Alligation Calculations in Pharmacy: Complete Math Guide

What Is Alligation in Pharmacy?

Alligation is a mathematical method used in pharmacy to determine the correct proportions for mixing two solutions or preparations of different concentrations to achieve a desired final concentration. This calculation method is essential for compounding pharmacies and appears regularly on the PTCB certification exam.

Whether you are preparing a custom topical cream, mixing IV solutions, or diluting a stock solution, alligation provides a straightforward approach to solving concentration problems. Mastering this technique will help you succeed on the PTCB exam and perform compounding calculations accurately in practice.

When to Use Alligation

Alligation is used whenever you need to mix two different concentrations of the same substance to produce a final product at a specific concentration. Common pharmacy scenarios include mixing two strengths of a cream or ointment, diluting a concentrated stock solution with a diluent, combining two different percentage solutions of the same drug, and preparing custom concentration IV admixtures.

The Alligation Medial Method

The alligation medial method is used to find the weighted average concentration when you know the quantities and concentrations of each component being mixed. The formula is straightforward: multiply each quantity by its concentration, add the products together, then divide by the total quantity.

For example, if you mix 200 mL of a 10% solution with 300 mL of a 5% solution, the calculation would be: (200 x 10) + (300 x 5) = 2000 + 1500 = 3500. Then divide by the total volume: 3500 / 500 = 7%. The resulting mixture has a concentration of 7%.

The Alligation Alternate Method

Step-by-Step Process

The alligation alternate method is more commonly tested on the PTCB exam. It determines the ratio of two components needed to achieve a desired concentration. Follow these steps to solve alligation alternate problems.

• Step 1: Draw a tic-tac-toe grid or cross pattern
• Step 2: Place the higher concentration in the upper left corner
• Step 3: Place the lower concentration (or zero for a diluent) in the lower left corner
• Step 4: Place the desired concentration in the center
• Step 5: Subtract diagonally (always subtract the smaller from the larger)
• Step 6: The differences give you the ratio of parts needed

Worked Example 1

You need to prepare 500 mL of a 7% solution using a 10% stock solution and a 3% solution. Set up the alligation cross: place 10% in the upper left, 3% in the lower left, and 7% in the center. Subtract diagonally: 7 minus 3 equals 4 parts of the 10% solution. Then 10 minus 7 equals 3 parts of the 3% solution. The total is 7 parts.

To find the actual volumes: for the 10% solution, calculate (4/7) x 500 mL = 285.7 mL. For the 3% solution, calculate (3/7) x 500 mL = 214.3 mL. Together these equal 500 mL of a 7% solution.

Worked Example 2: Using a Diluent

You need to prepare 120 g of a 2% hydrocortisone cream using a 5% stock cream and a plain cream base (0% concentration). Place 5% in the upper left, 0% in the lower left, and 2% in the center. Subtract diagonally: 2 minus 0 equals 2 parts of the 5% cream. Then 5 minus 2 equals 3 parts of the plain base. The total is 5 parts.

Calculate volumes: for the 5% cream, (2/5) x 120 g = 48 g. For the plain base, (3/5) x 120 g = 72 g. Mix 48 g of the 5% cream with 72 g of the plain base to get 120 g of 2% hydrocortisone cream.

Common Mistakes to Avoid

Students frequently make avoidable errors with alligation calculations. The most common mistake is subtracting in the wrong direction. Always subtract the smaller number from the larger number when working diagonally across the grid. Another frequent error is forgetting that a diluent has a concentration of zero percent.

• Always double-check that your desired concentration falls between the two starting concentrations
• Make sure the parts add up correctly before calculating volumes
• Verify that your final volumes add up to the total volume needed
• Remember that the number of parts from the diagonal subtraction corresponds to the opposite concentration

Practice Tips for the PTCB Exam

On the PTCB exam, alligation problems may appear in different formats. Sometimes you will be given the total volume needed and asked to find individual volumes. Other times you may be given a ratio and asked to find the final concentration. Practice both the medial and alternate methods until you can solve problems quickly and confidently.

Drawing the alligation cross on your scratch paper during the exam helps organize the information and reduces calculation errors. Work through at least 20 practice problems before exam day to build speed and accuracy. Remember that alligation is simply a tool for finding ratios, and once you have the ratio, the rest is basic proportion math.

Real-World Applications

In daily pharmacy practice, alligation calculations are most commonly used in compounding settings. Hospital pharmacies use alligation when preparing custom IV solutions, and community compounding pharmacies use it when mixing creams, ointments, and suspensions. Understanding this method makes you a more valuable and versatile pharmacy technician.