Last updated: July 3, 2026
Alligation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The alligation calculator computes the mixing proportions for two stock solutions. It uses parts_high = C_desired − C_low and parts_low = C_high − C_desired, then reports the simplified high:low ratio and fraction of the total mixture that should come from the high-strength stock.
Alligation finds a two-solution mixing ratio by subtracting diagonally. Parts of high stock equal desired concentration minus low concentration, and parts of low stock equal high concentration minus desired concentration.
Key Takeaways
- Alligation finds the high:low mixing ratio for two stock solutions and one target concentration.
- Use parts_high = C_desired − C_low and parts_low = C_high − C_desired.
- The target must lie between the low and high concentrations for a valid two-stock mixture.
- fractionHigh equals parts_high divided by total parts, so it scales directly to any batch size.
- All inputs must use the same concentration unit and basis before applying the shortcut.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
parts_high = C_desired - C_low; parts_low = C_high - C_desired; fraction_high = parts_high / (parts_high + parts_low)
Where:
- C_h=High-strength stock concentration(% or same concentration unit)
- C_l=Low-strength stock concentration(% or same concentration unit)
- C_d=Desired final concentration(% or same concentration unit)
- p_h=Parts of high-strength solution(parts)
- p_l=Parts of low-strength solution(parts)
- f_h=Fraction of total mixture from high stock(dimensionless)
Worked Examples
Mix 70% and 20% stocks to make 40%
Alligation alternate shows that the stronger stock contributes two parts for every three parts of weaker stock.
- 1Find high-stock parts from the opposite difference: parts_high = 40 − 20 = 20.
- 2Find low-stock parts: parts_low = 70 − 40 = 30.
- 3Simplify 20:30 to 2:3, and compute fraction high = 20 ÷ (20 + 30) = 0.4.
Mix 90% and 10% stocks to make 30%
A lower target close to the weak stock needs a smaller share of the stronger stock.
- 1parts_high = 30 − 10 = 20.
- 2parts_low = 90 − 30 = 60.
- 3The ratio 20:60 simplifies to 1:3, and fraction high = 20 ÷ 80 = 0.25.
Dilute 50% stock with 0% solvent to make 20%
When the low concentration is pure diluent, alligation is the same dilution balance in ratio form.
- 1parts_high = 20 − 0 = 20.
- 2parts_low = 50 − 20 = 30.
- 3Simplify to 2:3; fraction high = 20 ÷ 50 = 0.4.
Introduction
Alligation is a fast mixture method for finding how much of a stronger solution and a weaker solution to combine to obtain a target concentration. It is especially useful when both stocks use the same unit, such as percent w/w, percent v/v, ppm, or mol/L. This calculator applies alligation alternate, checks that the target lies between the two stocks, and reports parts high, parts low, the simplified high:low ratio, and the high-stock fraction. For single-stock dilution compare the bleach dilution calculator or the broader concentration calculator. The concentration arithmetic follows standard mixture-balance principles described in LibreTexts solution stoichiometry and IUPAC concentration terminology such as amount concentration.
What is alligation?
Alligation is a rule-of-mixtures shortcut. Instead of writing a full mass-balance equation every time, you subtract diagonally: the desired concentration minus the low concentration gives the parts of high-strength solution, and the high concentration minus the desired concentration gives the parts of low-strength solution.
Use C_low ≤ C_desired ≤ C_high.
All concentrations must be expressed in the same unit and basis.
The ratio is high stock:low stock, not high stock:final mixture.
The method assumes ideal mixing and additive amounts for the chosen basis.
Alligation formula
Let C_high be the stronger stock, C_low the weaker stock, and C_desired the target. Then parts_high = C_desired − C_low and parts_low = C_high − C_desired. The high-stock fraction is parts_high divided by total parts. The low-stock fraction is one minus that value.
If the desired concentration equals the low stock, use no high stock. If it equals the high stock, use no low stock.
