Alligatoon method

Mastery of the Alligation Method

Overview

• Presented by: Fuad Conteh and group members.

Understanding Alligation: From Antiquity to Modern Pharmacy

Learning Objectives

• At the end of this presentation, the audience will:

  • Define Allegation Method

  • Understand Its Objectives

  • Learn Its Core Principles

  • Identify Types (Medial and Alternate)

  • Analyze Advantages And Limitations Of The Method

  • Apply This Method Into Real World Pharmacy Compounding

Etymology

• The term "Alligation" originates from the Latin word "alligatio", meaning "to tie together."

• In pharmaceutical compounding, it refers to a mathematical method used to determine the proportions of various components required to achieve a desired strength or concentration of a mixture.

Historical Context

• Initially used to adjust "temperaments" or proportions in various mixtures, reflecting early scientific and medicinal practices (Moore, 1650).

Objectives of the Alligation Method

• Ensuring Accuracy: Precisely calculating component ratios to achieve exact target concentrations, critical for therapeutic efficacy.

• Enhancing Patient Safety: Minimizing dosage errors and adverse events by standardizing compounding calculations.

• Simplifying Complexity: Transforming intricate linear equations into a straightforward, visual, and intuitive method for pharmacists.

Core Principles

1. Conservation of Mass:

   • The total amount of active pharmaceutical ingredient (API) remains constant throughout the mixing process.

2. Proportionality:

   • The method relies on inverse proportionality: the closer a component's strength is to the target, the more of it will be needed, and vice versa.

3. Law of Mixing:

   • When two or more substances of different strengths are mixed, the resultant mixture's strength will fall between the highest and lowest strengths of the individual components.

Key Terminology

• Higher Strength Component: An ingredient with a concentration above the desired strength.

• Lower Strength Component: An ingredient with a concentration below the desired strength.

• Desired Strength: The target concentration required in the final preparation.

Types of Allegation Method

1. Allegation Medial

   • Employed when combining components of known strengths and quantities to determine the final concentration of the mixture.

2. Allegation Alternate

   • Utilized to determine the proportions or ratios of two substances of different strengths required to obtain a mixture of a desired intermediate strength.

Allegation Medial

Definition

• A method used to determine the final strength or concentration when various quantities of ingredients with different strengths are mixed together.

Primary Application

• Ideal for calculating the average concentration of a batch after compounding or for verifying existing mixtures.

Key Concept

• Represents a weighted average, where each component's strength is weighted by its quantity or volume.

The Allegation Medial Formula

• $C<em>f = \frac{C</em>1V<em>1 + C</em>2V<em>2 + \dots + C</em>nV<em>n}{V</em>1 + V<em>2 + \dots + V</em>n}$

• Where:

  • $C_f$ = Final Concentration

  • $V_n$ = Volume of component n

  • $C_n$ = Concentration of component n

Step-by-Step Procedure for Allegation Medial

1. Identify Denominations: List the concentration (percentage or ratio) and quantity (volume or weight) of each component to be mixed.

2. Multiply Parts: For each component, multiply its concentration by its quantity to find the "parts of API" contributed by that component.

3. Aggregate Totals: Sum all the "parts of API" from Step 2, and then sum all the quantities (volumes or weights) from Step 1.

4. Calculate Final Concentration: Divide the total "parts of API" by the total quantity to obtain the final concentration of the mixture.

Worked Example: Mixing Alcohol Solutions

Problem Statement

• A pharmacist needs to mix:

  • 100 mL of 40% alcohol

  • 200 mL of 60% alcohol

  • 50 mL of 70% alcohol.

• What is the final alcohol concentration?

Solutions:

1. Identify Components:

   • 100 mL of 40% alcohol

   • 200 mL of 60% alcohol

   • 50 mL of 70% alcohol

2. Calculate API Parts:

   • 40% x 100 mL = 4000

   • 60% x 200 mL = 12000

   • 70% x 50 mL = 3500

3. Aggregate Totals:

   • Total API Parts = 4000 + 12000 + 3500 = 19500

   • Total Volume = 100 mL + 200 mL + 50 mL = 350 mL

4. Final Concentration:

   • $\frac{19500}{350} = 55.71\%$

Clinical Logic (Medial for Batch Verification)

• Quality Assurance:

  • Allegation Medial provides a rapid and reliable method for pharmacists to verify the concentration of a compounded batch after mixing, ensuring it meets specified standards.

• Error Detection:

  • By comparing the calculated medial concentration with the expected target, discrepancies in ingredient measurement or mixing can be quickly identified and corrected.

• Regulatory Compliance:

  • Essential for adherence to compounding guidelines and regulations set by bodies like USP, ensuring patient safety and product integrity (Ansel and Stockton, 2020).

Allegation Alternate

Definition

• Determines the proportions or ratios of two substances of different strengths required to obtain a mixture of a desired intermediate strength.

Primary Use

• Ideal for diluting a higher concentration product or concentrating a lower one by mixing with a diluent or a more potent substance.

• Key Outcome: Results in "parts" of each ingredient, which then need conversion to specific weights or volumes for compounding.

