Alligation

1. Alligation: Rapid Arithmetic for Compounding Calculations

Alligation is a rapid arithmetic method used in pharmacy practice to calculate the proportional parts of two or more solutions or powders of known strength needed to prepare a mixture of a desired intermediate strength.[1] It is one of the most heavily tested calculation topics on the Pharmacy Technician Certification Exam (PTCE) and appears routinely in compounding, reconstitution, and dilution scenarios.[2]

Mastering alligation allows pharmacy technicians to verify formulas, adjust batch sizes, and check the accuracy of compounded preparations without relying on guesswork. The technique is especially valuable in hospital and community settings where products must be mixed precisely to ensure patient safety and therapeutic efficacy.[3]

2. Essential Alligation Terminology and Concentration Formats

  • Alligation Medial – A method used to determine the average strength of a mixture when the strengths and quantities of the individual components are known.[1]
  • Alligation Alternate – A method used to determine the proportion of two or more components needed to produce a mixture of a desired target strength.[1]
  • Strength – The concentration of an ingredient expressed as a percentage, ratio, or decimal (e.g., 10% w/w, 1:1000).
  • Target Strength – The desired final concentration of the mixture.
  • Higher Strength (HS) – The component with a concentration greater than the target.
  • Lower Strength (LS) – The component with a concentration less than the target.
  • Parts – The relative proportion of each component, expressed as a simple ratio (e.g., 3:2).
  • w/w, w/v, v/v – Common concentration expressions: weight per weight, weight per volume, volume per volume.

3. Procedural Steps for Medial and Alternate Methods

3.1 Alligation Medial (Finding Average Strength)

Use this method when you already know the strengths and volumes (or weights) of each component and need to find the final concentration of the blend.[1]

  1. Multiply the strength of each component by its quantity (volume, weight, or count).
  2. Sum the results to get the total amount of active ingredient.
  3. Sum the total quantities of all components.
  4. Divide the total active ingredient by the total quantity.
  5. Express the result as a percentage, ratio, or decimal strength.

Formula:

Mean Strength = (Sum of [Strength × Quantity]) ÷ (Sum of Quantities)

3.2 Alligation Alternate (Finding Proportions for a Target Strength)

Use this method when you need to mix two or more components to achieve a specific intermediate strength.[1]

  1. Draw a tic-tac-toe grid or a simple cross-diagram.
  2. Place the target strength in the center.
  3. Place the higher strength in the upper-left corner.
  4. Place the lower strength in the lower-left corner.
  5. Subtract diagonally: target minus lower (write result in upper-right), higher minus target (write result in lower-right).
  6. The numbers in the right column represent the parts of each component needed.
  7. Express the ratio and, if needed, scale to the desired batch size.

Diagram:

Higher Strength
(e.g., 10%)		 Parts of Higher Strength
(Target − Lower)
Target
(e.g., 6%)
Lower Strength
(e.g., 2%)		 Parts of Lower Strength
(Higher − Target)

4. Worked Examples: Dextrose and Cream Mixture Problems

Example 1: Alligation Alternate (Two Components)

Scenario: A pharmacist needs to prepare 500 mL of a 6% dextrose solution using a 10% dextrose solution and a 2% dextrose solution. How many mL of each are needed?[1]

  1. Higher Strength (HS) = 10%, Lower Strength (LS) = 2%, Target = 6%.
  2. Parts of HS = 6 − 2 = 4 parts.
  3. Parts of LS = 10 − 6 = 4 parts.
  4. Total parts = 8.
  5. mL of HS = (4/8) × 500 = 250 mL.
  6. mL of LS = (4/8) × 500 = 250 mL.

Example 2: Alligation Medial (Average Strength)

Scenario: You mix 100 mL of 20% cream with 200 mL of 5% cream. What is the final concentration?[1]

  1. Ingredient from 20% cream: 0.20 × 100 mL = 20 mL (active).
  2. Ingredient from 5% cream: 0.05 × 200 mL = 10 mL (active).
  3. Total active = 30 mL.
  4. Total volume = 300 mL.
  5. Mean strength = 30 ÷ 300 = 10%.