How to calculate a mixing ratio
First verify that the target is between the stock concentrations. Second, take the two diagonal differences. Third, simplify the parts as a ratio. Finally, convert the parts to fractions or volumes for your batch size.
Example: 70%, 20%, target 40% gives 20 parts high and 30 parts low.
Simplify 20:30 to 2:3.
For a 500 mL batch, high volume = 0.4 × 500 = 200 mL.
Low volume = 0.6 × 500 = 300 mL.
Using the fractionHigh output
The primary output is fractionHigh because it scales easily. Multiply fractionHigh by any desired final amount to get the amount of high-strength stock. Multiply one minus fractionHigh by the final amount to get the amount of low-strength stock.
For targets expressed in mol/L, you can prepare stocks with the molarity calculator and then use alligation to blend them.
Common chemistry applications
Alligation is used for preparing intermediate reagent strengths, blending acids or alcohol solutions, diluting stock standards with solvent, and combining two lots of different assay or concentration. It is a mental-math companion to the mass balance used in titration and formulation work.
| Use case | High stock | Low stock | Target |
|---|---|---|---|
| Alcohol blending | 70% ethanol | 20% ethanol | 40% ethanol |
| Acid dilution | 10 M acid | 1 M acid | 3 M acid |
| Assay adjustment | 95% purity lot | 80% purity lot | 90% blend |
| Solvent dilution | 50% stock | 0% solvent | 20% working solution |
Limitations and unit checks
Alligation is not a substitute for chemical compatibility, density correction, heat-of-mixing checks, or activity-coefficient corrections. Percent by mass, percent by volume, ppm, and molarity are not interchangeable unless the problem explicitly defines a common basis. For acid-base endpoint planning, use the titration calculator; for pH-sensitive mixtures, check the pH calculator.
Do not mix incompatible chemicals simply because the arithmetic works.
Use the same concentration basis for every input.
Non-ideal volume contraction can matter for concentrated alcohols and acids.
Analytical preparations may require calibrated glassware and temperature control.
Quick Reference Card
Alligation — Quick Reference
Quick reference • Alligation Calculator
parts_high = C_desired − C_low; parts_low = C_high − C_desired; fractionHigh = parts_high/(parts_high+parts_low)Valid range: C_low ≤ C_desired ≤ C_high, with all concentrations in the same unit and basis.
Common Values
⚠ Watch Out
- •Do not use different concentration bases in the same calculation.
- •The target must be between the two stock concentrations.
- •Check chemical compatibility and safety before physically mixing solutions.
- •Non-ideal density or volume changes can affect concentrated acid, alcohol, and salt mixtures.
Pro Tips
- →Simplify parts with a greatest common divisor when inputs are whole-number percentages.
- →Use fractionHigh × final batch size to get the high-stock amount.
- →Use calibrated glassware for analytical or regulated preparations.
- →When one stock is pure solvent, compare the result with C1V1 = C2V2 as a sanity check.
FAQs
What does an alligation ratio mean?
The ratio is the amount of high-strength stock to low-strength stock. A 2:3 ratio means mix 2 parts high stock with 3 parts low stock, for 5 total parts.
Why are the differences taken diagonally?
The diagonal differences are the distances from the desired concentration to the opposite stock. They are exactly the weights that balance the mixture average at the target concentration.
Can I use percent, ppm, or molarity?
Yes, provided all three concentrations use the same unit and basis. Do not mix percent w/w with percent v/v or mol/L unless you first convert them consistently.
What happens if the target is outside the two stock concentrations?
A mixture of only those two stocks cannot reach a target below the low stock or above the high stock. The calculator returns zeros and an empty ratio for invalid ranges.
How do I scale the ratio to a final batch size?
Multiply the total batch size by fractionHigh for the high-stock amount. Use the remaining amount, or total × (1 − fractionHigh), for the low stock.
Is alligation the same as C1V1 = C2V2?
It is the same conservation idea expressed for two starting solutions. C1V1 = C2V2 is the special case where the low concentration is zero or negligible diluent.