C₁V₁ + C₂V₂ = Cf(V₁ + V₂)
When mixing two solutions of different concentrations to achieve a target, the mass balance gives C₁V₁ + C₂V₂ = Cf(V₁+V₂). Solving: V₁/V₂ = (Cf−C₂)/(C₁−Cf). This is the alligation method used in pharmacy and chemistry. The target concentration must lie between the two starting concentrations. Applications include: preparing IV fluids in hospitals, blending ethanol solutions to desired proof, mixing paint colors, preparing laboratory reagents, and industrial chemical processing. When densities differ significantly, mass-based calculations are more accurate than volume-based ones. Always verify by checking that the final mixture satisfies the mass balance equation.