What this calculator does
Alligation finds how much of a strong and a weak solution to combine for a target strength. Mixing 70% and 20% to reach 40% takes 200 mL of the strong and 300 mL of the weak for a 500 mL total.
The trick is that each proportion comes from the gap on the opposite side. The strong solution share is the distance from the target down to the weak concentration, and the weak share is the distance from the strong concentration down to the target. Crossing them over like this is what the method is named for.
The formula
The share of the higher-concentration solution is the target minus the lower concentration; the share of the lower is the higher concentration minus the target. Those two numbers form the ratio, which is scaled to the total volume required. The method assumes the volumes add, which is close enough for most aqueous work and not exact for alcohol and water.
| Term | Meaning |
|---|---|
| Alligation | The cross-subtraction method for two-component mixing. |
| Target concentration | Must lie between the two source concentrations for a solution to exist. |
| Mixing ratio | The proportions of the two sources, before scaling to a volume. |
| Volume contraction | The reason alcohol and water do not quite add, which the method ignores. |
The inputs explained
| Field | What to enter |
|---|---|
| Higher concentration (%) | The stronger solution concentration. |
| Lower concentration (%) | The weaker solution concentration. Use 0 for pure solvent. |
| Target concentration (%) | Target concentration, which must lie between the two. |
| Total volume needed (mL) | Total volume you need to end up with. |
When to use it
Diluting a stock to a working strength
Where the diluent is not pure solvent but a weaker solution already on hand.
Pharmacy compounding
Alligation is a standard pharmacy calculation for preparing a strength between two available products.
Blending fuels or feeds
Any two-component blend to a specification follows the same arithmetic.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What mix does each target need?
A range of target concentrations between the two sources.
| Target concentration | Mixing ratio (higher : lower) | Volume of higher-concentration solution | Volume of lower-concentration solution |
|---|---|---|---|
| 25% | 5.00 : 45.00 | 50.0 mL | 450.0 mL |
| 40% | 20.00 : 30.00 | 200.0 mL | 300.0 mL |
| 55% | 35.00 : 15.00 | 350.0 mL | 150.0 mL |
| 70% | 50.00 : 0.00 | 500.0 mL | 0.0 mL |
Questions
What if my target is outside the two concentrations?
There is no answer. Mixing two solutions always gives something between them, so a target above the stronger or below the weaker is unreachable. You would need a stronger stock or a more dilute diluent, including pure solvent at 0%.
Why is it called alligation?
From the Latin for binding or tying, describing the crossed lines drawn between the concentrations in the traditional layout. The visual crossing is what gives each component its proportion from the opposite gap.
Does this work for more than two solutions?
Not directly. Alligation handles two components. Three or more have many possible solutions rather than one, so the problem becomes a system of equations with a constraint rather than a single arithmetic result.
Do the volumes really add?
Approximately, for dilute aqueous solutions. Mixing alcohol and water contracts noticeably, so 500 mL of one plus 500 of the other gives less than a litre. Where that matters, work in mass rather than volume.
For a simple dilution with pure solvent, see the dilution calculator. For repeated dilution steps, see the serial dilution calculator.