Alligation Calculations

Alligation Calculations

Master alligation calculations with free flashcards and spaced repetition practice. This lesson covers alligation methods for mixing concentrations, calculating proportions for desired strengths, and solving complex pharmacy compounding problems—essential concepts for the NAPLEX exam and safe medication preparation.

Welcome 🧮

Alligation is a powerful mathematical method pharmacists use to determine the proportions needed when mixing two or more preparations of different strengths to achieve a desired concentration. This technique is indispensable in compounding practice, where you'll frequently need to combine stock solutions, dilute concentrations, or create custom formulations. While the concept might seem complex at first, mastering alligation will save you valuable time on calculations and reduce the risk of medication errors.

Core Concepts

What is Alligation?

Alligation is a mathematical technique that calculates the relative quantities of ingredients of different strengths required to produce a mixture of a desired intermediate strength. The method comes from the Latin word "alligare," meaning "to bind together."

There are two main types of alligation:

1. Alligation Medial: Finding the weighted average concentration when quantities and concentrations are known
2. Alligation Alternate: Finding the proportions needed to achieve a desired concentration (most commonly used in pharmacy)

When to Use Alligation

Alligation is ideal when:

• Mixing two concentrations to obtain an intermediate strength ✅
• The desired concentration falls between the two available strengths ✅
• You need to determine proportions or parts, not exact quantities ✅

💡 Tip: If the desired strength doesn't fall between your two available strengths, alligation won't work. You'll need dilution or fortification formulas instead.

The Alligation Grid (Tic-Tac-Toe Method)

The most visual and error-resistant method uses a grid pattern:

┌─────────────────────────────────────────┐
│   ALLIGATION GRID SETUP                 │
├─────────────────────────────────────────┤
│                                         │
│   Higher %        Parts of              │
│   (H)      ╲     HIGHER                 │
│             ╲    needed                 │
│              ╲                          │
│            Desired %                    │
│              (D)                        │
│             ╱                           │
│            ╱     Parts of               │
│   Lower % ╱      LOWER                  │
│   (L)            needed                 │
│                                         │
└─────────────────────────────────────────┘

Step-by-step process:

1. Place the desired concentration in the center of the grid
2. Place the higher concentration in the upper left corner
3. Place the lower concentration in the lower left corner
4. Subtract diagonally:
   • Parts of HIGHER = Desired - Lower
   • Parts of LOWER = Higher - Desired
5. Reduce to simplest ratio if needed

The Mathematical Formula

For those who prefer equations over grids, the underlying principle is:

Parts of Higher : Parts of Lower = (D - L) : (H - D)

Where:

• H = Higher concentration
• L = Lower concentration
• D = Desired concentration

Alligation Medial (Weighted Average)

When you know the quantities and need to find the resulting concentration:

Formula:

Final Concentration = (Q₁ × C₁ + Q₂ × C₂) / (Q₁ + Q₂)

Where:

• Q₁, Q₂ = Quantities of each ingredient
• C₁, C₂ = Concentrations of each ingredient

Critical Rules and Limitations

⚠️ Must-Know Constraints:

1. Desired concentration MUST be between the two starting concentrations

   • Can't use alligation to get 10% from 5% and 7%
   • Can't use alligation to get 95% from 60% and 80%

2. All concentrations must be expressed in the same units

   • Don't mix % w/v with % w/w
   • Don't mix mg/mL with %

3. Alligation gives PROPORTIONS (parts), not absolute quantities

   • You may need to scale up/down based on the total amount needed
   • Always calculate the total parts first, then find what each part equals

4. The method assumes simple mixing with no volume contraction or expansion

   • For alcohol-water mixtures, use special tables
   • For significant volume changes, use specific gravity calculations

Converting Parts to Actual Quantities

Once you have your ratio from alligation, follow these steps:

Detailed Examples

Example 1: Basic Alligation - Mixing Alcohol Solutions 🧪

Problem: A pharmacist needs to prepare 500 mL of 70% isopropyl alcohol. In stock are 90% and 50% isopropyl alcohol. How many milliliters of each are needed?

Solution using the Alligation Grid:

      90% ───────╲
                  ╲
                   ╲    Parts of 90%
                    ╲   = 70 - 50 = 20 parts
                  70%
                   ╱
                  ╱     Parts of 50%
                 ╱      = 90 - 70 = 20 parts
      50% ──────╱

Answer: Mix 250 mL of 90% alcohol with 250 mL of 50% alcohol.

