Alligation Calculator
Calculate the mixing ratio and volumes needed to combine two solutions of different concentrations to achieve a desired intermediate concentration.
🧪 Alligation Method Calculator
✅ Result
What Is the Alligation Method?
The alligation method is a mathematical technique used to determine the correct proportions for mixing two solutions of different concentrations to produce a solution with a desired intermediate concentration. The word "alligation" comes from the Latin alligatio, meaning "binding together."
This method is widely used in pharmacy, chemistry, and food science where precise mixing of solutions is critical. Pharmacists routinely use alligation to compound medications at specific concentrations, while chemists use it for preparing reagent solutions and dilutions.
Alligation Formula
Given two solutions with concentrations CH (higher) and CL (lower), and a desired concentration CD:
Parts of Lower = CH − CD
Ratio = (CD − CL) : (CH − CD)
Volume Calculations
If you need a specific total volume Vtotal:
VL = [Parts of Lower / (Parts of Higher + Parts of Lower)] × Vtotal
How the Alligation Method Works
The alligation method works by setting up a simple cross-difference calculation:
- Place concentrations: Write CH (higher) at the top left and CL (lower) at the bottom left. Write CD (desired) in the center.
- Calculate differences: Take the absolute difference between each outer concentration and the desired concentration:
- Top right: CD − CL = parts of higher solution needed
- Bottom right: CH − CD = parts of lower solution needed
- Express as ratio: The two "parts" values give you the mixing ratio.
- Calculate volumes (if needed): Distribute the total required volume proportionally according to the ratio.
Mix a 22% solution and a 15% solution to make a 19% solution.
Step 1: Parts of Higher = 19 − 15 = 4
Step 2: Parts of Lower = 22 − 19 = 3
Step 3: Ratio = 4 : 3
For every 4 parts of the 22% solution, use 3 parts of the 15% solution.
Prepare 120 mL of an 8 M solution from 13 M and 4 M stock solutions.
Step 1: Parts of Higher = 8 − 4 = 4
Step 2: Parts of Lower = 13 − 8 = 5
Step 3: Ratio = 4 : 5, total parts = 9
Step 4: VH = (4/9) × 120 = 53.33 mL of the 13 M solution
Step 5: VL = (5/9) × 120 = 66.67 mL of the 4 M solution
Alligation in Pharmacy
The alligation method is a cornerstone of pharmaceutical compounding. Common applications include:
- Compounding creams and ointments: Mixing a high-strength and low-strength preparation to achieve a prescribed concentration.
- IV fluid preparation: Combining solutions of different dextrose or saline concentrations to match patient needs.
- Alcohol dilution: Preparing solutions of specific alcohol percentages from higher and lower concentration stocks.
- Pediatric formulations: Diluting adult-strength medications to appropriate pediatric concentrations.
Alligation Medial vs. Alligation Alternate
| Alligation Medial | Alligation Alternate |
|---|---|
| Finds the resulting concentration when given known volumes and concentrations | Finds the ratio needed to achieve a desired concentration |
| All quantities are known beforehand | The mixing ratio is the unknown |
| Formula: Cresult = (C1V1 + C2V2) / (V1 + V2) | This calculator uses alligation alternate |
Special Cases
Dilution with Pure Solvent
If you need to dilute a solution using pure water (0% concentration), set CL = 0. The alligation method works the same way, telling you how much stock solution and how much pure solvent to combine.
Mixing with Pure Substance
If mixing a solution with pure (100%) substance, set CH = 100. This is common when adding pure alcohol, acid, or other neat substances to a dilute solution.
Frequently Asked Questions
Can I use alligation with more than two solutions?
The basic alligation alternate method is designed for two solutions. For three or more solutions, you can apply the method iteratively (mix two solutions first, then use the result as one of the inputs for the next step) or use a system of linear equations for a more direct solution.
Does the alligation method account for volume changes on mixing?
The standard alligation method assumes that volumes are additive (Vtotal = V1 + V2). In most dilute aqueous solutions, this is a reasonable approximation. For concentrated solutions or mixtures of different solvents (e.g., ethanol-water), volumes may not be perfectly additive due to molecular interactions, and small corrections may be needed.
What if the desired concentration equals one of the starting concentrations?
If CD = CH, then the parts of lower = 0, meaning you need only the higher-concentration solution and none of the lower. Similarly, if CD = CL, you only need the lower-concentration solution. The calculator handles these edge cases.
What units can I use?
The alligation method works with any concentration units as long as all three values (CH, CL, CD) use the same unit. Common units include percent (%, w/v or w/w), molarity (M), mass per volume (g/L), and parts per million (ppm).