Mixture Problem

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Compute the concentration of a combined mixture from the concentrations and amounts of its components.

When two solutions with volumes V_1 and V_2 and concentrations c_1 and c_2 are combined, the resulting mixture has volume V_1 + V_2 and concentration c_final = (c_1 · V_1 + c_2 · V_2) / (V_1 + V_2). This is simply a weighted average of the input concentrations, with each concentration weighted by its volume. The same logic extends to three or more components: multiply each component's concentration by its amount, sum, and divide by the total amount.

c_final = Σ(c_i · V_i) / Σ V_i

• Mix 2 L of 10% saline with 3 L of 20% saline → c = (0.10·2 + 0.20·3)/5 = 0.16 = 16%
• Blend 500 g of 40% silver alloy with 300 g of 60% alloy → c = (0.40·500 + 0.60·300)/800 = 47.5%
• Dilute 1 L of 25% sugar solution with 3 L of water (0%) → c = (0.25·1 + 0·3)/4 = 6.25%

How to recognize it

• Two or more amounts with different concentrations are being combined
• Keywords: 'mix', 'blend', 'dilute', 'alloy', 'solution'
• The answer is a combined concentration or a required amount to reach a target

Common mistakes

• Averaging concentrations directly (without weighting by volume)
• Forgetting that adding pure solvent is a component with concentration 0
• Mixing units (mL with L, g with kg)

Practice guidance

Numerical Reasoning