Free · Mixtures and solutions calculators ⚗️

Two-Solution Alligation Mixture

Find the volumes and reduced mixing ratio for two compatible solutions with additive volumes and a shared volume-based concentration scale.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

To make 12 mL at concentration 4 from solutions at 10 and 1, mix 4 mL of the stronger solution with 8 mL of the weaker solution: a 1:2 volume ratio.

H Vh + L Vl = R(Vh + Vl); Vh/Vl = (R−L)/(H−R); Vh = V(R−L)/(H−L).

Worked example

Enter these known values and leave the other values blank.

Higher concentration
10
Lower concentration
1
Required concentration
4
Required concentration volume
12 mL
Higher concentration volume
4 mL
Lower concentration volume
8 mL

Assumptions and limitations

  • Both concentrations describe the same constituent on the same volume-based scale; volumes add without reaction, contraction, loss or constituent generation.
  • The source requires strictly positive concentrations and volumes, with higher > required > lower. Pure-solvent and endpoint mixtures are outside this captured contract.
  • The entered raw expression must be exactly representable finite decimal arithmetic. Input magnitude is bounded from10^-10000 through less than10^10001 before and after volume-unit conversion; an unsupported value is rejected rather than rounded into a different ratio.
  • Exact ratio work permits at most256 canonical characters, a256-place relative decimal alignment span and positive integer-scaled gaps at most10^10. The Euclidean helper has a68-division bound.
  • A concentration inferred from a volume inverse may contain a finite working residual when the entered volume is a rounded decimal re-expression in another unit. Such a result can exceed the exact-ratio proof budget and be refused even when a nearby ideal rational ratio is simple. No fuzzy rounding is used to label that residual as an exact integer ratio.
  • The volume formulas retain the captured inverses and ordinary finite working precision. This is not a recommendation for clinical dosing, reactive chemicals or hazardous mixing.

Common questions

Which side of the ratio is the stronger solution?

The first number is the volume part for the solution labelled Higher concentration; the second belongs to Lower concentration.

Can I enter mass fractions instead?

Only if you first establish an appropriate common volume-based concentration and additive-volume model. A weighted mass-fraction balance uses masses, so its numbers cannot simply be used as solution volumes.

Why can a mathematically valid ratio be refused?

This implementation proves the reduced ratio with bounded integer arithmetic. Very long inputs or widely separated decimals can exceed that proof budget; the calculator reports the limitation instead of inventing a rounded integer ratio.

References

Replace the captured potentially nonadvancing ratio search with bounded exact decimal and integer normalization. Reduce the displayed integer parts directly rather than multiplying a rounded reciprocal scale. Preserve all ten volume inverse expressions and eleven volume choices; use exact US liquid-gallon and US fluid-ounce definitions at the six captured selector occurrences. Keep whole-source raw-input guards and scientific fallback for nonzero values beyond ordinary decimal-place display.

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