Alligation Calculator
Find the ratio of a low and high concentration to mix into a target concentration, using the classic alligation rule - optionally convert to actual volumes.
➗ What is Alligation?
Alligation (also called the alligation alternate method) is a classic mixture rule for finding the exact ratio in which two solutions of different concentrations must be combined to produce a solution at a desired target concentration. It answers a question that comes up constantly in pharmacy, chemistry, and everyday mixing problems: "I have a weak solution and a strong solution, how much of each do I mix to get something in between?"
The rule uses a simple cross-multiplication pattern: parts of C1 : parts of C2 = (C2 − Ct) : (Ct − C1). Each source concentration's required proportion pairs with the "opposite" distance, C1's parts come from the distance between C2 and the target, and C2's parts come from the distance between the target and C1.
This calculator computes that ratio directly, automatically simplifying it to lowest whole-number terms when possible, and can optionally convert the ratio into actual volumes if you specify a total desired volume. The result is exact, not approximate, for any linear concentration measure (mass percent, molarity, alcohol proof, and similar measures all mix linearly across the typical ranges used in practice).
Alligation is taught extensively in pharmacy compounding courses (diluting or strengthening a stock preparation to a prescribed strength) and general chemistry (blending stock solutions), and the same underlying math applies to any two-value mixture-to-target problem, from blending fuel grades to averaging commodity prices.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Blending to a 500 mL target volume
C1 = 10%, C2 = 50%, Ct = 25%, total volume = 500 mL.
Example 2 — Ratio only, no volume
C1 = 20%, C2 = 90%, Ct = 60%, no total volume specified.
Example 3 — Blending from a pure source (C1 = 0)
C1 = 0%, C2 = 100%, Ct = 35%, no total volume specified.
Example 4 — Alcohol proof blending to 1 litre
C1 = 40%, C2 = 95%, Ct = 70%, total volume = 1000 mL.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the alligation rule?
The alligation rule (also called the alligation alternate method) is a shortcut for finding the ratio in which two solutions of different concentrations must be mixed to produce a solution of a desired intermediate concentration. It states: parts of C1 : parts of C2 = (C2 - Ct) : (Ct - C1), where C1 and C2 are the two source concentrations and Ct is the target concentration.
Why does the alligation rule use 'crossed' differences?
The rule works because each source's required proportion is inversely related to how far it is from the target: a source further from Ct needs to contribute less, and a source closer to Ct needs to contribute more, but the algebra works out so that the amount of C1 pairs with the distance (C2 - Ct) and the amount of C2 pairs with the distance (Ct - C1), the 'crossed' pairing that gives the method its name.
What happens if Ct is not between C1 and C2?
No mixture of C1 and C2 alone can produce a target concentration outside the range [C1, C2], since mixing two concentrations can only ever produce something between them (or equal to one of them). This calculator shows an error if Ct is not strictly between C1 and C2.
How do I convert the alligation ratio into actual volumes?
Once you have the ratio (parts of C1) : (parts of C2), divide your desired total volume proportionally: Volume of C1 = Total x parts(C1) / (parts(C1) + parts(C2)), and Volume of C2 = Total x parts(C2) / (parts(C1) + parts(C2)). This calculator does this automatically when you enter a total volume.
Can alligation be used for things other than chemical solutions?
Yes. Alligation is a general mixture-ratio technique used anywhere two known values must be blended to hit a target average: mixing two grades of alloy, blending two grain prices to hit a target average price, or combining two milk-fat percentages to hit a target fat content. Anywhere a linear mixture average applies, the same ratio formula works.
What is the difference between alligation and a simple average?
A simple average assumes equal parts of each component. Alligation instead finds the specific unequal ratio needed to hit a target that is not exactly halfway between C1 and C2, weighting each component's contribution by how close the target is to the other component.
Why does the calculator require C2 to be greater than C1?
By convention this calculator treats C1 as the lower concentration and C2 as the higher one, which keeps both computed parts positive and the formula unambiguous. If your two known concentrations are in the opposite order, simply swap which one you enter as C1 and which as C2.
What does it mean if the ratio simplifies to a whole number like 5:3?
It means the concentrations and target were chosen (or happen to work out) so that the parts needed reduce to a simple whole-number ratio, for example, 5 parts of the low concentration for every 3 parts of the high concentration. This calculator automatically reduces ratios like this to lowest terms when both parts are whole numbers.
How accurate is the alligation method?
It is exact, not an approximation, as long as the concentration measure being mixed behaves linearly under mixing (mass percent, molarity, and similar measures generally do, for typical dilute-to-moderate concentration ranges). It is the standard method taught in pharmacy and chemistry courses for solving concentration-blending problems by hand.
Can I use alligation with more than two source concentrations?
The basic alligation alternate method as implemented here handles exactly two source concentrations blending to one target. Problems with three or more source concentrations require an extended alligation method or a system of equations, which is a more advanced technique beyond this calculator's scope.