Chapter 28: Alligations

Introduction to Alligations

Alligations are essential principles utilized in calculations for mixing ingredients of differing concentrations. They are particularly useful in determining how to combine two products that contain the same active component at different percentages. The key outcome when employing alligations is that the strength of the resulting mixture will always be between the strengths of the starting products.

Objectives

• The main objective of utilizing alligation methods is to accurately calculate the volume (in milliliters) or weight (in grams) of each ingredient required to achieve a specific final concentration or strength.

Measurement Essentials

• Measurements for liquid forms (such as solutions, syrups, and lotions) are typically expressed in milliliters (mL).

• For solid forms like powders and creams, measurements are expressed in grams (g).

• Conversion Standards:

  • 1 fluid ounce is approximately equivalent to 30 mL.

  • 1 avoirdupois ounce is also approximately equivalent to 30 g.

The Alligation Grid

The alligation grid is structured similarly to a tic-tac-toe board and is used to visually facilitate the calculations involved.

Structure of the Alligation Grid

• Top Left Box: This is where the higher concentration strength is placed.

• Bottom Left Box: This is where the lower concentration strength is placed.

• Center Box: The desired strength and the total quantity necessary for the final mixture are placed here.

Filling in the Alligation Grid

An example in filling out the grid may include:

• Higher strength: 2.5

• Desired strength: [To be filled out based on requirements]

• Lower strength: 1

Calculation Steps

The calculation joins the upper and lower strengths with the desired strength through a series of steps:

1. Subtract the center value (desired strength) from the top left value (higher strength) to determine the value that goes in the bottom right box.

2. Subtract the center value from the bottom left value to determine the value that appears in the top right box.

Interpreting and Applying Proportions

• The right-hand side of the grid indicates the required ratios or parts of each component in the final mixture.

• The total parts are calculated by summing up the values in the right column.

Examples of Part Proportions

• For a 2.5% ointment:

  • The proportion might be expressed as: 1 part from the lower strength to 1.5 parts from the higher strength.

• For a 1% ointment:

  • The ratio would be 0.5 parts from the lower strength to 1.5 parts from the higher strength.

Practical Application & Safety

To determine the correct amounts of each ingredient to mix according to the identified proportions, practical calculations follow:

• Example Calculation: For a 2.5% ointment in a total combination of 120 g, the formula to derive is:
  $(120g/1.5)×1=80 g$

• Safety Reminder: Always use leading zeros with decimals to prevent dosage errors, for example, writing 0.25% instead of .25%.

Compounding Example

Example 1 (1 of 3)

In the first scenario, you need to create 500 mL of a 45% dextrose solution in water. You have two solutions available: 5% and 70% dextrose in water.

• The method of alligation can be used:

  • Place the strengths into the grid: 70 on the top left, 5 on the bottom left, and 45 in the center.

• Calculation follows: $ext{Desired Strength} - ext{Lower Strength} = 45 - 5 = 40$ $ext{Higher Strength} - ext{Desired Strength} = 70 - 45 = 25$

  • The total parts needed are $(40 + 25 = 65)$.

Example 2 (2 of 3)

Step 2a requires setting up proportions to find amounts needed for each solution.

Example 3 (3 of 3)

Step 2b continues this approach to conclusively determine the necessary amounts of each solution.

Additional Compounding Scenarios

Several additional exercises require students to determine specific volumes needed based on given concentrations:

• Prepare 200 mL of a 2% w/v lidocaine solution with a 10% w/v solution and 0% w/v.

• Prepare 120 mL of a 5% w/v hydrocortisone cream with a 10% w/v and 0% w/v base.

• Prepare 600 mL of a 15% v/v ethanol solution, and more similar scenarios.

Practice Simulations

Three fictitious prescriptions will require students to use the alligation alternate method with placeholders for desired concentrations and amounts, using colored waters as placebos.
Examples include preparing concentrations ranging from 25% to 12%, all requiring careful calculation of parts and volumes required from the starting mixtures.

Example Calculation Problem

A final example demonstrates how to use alligation in practice:

• Scenario: A pharmacist needs to create 200 mL of a 4% salicylic acid solution using a 2% and a 6% solution.

1. Set up the necessary equations:

   • 2 ext{%}x + 6 ext{%}y = 4 ext{%}*200

   • $x + y = 200$

2. Solve for values x and y, leading to the final amounts required of each solution to achieve the desired concentration.

• The final answer concludes with a recommendation to mix 100 mL of the 2% solution and 100 mL of the 6% solution to produce 200 mL of 4% salicylic acid solution.