Verification (always verify!):

(250 mL × 90%) + (250 mL × 50%) = (250 × 0.90) + (250 × 0.50)
                                 = 225 + 125 = 350
350 ÷ 500 mL = 0.70 = 70% ✓

💡 Tip: Notice the ratio was 1:1 (20:20). This happens when the desired concentration is exactly halfway between the two starting concentrations.

Example 2: Unequal Ratios - Zinc Oxide Ointment 💊

Problem: Prepare 120 g of 10% zinc oxide ointment using 20% and 5% zinc oxide ointments.

Solution:

      20% ───────╲
                  ╲
                   ╲    Parts of 20%
                    ╲   = 10 - 5 = 5 parts
                  10%
                   ╱
                  ╱     Parts of 5%
                 ╱      = 20 - 10 = 10 parts
      5% ───────╱

Answer: Mix 40 g of 20% ointment with 80 g of 5% ointment.

Verification:

(40 g × 20%) + (80 g × 5%) = (40 × 0.20) + (80 × 0.05)
                            = 8 + 4 = 12 g zinc oxide
12 g ÷ 120 g = 0.10 = 10% ✓

🧠 Memory Device: The ratio is 5:10, which simplifies to 1:2. Notice that you need TWICE as much of the weaker concentration because the desired strength (10%) is closer to the weaker one (5%) than to the stronger one (20%).

Example 3: Three-Component Mixture - Advanced Application 🎯

Problem: A physician orders 240 mL of a 15% dextrose solution. The pharmacy stocks 5%, 10%, and 50% dextrose solutions. Calculate one way to prepare this using all three concentrations.

Solution: This requires a two-step approach:

Step 1: Use alligation to mix 10% and 50% to get close to 15%:

      50% ───────╲
                  ╲
                   ╲    Parts of 50%
                    ╲   = 15 - 10 = 5 parts
                  15%
                   ╱
                  ╱     Parts of 10%
                 ╱      = 50 - 15 = 35 parts
      10% ──────╱

Ratio of 50% to 10% = 5:35 = 1:7

Step 2: Use alligation to mix 5% and the 50% to get 15%:

      50% ───────╲
                  ╲
                   ╲    Parts of 50%
                    ╲   = 15 - 5 = 10 parts
                  15%
                   ╱
                  ╱     Parts of 5%
                 ╱      = 50 - 15 = 35 parts
      5% ───────╱

Ratio of 50% to 5% = 10:35 = 2:7

Option A (using 10% and 50%):

• Total parts = 1 + 7 = 8 parts
• 240 mL ÷ 8 = 30 mL per part
• 50% solution: 1 × 30 = 30 mL
• 10% solution: 7 × 30 = 210 mL

Option B (using 5% and 50%):

• Total parts = 2 + 7 = 9 parts
• 240 mL ÷ 9 = 26.67 mL per part
• 50% solution: 2 × 26.67 = 53.34 mL
• 5% solution: 7 × 26.67 = 186.66 mL

💡 Clinical Pearl: Multiple solutions exist! Choose based on stock availability and which minimizes waste. The 10% + 50% option uses less of the concentrated solution.

Example 4: Working with mg/mL Concentrations 💉

Problem: Prepare 60 mL of morphine sulfate 8 mg/mL using 15 mg/mL and 4 mg/mL solutions.

Solution:

      15 mg/mL ──╲
                  ╲
                   ╲    Parts of 15 mg/mL
                    ╲   = 8 - 4 = 4 parts
                 8 mg/mL
                   ╱
                  ╱     Parts of 4 mg/mL
                 ╱      = 15 - 8 = 7 parts
      4 mg/mL ──╱

Answer: Mix 21.8 mL of 15 mg/mL solution with 38.2 mL of 4 mg/mL solution.

Verification:

Total morphine = (21.8 × 15) + (38.2 × 4) = 327 + 152.8 = 479.8 mg
Concentration = 479.8 mg ÷ 60 mL = 8 mg/mL ✓

⚠️ Safety Note: Always double-check calculations for controlled substances like morphine. A 1% error in concentration could have serious clinical consequences.

