Alligation is a shortcut method for figuring out how to mix two (or more) solutions of different strengths to get a desired concentration. Instead of setting up algebra equations, you use a simple grid that gives you the ratio of each ingredient you need. It’s widely taught in pharmacy programs, but the technique works for any mixing problem, from chemistry labs to industrial manufacturing.
There are two main types: alligation alternate, which tells you how much of each ingredient to combine, and alligation medial, which tells you the final concentration when you already know the amounts. Most people searching for “how to do alligation” need the alternate method, so that’s where we’ll focus first.
How Alligation Alternate Works
Alligation alternate solves one specific problem: you have a higher-strength solution and a lower-strength solution, and you want to mix them to create a target strength in between. The method gives you the ratio of parts for each solution.
The setup uses a grid that looks like a tic-tac-toe board. Here’s how to fill it in:
- Top left: Write the higher concentration.
- Bottom left: Write the lower concentration.
- Center: Write the desired (target) concentration.
- Top right: Subtract the desired concentration from the higher concentration. This tells you the parts of the lower-strength solution you need.
- Bottom right: Subtract the lower concentration from the desired concentration. This tells you the parts of the higher-strength solution you need.
The key detail that trips people up: the subtraction results cross diagonally. The number in the top right (higher minus desired) corresponds to the lower-strength solution, and the number in the bottom right (desired minus lower) corresponds to the higher-strength solution. This feels counterintuitive at first, but it’s the whole trick of the method.
A Simple Example, Step by Step
Say you need to prepare a solution with a concentration of 10%, and you have a 50% solution and a 5% solution available. Here’s the grid:
Top left: 50 (higher concentration)
Center: 10 (desired concentration)
Bottom left: 5 (lower concentration)
Now do the diagonal subtractions:
Top right: 10 − 5 = 5 (parts of the 50% solution)
Bottom right: 50 − 10 = 40 (parts of the 5% solution)
The ratio is 5 parts of the 50% solution to 40 parts of the 5% solution, which simplifies to 1:8. For every 1 part of the strong solution, you need 8 parts of the weak one. That makes intuitive sense: since 10% is much closer to 5% than to 50%, you’d expect to use far more of the weaker solution.
Converting Parts to Actual Volumes
The grid gives you a ratio, but most real problems ask for specific volumes. To get there, add the parts together and use that total to calculate each volume.
Suppose you need 500 mL of 4 M hydrochloric acid and you have 2.3 M and 8.9 M solutions. Set up the grid:
Higher: 8.9, Desired: 4, Lower: 2.3
Top right (parts of 2.3 M): 4 − 2.3 = 1.7
Bottom right (parts of 8.9 M): 8.9 − 4 = 4.9
Wait, remember the diagonal cross. Let’s be precise:
Parts of the higher (8.9 M) solution: desired minus lower = 4 − 2.3 = 1.7
Parts of the lower (2.3 M) solution: higher minus desired = 8.9 − 4 = 4.9
Total parts: 1.7 + 4.9 = 6.6
Volume of 8.9 M solution: (1.7 / 6.6) × 500 = 128.8 mL
Volume of 2.3 M solution: (4.9 / 6.6) × 500 = 371.2 mL
These add up to 500 mL, which confirms the math checks out.
A Pharmacy Example With Dextrose
This is a common real-world scenario. A pharmacy has a parenteral nutrition bag containing 150 mL of Dextrose 20%, and the physician wants the concentration increased to 25%. You can add Dextrose 50% to the bag. How much do you add?
Set up the grid with 50% as the higher concentration, 20% as the lower, and 25% as your target:
Parts of 50% solution: 25 − 20 = 5
Parts of 20% solution: 50 − 25 = 25
The ratio is 5:25, or 1:5. For every 5 parts of the existing 20% solution, you add 1 part of the 50% solution. Since you have 150 mL of the 20% solution: 150 / 5 = 30 mL of Dextrose 50% to add.
Working With Ratio Strengths
Some problems express concentrations as ratio strengths (like 1:1000) instead of percentages. Before you can use the alligation grid, convert everything to the same units. The conversion is straightforward: divide 100 by the ratio number. A 1:500 solution is 100 / 500 = 0.2%. A 1:1000 solution is 100 / 1000 = 0.1%.
Once everything is in the same format, whether percentages, ratios, or molar concentrations, the grid works identically.
Alligation Medial: Finding the Final Concentration
Alligation medial solves the reverse problem. You already know how much of each solution you’re mixing, and you want to find the resulting concentration. This one is pure weighted averaging.
Multiply each volume by its concentration, add those products together, then divide by the total volume. For example, mixing 500 mL of 40% alcohol, 200 mL of 60% alcohol, and 50 mL of 70% alcohol:
500 × 40 = 20,000
200 × 60 = 12,000
50 × 70 = 3,500
Total = 35,500
Total volume: 500 + 200 + 50 = 750 mL
Final concentration: 35,500 / 750 = 47.3%
Notice that the result is closer to 40% than to 70%, which makes sense because the 40% solution makes up the largest share of the mixture.
How to Check Your Work
The quickest sanity check is intuitive: your answer must fall between the two original concentrations, never outside them. If your target concentration is closer to the weaker solution, you should be using more of the weak one, and vice versa.
For a numerical check, use alligation medial on your answer. Take the volumes you calculated, multiply each by its concentration, add them up, and divide by total volume. The result should equal your target concentration. If it doesn’t, go back and check whether you crossed the diagonal subtractions correctly.
Common Mistakes to Avoid
The most frequent error is forgetting the diagonal cross. The difference between the higher concentration and the target gives you the parts of the lower solution, not the higher one. Mixing this up flips your ratio and gives you the wrong volumes.
Another common mistake is using mismatched units. If one concentration is given as a percentage and another as a ratio strength, converting both to the same unit before setting up the grid prevents errors that are otherwise hard to spot.
Finally, some students forget that when one of the “solutions” is a pure diluent (like water or a base with 0% active ingredient), you enter 0 as the lower concentration. The grid still works. For instance, if you need 6% from a 16% solution diluted with water: parts of the 16% solution = 6 − 0 = 6, and parts of water = 16 − 6 = 10, giving a ratio of 6:10 or 3:5.
Mixing Three or More Ingredients
When a problem involves three different concentrations, the approach gets slightly more complex. You pair each concentration that’s above the target with one that’s below the target and run separate alligation grids. If you have solutions at 50%, 20%, and 5% and need 10%, you’d pair the 50% with the 5% in one grid and the 20% with the 5% in another (or pair differently, since multiple valid combinations exist). The results give you one possible set of ratios, though the answer isn’t unique the way two-ingredient problems are.
For most pharmacy coursework and certification exams, two-ingredient alligation is what you’ll encounter. The three-ingredient version appears occasionally, but the core grid method is the same for each pair.