Notes on the Use of Alligation Method in Pharmaceutical Practice

Alligation is a rapid arithmetic method used to solve problems that involve mixing two products of different strengths to form a product having a desired intermediate strength. The term alligation has its origin in Latin, alligatio which means the art of attaching and therefore refers to lines drawn during calculation to bind quantities together.

Alligation medial is used in calculating the strength of the mixture of two or more components of different strengths. It gives the weighed average percentage strength of a mixture resulting from components of known quantities and concentration.

Alligation method is used to calculate:

(a) The amount of diluent that must be added to a given amount of higher strength preparation to make a desired lower strength.

(a) The amounts of active ingredient that must be added to a given amount of lower strength preparation to make a higher strength.

(a) The amount of higher and lower strength preparations that must be combined to make a desired amount of an intermediate strength.

It is often more practical to dilute a known strength preparation than it would be to compound an entire preparation. Sometimes, a simple calculation using alligation allows us to calculate the amount of diluent to be added to an already prepared higher strength preparation to form the strength desired.

Sometimes, it is necessary to increase the strength of a preparation by adding an active ingredient for example if 1% coal tar ointment is available and required strength is 2 %, it can be accomplished by adding an unknown amount of coal tar (100 percent).The unknown amount may be found by using alligation method.

Procedure of Calculation

(a) Draw a problem matrix

(a) Insert quantities as shown

(a) Subtract along the diagonals

(a) Read along the horizontals

The desired strength always goes in the center square of the matrix. The desired strength is the strength of the preparation that one wishes to make.

Example:

In what proportions should a preparation containing 15% of drug be mixed with one containing 30% of drug to prepare a mixture of 20% strength.

Thus 5 parts of 30% and 10 parts of 15% drug mixed together will give a mixture of 20% drug strength.

Proof of Alligation Method

Suppose a product of X percentage strength is to be prepared by mixing constituents of percentage Y (Higher) and Z (lower).

Hence desired percentage strength of X can be obtained by mixing together (X- Z) parts of Y and (Y-X) parts of Z

(X-Z) x Y + (Y-X) x Z = [(X-Z) + (Y-X)] x X

Solving the equation

XY-ZY +YZ-XZ = [X-Z + Y-X] x X

XY – XZ =[Y-Z]xX = XY -XZ

This equation proves the line diagram method of alligation is correct and most suitable in such calculation

Example 1

What is the % of alcohol in a mixture obtained by mixing 5 L of 25%, 1L of 50% and 2 L of 95% alcohol?

Solution

(1 Litre = 1000 ml)

5000 mix 25%= 1250

1000 ml x 50% = 500

2000 ml x 95% = 1900

8000 ml

Final mixture concentration = 45.6%

Example 2

What is the strength of zinc oxide in an ointment prepared by mixing 400 g of 10%, 100 g of 20% and 50 g of 5% ointment?

400 g x 10% = 40

100 g x 20% = 20

50 g x 5% = 2.5

550 ml 62.50 62.50 : 550 : : X x 100

Final mixture concentration = 45.6%

Example 3

How many parts of 70%, 60% 40% and 30% alcohol should be mixed to get 50% alcohol. Calcula

20:10:10:20 or 10:20:20:10

Example 4

In what proportion should 12%, 8% and 3% alcohol be mixed to get 5% alcohol”?

Solution

7+3 =10