Common Mistakes to Avoid ⚠️

Mistake 1: Placing Numbers in Wrong Grid Positions

❌ Wrong: Putting the desired concentration in an upper or lower corner
✅ Correct: Desired concentration ALWAYS goes in the center

Mistake 2: Subtracting in the Wrong Direction

❌ Wrong: Calculating "Higher - Lower" for parts
✅ Correct: Always subtract DIAGONALLY across the center:

• Parts of higher = Desired - Lower
• Parts of lower = Higher - Desired

Mistake 3: Forgetting to Check if Alligation Applies

❌ Wrong: Trying to use alligation when desired strength is outside the range
✅ Correct: Verify that Lower < Desired < Higher before starting

Example of when NOT to use alligation:

• Want: 25% solution
• Have: 10% and 15% solutions
• Problem: 25% > both available strengths → Use fortification instead!

Mistake 4: Mixing Up Parts with Final Quantities

❌ Wrong: Reporting "5 parts and 10 parts" as the final answer
✅ Correct: Convert to actual quantities based on total volume/weight needed

Mistake 5: Using Different Concentration Units

❌ Wrong: Mixing % w/v with % w/w or mg/mL
✅ Correct: Convert everything to the same unit system first

Mistake 6: Not Simplifying Ratios

💡 Tip: While not technically "wrong," ratios like 20:40 should be simplified to 1:2 for easier calculation and reduced error risk.

Mistake 7: Skipping Verification

❌ Wrong: Assuming your calculation is correct and moving on
✅ Correct: ALWAYS verify using weighted average formula

🧠 Remember: "Trust, but verify" - especially on the NAPLEX where one calculation error can cost points!

Mistake 8: Rounding Too Early

❌ Wrong: Rounding each intermediate step
✅ Correct: Keep full precision until the final answer, then round appropriately

Example:

240 ÷ 11 = 21.818181...
Use 21.818 in subsequent calculations, not 21.8
Only round final answer: 21.82 mL (to hundredths)

Advanced Applications

Alcohol Dilution Special Case

When diluting alcohol with water, remember that volumes don't add linearly due to contraction. For precise work, use official alcohol dilution tables. However, for NAPLEX purposes, simple alligation is usually acceptable unless the question specifically mentions volume contraction.

Specific Gravity Considerations

If ingredients have significantly different specific gravities and the question asks for weights (not volumes), you may need to:

1. Perform alligation calculation
2. Convert volumes to weights using specific gravity
3. Or vice versa, depending on what's requested

Formula: Weight (g) = Volume (mL) × Specific Gravity

Stock Solution Calculations Combined with Alligation

Some complex problems require both stock solution dilution AND alligation:

1. First, determine if you can make needed concentrations from stock
2. Then use alligation to determine proportions
3. Finally, calculate actual quantities

Key Takeaways 🎯

Test Yourself 🔧

Try this: You need 180 g of 12% hydrocortisone cream. Available: 2.5% and 20% hydrocortisone cream. Calculate the quantities needed.

      20% ────╲
               ╲
                ╲    12 - 2.5 = 9.5 parts of 20%
              12%
               ╱
              ╱     20 - 12 = 8 parts of 2.5%
      2.5% ──╱

Visual Summary

┌────────────────────────────────────────────────────────┐
│           ALLIGATION DECISION TREE                     │
└────────────────────────────────────────────────────────┘

        Is desired concentration between
          the two available strengths?
                     │
        ┌────────────┴────────────┐
        │                         │
      ┌─┴─┐                     ┌─┴─┐
      │YES│                     │NO │
      └─┬─┘                     └─┬─┘
        │                         │
        ▼                         ▼
  ┌─────────────┐         ┌──────────────┐
  │Use Alligation│         │Use different │
  │   Method     │         │method:       │
  └──────┬───────┘         │- Dilution    │
         │                 │- Fortification│
         ▼                 └──────────────┘
  1. Set up grid
  2. Diagonal subtraction
  3. Calculate total parts
  4. Find value per part
  5. Multiply to get quantities
  6. VERIFY answer!

📚 Further Study

For deeper understanding and practice:

1. Pharmaceutical Calculations by Howard C. Ansel - Comprehensive textbook with extensive alligation practice problems https://www.amazon.com/Pharmaceutical-Calculations-Howard-C-Ansel/dp/1609137183

2. RxCalculations Practice Problems - Free online calculator and practice questions https://www.rxcalculations.com

3. ASHP Pharmacy Calculations Course - Interactive modules with video explanations https://www.ashp.org/professional-development/learning-center

💡 Final Exam Tip: On the NAPLEX, alligation problems are moderate-difficulty questions that separate average candidates from high scorers. Master this technique, practice until it becomes automatic, and you'll gain confidence and save precious test time. The 2-3 minutes you invest practicing alligation daily will pay dividends on exam day!