Alligation is an old pharmacy and trade method for mixing two stock preparations of different strength to get one of intermediate strength. Pharmacists use it to make a 10% ointment from 20% and 5% stock, chemists to make 9 M acid from 12 M and 5 M stock, and jewellers to make an alloy of a set fineness. Unlike a dilution, where one ingredient is pure solvent, both ingredients here contain the active substance.
The trick is a simple cross subtraction known as the alligation grid or “tic-tac-toe” method. Write the higher strength top left, the lower strength bottom left and the target in the middle; subtract across the diagonals and you get the number of parts of each stock. The parts of the higher strength equal the distance from the target to the lower strength, and vice versa – so the stock closer to the target is used in larger amount.
The method works with any concentration unit (%, mg/mL, molarity, proof) as long as all three values use the same one, and it assumes the amounts are additive, which is close enough for most dilute solutions and for mass-based mixing.
How to use the Alligation Calculator
- Higher concentration — Any unit – %, M, mg/mL – but the same for all three concentrations.
- Lower concentration (0 for pure diluent)
- Required concentration
- Amount I know
- Known amount — Volumes are assumed additive; for masses pick any unit and read the result in the same unit.
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the higher, lower and required concentrations. The calculator subtracts across the alligation grid to get the parts, reduces them to the simplest whole-number ratio and, if you give one amount (the total or either ingredient), scales the parts into actual volumes or masses.
Worked example
You need 612.5 mL of 9 M solution from 12 M and 5 M stock. Parts of 12 M = 9 − 5 = 4; parts of 5 M = 12 − 9 = 3; total 7 parts. Volume of 12 M = 612.5 × 4/7 = 350 mL and volume of 5 M = 612.5 × 3/7 = 262.5 mL. Check: (350 × 12 + 262.5 × 5) / 612.5 = (4,200 + 1,312.5) / 612.5 = 9 M.
For 120 mL of 8 M solution from 13 M and 4 M stock the parts are 4 and 5, so you mix 53.33 mL of 13 M with 66.67 mL of 4 M.
The alligation grid
Draw a 3 × 3 grid. Put the higher strength top left, the lower strength bottom left and the target in the centre. Subtract along each diagonal, ignoring signs: the number that lands top right is the parts of the higher strength, and the number bottom right is the parts of the lower strength. Add the two to get the total parts, then divide your total amount by the total parts to find the size of one part.
Tips
- Use the final volume of the solution, not the volume of solvent added, for molarity.
- For dilutions, C₁V₁ = C₂V₂ only holds when both concentrations use the same unit.
- Add acid to water, never water to acid.
- Everything is calculated in your browser.
Common mistakes to avoid
- Mixing w/w, w/v and v/v percentages.
- Forgetting that pH is logarithmic – one unit is a tenfold change in [H⁺].
- Using pKa where Ka is asked for (or the other way round).
Inputs explained
| Field | What to enter |
|---|---|
| Higher concentration | A number. Any unit – %, M, mg/mL – but the same for all three concentrations. |
| Lower concentration (0 for pure diluent) | A number |
| Required concentration | A number |
| Amount I know | Choose one: Total amount to make, Amount of the higher-strength stock, Amount of the lower-strength stock, None – just the ratio |
| Known amount | Units: mL, L, µL, cm³, m³, fl oz (US) …. Volumes are assumed additive; for masses pick any unit and read the result in the same unit. |
What each result means
- Simplest ratio – higher = [Parts of the higher concentration]÷[g]
- Simplest ratio – lower = [Parts of the lower concentration]÷[g]
- Parts of lower per 1 part of higher = [Parts of the lower concentration]÷[Parts of the higher concentration]
- Share of the higher-strength stock = ([Parts of the higher concentration]÷([Parts of the higher concentration] + [Parts of the lower concentration]))×100
- Amount of higher-strength stock = [Total amount]×[Parts of the higher concentration]÷([Parts of the higher concentration] + [Parts of the lower concentration])
- Amount of lower-strength stock = [Total amount]×[Parts of the lower concentration]÷([Parts of the higher concentration] + [Parts of the lower concentration])
Example values
How the main result changes when Higher concentration changes and everything else stays at the example values above:
| Higher concentration | Parts of lower per 1 part of higher |
|---|---|
| 12 | 0.75 |
| 15 | 1.5 |
| 18 | 2.25 |
| 24 | 3.75 |
Formula
Parts of higher = required − lower · Parts of lower = higher − required · V_high = V_total × parts_high ÷ (parts_high + parts_low) · V_low = V_total − V_highWorked example
Inputs — Higher concentration: 12; Lower concentration (0 for pure diluent): 5; Required concentration: 9; Amount I know: Total amount to make; Known amount: 612.5 mL.
Results — Parts of the higher concentration = 4; Parts of the lower concentration = 3; Simplest ratio – higher = 4; Simplest ratio – lower = 3; Parts of lower per 1 part of higher = 0.75; Share of the higher-strength stock = 57.143 %; Total amount = 612.5 mL; Amount of higher-strength stock = 350 mL; Amount of lower-strength stock = 262.5 mL.
Key terms
- Alligation medial
- Finding the strength of a mixture from known amounts and strengths (a weighted average).
- Alligation alternate
- Finding the amounts of stocks of known strength needed for a desired strength – what this calculator does.
- Parts
- Relative amounts of each ingredient; only their ratio matters.
Frequently asked questions
How does alligation work?
Subtract across the grid: parts of the stronger stock = target − weaker, parts of the weaker stock = stronger − target. Mixing 12 M and 5 M to make 9 M needs 4 parts of 12 M and 3 parts of 5 M.
What is the difference between alligation and dilution?
In a dilution the second ingredient is plain solvent (concentration 0) and C₁V₁ = C₂V₂ is enough. Alligation handles two stocks that both contain the substance; with a lower concentration of 0 it reduces to an ordinary dilution.
How do I mix 20% and 5% ointment to make 10%?
Parts of 20% = 10 − 5 = 5 and parts of 5% = 20 − 10 = 10, so the ratio is 1 : 2. For 30 g you need 10 g of 20% and 20 g of 5%.
Why must the required concentration lie between the two stocks?
Mixing can only give a strength between the two ingredients. If the target is outside that range you need a stronger stock, or you must add pure solvent instead.
Can I use alligation with volumes and percent w/w?
The method is exact for amounts measured in the same basis as the concentration – masses for w/w %, volumes for w/v % or molarity. Mixing by volume with w/w strengths is approximate because densities differ.
What is alligation alternate?
The same grid used with more than two stocks, pairing each stock above the target with one below it. For two stocks it is identical to this calculation.
What formula does the Alligation Calculator use?
Parts of higher = required − lower · Parts of lower = higher − required · V_high = V_total × parts_high ÷ (parts_high + parts_low) · V_low = V_total − V_high.