The Alligation Grid: Noughts and Crosses

• A visual aid that simplifies the Alligation Alternate method.

• Systematically organizes the strengths and the desired concentration, making the subtraction process intuitive and reducing errors (Ansel and Stockton, 2020).

Instructions for Using the Alligation Grid

1. Place the higher strength in the top-left.

2. Place the lower strength in the bottom-left.

3. The desired strength goes in the center.

4. Diagonally subtract the smaller from the larger values.

Diagonal Subtraction & Absolute Values

• Subtracting diagonally across the grid yields the "parts" of each component needed.

• The absolute difference is crucial, ensuring positive values for proportions.

Converting "Parts" to Specific Weights/Volumes

• After determining the parts, use the "Rule of Three" (Ansel and Stockton, 2020):

  • If a certain number of parts equals a known quantity, another number of parts can be found by direct proportion.

Rule of Three Examples

• Quantity of each ingredient proportional to the difference between opposite parts:

  • $\frac{Quantity \ of \ A}{Quantity \ of \ B} = \frac{b}{a}$

• Key Rule: Each ingredient’s quantity is proportional to the opposite part.

Percentage of a Mixture from Its Components

• If a mixture is made by combining components with known percentages:

  • Let Component 1 = $P<em>1 \%$, quantity = $Q</em>1$

  • Component 2 = $P<em>2 \%$, quantity = $Q</em>2$

• Key Point: Mixture percentage is the weighted average of the component percentages.

Worked Example 1

Problem Statement

• A 50 L mixture contains Alcohol (80% pure) and Water (0% pure). The mixture is 32% alcohol. Find:

  • The ratio (parts) of Alcohol and Water.

  • The quantity of each component.

Solutions:

1. Find the ratio:

   • $Alcohol : Water = (32 - 0) : (80 - 32) = 32 : 48 = 2 : 3$

2. Find total parts:

   • Total parts = 2 + 3 = 5

3. Find quantities:

   • $Alcohol = \frac{2}{5} \times 50 = 20 L$

   • $Water = \frac{3}{5} \times 50 = 30 L$

• Final Answer:

  • Parts (Alcohol: Water) = 2 : 3

  • Alcohol = 20 L

  • Water = 30 L

Worked Example 2

Problem Statement

• A mixture of 40 L contains Milk and Water:

  • Milk costs ₹60/L and Water costs ₹0/L. The cost of the mixture is ₹24/L. Find:

  • The ratio (parts) of Milk and Water.

  • The quantity of Milk and Water.

Solutions:

1. Find the ratio:

   • $Milk : Water = (24 - 0) : (60 - 24) = 24 : 36 = 2 : 3$

2. Find total parts:

   • Total parts = 2 + 3 = 5

3. Find quantities:

   • $Milk = \frac{2}{5} \times 40 = 16 L$

   • $Water = \frac{3}{5} \times 40 = 24 L$

• Final Answer:

  • Parts (Milk: Water) = 2 : 3

  • Milk = 16 L

  • Water = 24 L

When Three Components are Involved

Example Scenario

• A hospital pharmacist wants to mix:

  • 50% zinc oxide ointment

  • 20% zinc oxide ointment

  • 5% zinc oxide ointment

• In which proportions should they be mixed to prepare a 10% zinc oxide ointment?

Relative Amounts Found

• 5 parts of 50% ointment

• 5 parts of 20% ointment

• 50 parts of 5% ointment

• The proportions are: 5 : 5 : 50 or 1 : 1 : 10

Combination of Different Alcohol Concentrations

Problem Statement

• The solvent for the extraction of a vegetable drug is 70% alcohol.

• In what proportion can 95%, 60%, and 50% alcohol be mixed to prepare a solvent of the desired concentration? (Assuming no contraction of volume after mixing)

Steps:

1. Pick Alligation Pairs:

   • Target Concentration: 70%

   • Available Concentrations: 95% and 60%, 95% and 50%

2. Perform Diagonal Subtraction:

   • Pair 1: 95% ↔ 60% (Target = 70%):

     • 95 − 70 = 25 parts (of 60%)

     • 70 − 60 = 10 parts (of 95%)

   • Pair 2: 95% ↔ 50% (Target = 70%):

     • 95 − 70 = 25 parts (of 50%)

     • 70 − 50 = 20 parts (of 95%)

3. Combine Like Components:

   • Parts of 95% = 10 + 20 = 30 parts

   • Parts of 60% = 25 parts

   • Parts of 50% = 25 parts

Preparing Mixtures with Multiple Strengths

Example Problem

• Prepare a mixture of 15% w/w drug concentration:

  • 30% w/w (High)

  • 25% w/w (High)

  • 10% w/w (Low)

  • 5% w/w (Low)

Solutions Steps:

1. Identify Components:

   • High strengths = 30%, 25%

   • Low strengths = 10%, 5%

   • Desired strength = 15%

2. Form High–Low Pairs:

   • Pair 1: 30% ↔ 10%

   • Pair 2: 25% ↔ 5%

3. Perform Diagonal Subtraction:

   • Pair 1: 30 − 15 = 15 parts (of 10%)

     • 15 − 10 = 5 parts (of 30%)

   • Pair 2: 25 − 15 = 10 parts (of 5%)

     • 15 − 5 = 10 parts (of 25%)

4. Compile All Parts:

   • Parts of 30% = 5 parts

   • Parts of 25% = 10 parts

   • Parts of 10% = 15 parts

   • Parts of 5% = 10 parts

Advantages & Disadvantages of the Alligation Method

Advantages

• Alligation is:

  • Fast

  • Visually intuitive

  • Reduces algebraic complexity

  • Offers a structure that allows pharmacists to quickly verify that calculated proportions are reasonable.

