (w/w) cream
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Use alligation medial to
validate your results from 133,3 g x 5 % = 666,5 p
+ +
alligation alternate: 116,7 g x 20 % = 2334 p
250 g ÷ 3000,5 p = 12%

Practical Application!
1. How many grams of zinc oxide ointment at 3% (w/w) and 15% (w/w) will be necessary to make
100 g of zinc oxide ointment at 10% (w/w)?
2. Again, regarding zinc oxide, what would be the final concentration, expressed as a percentage
(w/w), obtained after mixing 100 g of zinc oxide at 3% (w/w), 200 g at 5% (w/w), and 300 g at 20%
(w/w)?
3. Which type of alligation is used to find the necessary quantities of different ingredients to obtain a
final product at a certain concentration: alligation alternate or alligation medial?
4. You have a tube of benzocaine ointment at 20% (w/w) and a base ointment with no active
ingredient. How much of each of these ingredients will you use to prepare 30 g of benzocaine at
2.5% (w/w)?
5. You are asked to mix 3 bottles of alcohol together. One contains ¼ L at 90% (v/v), another 210 mL
at 20% (v/v), and the last 125 mL at 30% (v/v). What will be the final concentration as a percentage
(v/v) after this mix?
6. In a non-pharmacy context, how many mL of 1% milk and 3.25% milk would be needed to obtain
½ L of 2% milk?
7. Returning to the field of pharmacy. How many mL of a syrup containing 85% (w/v) sugar and
another containing 60% (w/v) sugar need to be mixed to obtain 600 mL of syrup containing 80%
(w/v) sugar?
8. How many mL of a phenobarbital elixir at 20 mg/5 mL and at 30 mg/5 mL need to be used to
prepare 1 L of elixir containing 4.6 mg of phenobarbital per mL (4.6 mg/mL)?
9. You have vials of metoclopramide at 5 mg/mL and sterile water for injection. You need to prepare
25 mL of metoclopramide at 0.5 mg/mL. How many mL of each will you use? Different methods
can be used to perform this calculation; to practice this type of calculation, use alternate alligation.

This document has been meticulously prepared by RXpharmaLAB.com, which strives for accurate and reliable information.
However, we cannot guarantee its infallibility and disclaim any responsibility for the consequences of its use.

1.
2.
3.
Use 41.7 g of 3% ointment and 58.3 g of 15% ointment.
The final concentration will be 12.17% (w/w).
Alligation alternate.
Solutions
4. Use 3.75 g of 20% ointment and 26.25 g of base without active ingredient.
5. The final concentration will be 52.05% (v/v).
6. Use 277.78 mL of 1% milk and 222.22 mL of 3.25% milk.
7. Use 480 mL of syrup at 85% and 120 mL of syrup at 60%.
8. Use 700 mL of the elixir at 20 mg/5 mL and 300 mL of the elixir at 30 mg/5 mL.
9. Use 2.5 mL of the solution at 5 mg/mL and 22.5 mL of sterile water.

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