Alligation Calculator

Mix a higher and lower concentration to a required intermediate concentration and volume.

Supports any consistent concentration unit and 11 volume units with full bidirectional solving.

Updated September 1, 2026

CreatorFrank Zhao

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What is alligation?

Alligation is a method for finding the right proportions to mix two solutions of different concentrations and obtain a target concentration somewhere in between. It has been a staple technique in pharmacy compounding, chemical manufacturing, and laboratory preparation for over a century.

The core idea is simple: the difference between each source concentration and the target tells you the proportion of the other solution you need. A bigger gap means you need more of the opposite solution to pull the blend toward your target.

This calculator applies the alligation method with full bidirectional solving. Enter any three concentration values and at least one volume, and the remaining fields are computed automatically. The alligation ratio $H:L$ is displayed in real time as you type, so you can see the mixing proportions before committing to specific volumes.

How to use this calculator

You only need to fill in the fields that matter to you. The calculator will automatically compute the rest. Here is the typical workflow:

1
Enter all three concentrations. Type the higher concentration, the lower concentration, and the required (target) concentration. All three must use the same unit (%, w/v, mg/mL, g/L, etc.) and the required concentration must fall strictly between the other two.
2
Check the ratio. The blue read-only field immediately shows the simplified mixing ratio $H:L$ , so you know the proportions at a glance.
3
Enter one volume. Provide any one of the three volume fields (higher concentration volume, lower concentration volume, or total required volume). The other two are calculated automatically. You can switch between 11 volume units, including mL, L, US/UK gallons, and fluid ounces.

Quick example: mixing salt solutions

A pharmacist has a 30% saline stock solution and a 10% saline solution. She needs 500 mL of a 20% solution.

Higher concentration: 30
Lower concentration: 10
Required concentration: 20

Ratio calculation:

H = 20 - 10 = 10

,

L = 30 - 20 = 10

\Rightarrow

H:L = 1:1

The ratio is 1:1, meaning equal parts of each solution. Entering the total required volume of 500 mL, the calculator shows 250 mL of the 30% solution and 250 mL of the 10% solution. The result can be verified:

C_{\mathrm{req}} = \frac{30 \times 250 + 10 \times 250}{500} = \frac{10{,}000}{500} = 20\%

Formula explanation

The alligation method relies on two groups of equations. The ratio equations determine proportions from concentrations, and the volume equations translate those proportions into actual amounts.

Ratio from concentrations

Given a higher concentration $C_{\mathrm{high}}$ , a lower concentration $C_{\mathrm{low}}$ , and a required concentration $C_{\mathrm{req}}$ :

H = C_{\mathrm{req}} - C_{\mathrm{low}}

,

L = C_{\mathrm{high}} - C_{\mathrm{req}}

The simplified ratio $H:L$ tells you how many parts of the higher-concentration solution to mix with parts of the lower-concentration solution.

Volume from ratio

Once you know the ratio and one volume, the remaining volumes follow from:

V_{\mathrm{high}} = \frac{H}{H+L}\,V_{\mathrm{req}}

V_{\mathrm{low}} = \frac{L}{H+L}\,V_{\mathrm{req}}

V_{\mathrm{req}} = V_{\mathrm{high}} + V_{\mathrm{low}}

Bidirectional solving: This calculator is not limited to one direction. Enter any two concentrations and one volume, and the third concentration is computed. Or supply all three concentrations and two volumes to find the missing volume. The smart solving engine figures out which field to calculate based on what you provide.

Real-world examples

Pharmacy compounding: diluting an antiseptic

A hospital pharmacy has a 12% chlorhexidine stock solution and a 2% ready-to-use solution. The protocol calls for a 4% wash for pre-surgical skin preparation, and 3 L is needed for the day.

H = 4 - 2 = 2

,

L = 12 - 4 = 8

\Rightarrow

H:L = 1:4

For every 1 part of the 12% stock, mix 4 parts of the 2% solution. With a total of 3,000 mL, that means 600 mL of the 12% stock and 2,400 mL of the 2% solution.

Industrial mixing: coolant concentration

A CNC machining shop uses a 60% glycol coolant concentrate and wants to prepare 20 US gallons of a 30% working solution for a new batch of machines.

H:L = (30 - 0):(60 - 30) = 1:1

Equal parts concentrate and water: 10 US gal of concentrate and 10 US gal of water. The calculator accepts US gallons directly, so no unit conversion is needed.

Reverse calculation: finding the target concentration

You have 150 mL of a 25% solution and 350 mL of a 5% solution, and you want to know what concentration you will get by mixing them. Enter all three concentration fields, then supply both volumes. The calculator will solve for the required concentration:

C_{\mathrm{req}} = \frac{25 \times 150 + 5 \times 350}{500} = \frac{5{,}500}{500} = 11\%

The bidirectional engine computes this automatically, making the calculator useful not just for planning mixes but also for checking what a known combination will yield.

Frequently asked questions

What if the required concentration is not between the two source concentrations?

Alligation only works when the target concentration lies strictly between the higher and lower concentrations. If the required value equals or exceeds either bound, the calculator will flag a validation error. In that case, you need to choose different source solutions or adjust your target.

Can I use different concentration units for the three fields?

No. All three concentration fields must use the same unit — for example, all in %, all in mg/mL, or all in g/L. Because the concentration fields have no built-in unit selector, you are responsible for ensuring consistency. Mixing units (e.g., % for higher but mg/mL for lower) will produce incorrect results.

Why is the ratio shown in blue?

The blue text in the ratio field indicates it is a derived value — it is automatically calculated from the three concentration inputs. You cannot edit it directly; change any concentration and the ratio updates instantly. This color convention is consistent across all derived fields in this calculator.

Limitations & disclaimers

•This calculator assumes ideal mixing: the volumes of the two solutions combine additively. In practice, some solutions exhibit volume contraction or expansion upon mixing, which can shift the final volume and concentration slightly.
•All three concentrations must use the same unit and the required concentration must fall strictly between the higher and lower values. The calculator does not verify unit compatibility — it is your responsibility to ensure consistency.
•Concentrations and volumes must be positive numbers. Zero or negative values are not supported.
•This tool provides calculation assistance for educational and planning purposes. For pharmaceutical compounding, industrial chemical mixing, or any application where accuracy is safety-critical, always verify results against approved protocols and consult a qualified professional.

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