Disadvantages and Limitations

1. A major limitation is that users may rely on the grid mechanically without understanding the underlying mathematics.

2. The method does not account for volume contraction, which can occur in some chemical mixtures (Thompson, 2013).

3. Potential for Human Error:

   • Errors often arise from misplacing values within the grid, necessitating a final reasonableness check before compounding.

4. The Zero Ingredient:

   • Pure bases such as water or syrup are assigned a concentration of zero percent, and failing to recognize this is a frequent source of mistakes.

Pharmacy Application

• Use in IV Admixtures:

  • Alligation is widely applied in preparing intravenous solutions, such as adjusting electrolyte concentrations in total parenteral nutrition or diluting highly concentrated sodium chloride solutions for pediatric uses.

• Compounding Semisolids:

  • Alligation aids in the dilution of potent creams in dermatological compounding. For example, a 1% hydrocortisone cream can be mixed with a zero-percent base such as petrolatum to produce a safer concentration for infants.

• Alcohol and Liquid Dilutions:

  • The method is also applicable to liquid preparations, such as adjusting high-proof alcohols to medicinal concentrations like 70% isopropyl alcohol used for disinfection.

References

1. Ansel, H.C. and Stockton, S.J. (2020) Pharmaceutical Calculations. 16th edn. Philadelphia: Wolters Kluwer.

2. Moore, J. (1650) Arithmetick: Or, the Art of Computation. London: Printed by R. and W. Leybourn.

3. Winfield, A.J., Richards, R.M. and Rees, J.A. (2009) Pharmaceutical Practice. 4th edn. Edinburgh: Churchill Livingstone/Elsevier.

4. Jones, D. (2008) Pharmaceutics: Dosage Form and Design. London: Pharmaceutical Press.

5. Thompson, J.E. (2013) A Practical Guide to Contemporary Pharmacy Practice. 3rd edn. Philadelphia: Lippincott Williams & Wilkins.

6. OpenAI (2026) ChatGPT (GPT-5.2). Available at: https://chat.openai.com (Accessed: January 2026).

Group Members List

1. Fuad Conteh (group leader) 13387

2. Umu Yanguba 13712

3. Hassanatu Jalloh 12782

4. Amadu W Barrie 12630

5. Adama Tholley 12575

6. Chernor Maju Mohamed Jalloh 12702

7. Jaka Sheriff 11712

8. Aruna Sesay 12940

9. Sheku Jakitay 12620

10. Mechiah Gborie 12718

11. Musu Wonneh 12190

12. Salamatu Sankoh 11808

13. Mabinty Dumbuya 12746

14. Milton Tamba Lansana 12698

15. Makalay P Bangura 13540

16. Isatu Bah 14144

17. Abdul Salam Kassim

18. Abdulainatu Kargbo 11883

19. Abdul Tawwab I.B Conteh 11706

20. Alysious S. Koroma 13052

21. Mohamed Lamin Meinday 12573

22. Isatu Kargbo 12563

23. Edwarda B Sapha 12494

24. Mary Kallon 11700

25. Bintu Jalloh 12677

26. Fuad Mansour 12655

27. Abigeal Allie 13846

28. Samuel Jide Chukwu Afamefuna Oduah 13242

29. Ibrahim Bai Koroma. 12812

30. Adama Sankoh Yillah 10458

31. Salieu Lamin Kamara 9337

32. Anthony Abdul Dabor 10915

33. Mohamed Kelvin Koroma 10122

34. Mohamed S. Jalloh 13732

35. Mariama D. Jalloh 12006

36. Idrissa Hassan Kabia 12546

37. Francis Eric Davowah 10449

38. Moses Charles Conteh 13729

39. Ibrahim Gbassay Kargbo. 12582

40. Saidu Sow 12556

41. Bockarie Ansumana 14048

42. Agnes Conteh 12584

43. Abdul Ormoray Conteh 11572

44. Kadijah Harry Kamara 12622

45. Adama Alpha Kebe 13520

46. Sheka Bangura 12604

47. Hawanatu Samuka 13668

48. Fanta Marah 11129

49. Abubakarr Sulaiman Kamara 13314

50. Maada Brima 13495.

51. Fatmatu Bah 11104

52. Princess Koroma 11922

53. Zainab Massaquoi 13432

54. Amadu Sewa Kamara 12371

55. Isata Turay 12136

56. Ngegba, Musa Mohamed 12578

57. Tenneh Tina Songa 11799

Thank You

• Date: January 2026

• Event: Allegation Method Presentation