Alligation

It is often more practical to dilute a known strength preparation than it would be to compound an entire preparation. Compounding may involve weighing, measuring, heating, levitating, and extensive mixing of all the ingredients to achieve the finished product. Sometimes, a simple calculation using alligation allows you to calculate the amount of diluent to be added to an already prepared higher strength preparation to form the desired strength. So, as you can see alligation is simply a method used to solve problems that involve mixing two products of different strengths to form a product having a desired intermediate strength.

To perform an alligation you must set up a problem matrix similar to a tic-tac-toe board.

First, draw a problem matrix (tic-tac-toe). Insert quantities as shown and subtract along the diagonals. Read along the horizontals to obtain the number of parts for both the higher and lower strength. Please note that the desired strength always goes in the center square of the matrix. The desired strength is the strength of the preparation that you want to make. MAKE is the key word in deciding the desired strength. Usually, the strength on the prescription will be the desired strength.

Let's try a real world situation: You have on hand a 70% alcoholic elixir and a 20% alcoholic elixir. The prescription calls for a 30% alcoholic elixir. In what proportion must the 70% elixir and the 20% elixir be combined to make a 30% elixir?

After drawing the problem matrix, inserting the quantities, and subtracting along the diagonals, we get 10 parts of the 70% elixir and 40 parts of the 20% elixir. This means that if (reducing) one part of 70% elixir is mixed with four parts of 20% elixir, it will yield five parts of 30% elixir. The dotted arrows in the matrix above are a reminder that the total number of parts always represents the desired strength.

Let's try another one! How many mL of water must be added to 300 mL of 70% alcohol solution to make a 40% alcohol solution?

Before the matrix can be formed, the problem has to be analyzed to see which method is best for working the problem. The key words which indicate that this is an alligation problem are "must be added to." Other key words indicating alligation as the best method are "Must be combined" and "must be mixed." In this problem, 40% is the desired strength and must be placed in the center of the matrix. The next step is to see if a higher or lower strength is given. 70% is a higher strength and must be placed in the upper left-hand corner. If no lower strength is given, it can be assumed to be 0%.

The relationship of parts of each percent strength to their combined final volume may be used as the first ratio of a proportion. To formulate the complete equation, place the known factors in the proper position on the matrix. Assign the X value first: The question asks, "How many milliliters of water?"; the X value is placed on the extended line opposite the percentage denoted by water (0%). The other known factor is that the water will be added to 300 ml of 70%. The 300 mL pertains to the 70% so it is placed on the line opposite the 70% on the matrix.

These values form the proportion: 4 /3 = 300 / x ; cross multiply: 4 x = 900; x = 225 mL of water (0%). Take note that when distilled water, ointment bases, or normal saline are used as diluents, they will contain zero percent (0%) active ingredient.

When solving for the amount of both ingredients when the final volume is known, solve for the higher first. For example: A prescription calls for an elixir that contains 45% alcohol. On hand, you have 10% alcoholic elixir and a 75% alcoholic elixir. How many mL of the 75% elixir must be combined to make 1000 mL of 45% alcoholic elixir?

To solve for the higher, place the X on the line opposite the 75% (see above). The only other known factor is that 1000 mL of the 45% must be prepared. The 1000 mL must be placed on the line opposite the 45%. The arrow indicates that the bottom line is the 45% line (see above). With these figures in place, construct you proportion and solve for X.

The volume of the other ingredient may be found by subtracting the known volume from the final volume. In this case we just found the volume of the 75% elixir to be 538.46 mL. Subtract this figure from the final volume of 1000 mL to get 461.54 mL; or, using the same matrix place the X opposite the 10% on the matrix and construct another proportion (see above)