5. Adapting Alligation for Diluents and Multiple Components

  • Three or more components: Combine components with similar strengths into two groups (above target and below target), then apply the standard alternate method.[3]
  • Using a diluent (0% strength): Treat the diluent as a component with 0% strength (e.g., water, petrolatum base, or lactose).
  • Using a pure drug (100% strength): Treat a pure powder as a component with 100% strength when mixing with a diluent.
  • Scaling to a specific batch size: After finding the parts ratio, divide the desired quantity by the total number of parts, then multiply each component's parts by that factor.

6. Strategic Exam Techniques for Alligation Problems

  • Memorize the cross-diagram: Practice drawing the tic-tac-toe grid until it becomes automatic. Most PTCE alligation questions can be solved in under 60 seconds with this method.
  • Double-check your subtraction order: Always subtract diagonally: target minus lower (for HS parts) and higher minus target (for LS parts). Reversing the order will produce incorrect ratios.
  • Keep units consistent: All strengths must be in the same scale (e.g., all percentages or all ratios) before you begin.
  • Watch for diluent questions: Distilled water, cream base, or tablet filler are at 0% strength — this is a frequent exam trap.
  • Use alligation medial to check your work: If you have time, verify an alternate result by performing a medial calculation on the final mixture.
  • Know when to use which method:
    • Use Alligation Alternate when you know all strengths and need to find how much of each component to mix.
    • Use Alligation Medial when you know the quantities and need to find the final strength of the mixture.
  • Practice reading prescription labels: Some exam questions embed the strengths and quantities in a label format — extract the numbers carefully before setting up your grid.

7. Avoiding Errors and Ensuring Safety in Alligation

  • Always verify your answer: A small arithmetic error in subtraction can lead to a clinically significant dosing mistake. If possible, ask a pharmacist to double-check the calculation before compounding.[4]
  • Never assume components are compatible: Alligation only handles concentration math — it does not account for chemical compatibility, stability, or physical interaction between components.[5]
  • Avoid mixing percentage scales: Do not mix w/w with w/v or v/v in the same alligation without first converting all strengths to the same basis.
  • Check for "target strength" outside the component range: The desired strength must fall between the highest and lowest component strengths. If it does not, alligation cannot produce the target — you would need a different approach (e.g., trituration with a higher-strength stock or further dilution).

8. Mnemonics and Quick-Reference Aids for Alligation

  • "Tic-Tac-Toe, Then Diagonally Go" – Reminds you to set up the grid and subtract across the diagonal.
  • "HS-LS on the left, Target in the middle, Parts on the right" – Quick setup guide.
  • "For Medial, Multiply and Sum" – Alligation medial is just a weighted average: multiply strength × quantity, sum, then divide by total quantity.
  • Flashcard-ready summary:
    • Alternate → find parts needed for target strength.
    • Medial → find final strength from known quantities.

9. References and Sources

  1. Ansel H.C., Prince S.J. (2022). Pharmaceutical Calculations: A Conceptual Approach. Wolters Kluwer. https://www.academia.edu/44789086/Pharmaceutical_Calculations
  2. Pharmacy Technician Certification Board (PTCB). (2023). PTCE Blueprint: Content Areas and Knowledge Statements. https://www.scribd.com/document/340251686/new-ptce-blueprint-pdf
  3. Moini J. (2021). Pharmacy Technician: Foundations and Practices (4th ed.). Pearson. https://www.pearson.com/en-us/subject-catalog/p/pharmacy-technician-foundations-and-practices/P200000001003
  4. American Society of Health-System Pharmacists (ASHP). (2022). ASHP Guidelines on Pharmacy Technician Training and Certification. https://www.ashp.org/-/media/assets/about-ashp/docs/PELA/ASHP-Pharmacy-Technician-Analysis-Full-Document-FINAL.pdf
  5. United States Pharmacopeia (USP). (2023). USP General Chapter <795>: Pharmaceutical Compounding—Nonsterile Preparations. https://www.usp.org/compounding/general-chapter-795
  6. Davis Company. (2021). Pharmacy Technician Certification Quick-Study Guide. F.A. Davis. https://pharmacylibrary.com/doi/book/10.21019/9781582123202

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