Mixture Problems UPCAT Reviewer: Concentration, Blending, and Replacement

TEACHER ABI UPCAT MATHEMATICS

Mixture Problems

Model mixture problems by tracking the conserved substance or total value, then distinguish dilution, evaporation, blending, and replacement situations.

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Track the amount that is being conserved

This is a Mathematics modeling topic. The liquid, alloy, food, or price context supplies quantities for a weighted equation; no Chemistry knowledge is required.

For concentration problems, track the amount of pure substance:

pure amount = concentration × total amount

For price blends, track total value:

total value = unit price × quantity

Then write:

amount before + amount added − amount removed = amount after

Concentration

Write a percent as a decimal before multiplying.

Dilution

Adding water changes total volume but not pure solute.

Evaporation

Only water leaves; the pure solute remains.

Price blend

Equate the sum of ingredient values to the blend’s total value.

Replacement

Removing mixture removes every component in the same current proportion.

DO IT FAST

Build a quantity–rate–pure amount table

For every ingredient, record:

quantity × concentration = pure amount

Add the pure-amount column, not the concentration column. If the total mixture is fixed, express one quantity as total−x.

For repeated replacement, multiply by the fraction retained at every step.

Why it works

Concentrations cannot usually be averaged directly. The conserved pure amount or total value determines the correct weighted result.

WORKED EXAMPLES

Five forms you should recognize

1. Combine two solutions

Problem: Two liters of a 20% salt solution are mixed with 3 liters of a 40% solution. Find the final concentration.

Setup: Add the pure salt, then divide by the combined volume.

0.20(2) + 0.40(3) = 1.6 L salt1.6÷(2 + 3) = 0.32 = 32%

Why: Concentrations are weighted by their quantities.

2. Dilute with water

Problem: How much water must be added to 10 L of a 30% solution to make it 20%?

Setup: The solution initially contains 0.30(10) = 3 L of pure substance. Water changes the total volume, not that 3 L.

final total = 3÷0.20 = 15 Lwater added = 15−10 = 5 L

3. Blend prices

Problem: A merchant has 20 kg of rice worth ₱80 per kilogram. How many kilograms worth ₱140 per kilogram must be added to produce a blend worth ₱100 per kilogram?

Setup: Total ingredient value equals total blend value.

80(20) + 140x = 100(20 + x)1600 + 140x = 2000 + 100x → x = 10 kg

4. Evaporation

Problem: A 25 L syrup mixture is 12% sugar. How much water must evaporate for it to become 15% sugar?

Setup: The 3 L of sugar remains; only water leaves.

0.12(25) = 3 L sugarfinal total = 3÷0.15 = 20 Lwater evaporated = 25−20 = 5 L

5. Remove and replace

Problem: A 20 L container holds a 30% salt solution. Five liters are removed and replaced with water. Find the new concentration.

Setup: Removing 5 of 20 liters removes one-fourth of the salt.

salt remaining = 0.30(20)(1520) = 4.5 Lnew concentration = 4.5÷20 = 22.5%

Why: Replacement restores the total volume but adds no salt.

COMMON TRAPS

Check before you commit

• Averaging concentrations without considering quantities
• Adding water to the pure-solute amount
• Treating evaporation as removal of the whole mixture
• Forgetting that removed mixture contains solute
• Using the original concentration during a second replacement step
• Mixing pesos per kilogram with total pesos

FIVE-FORM SKILL CHECK

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Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

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Apply the competency in unfamiliar representations.

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QUICK ANSWERS

Mixture Problems FAQ

When can I simply average two concentrations?

Only when equal quantities are mixed. Otherwise use a weighted average based on quantity.

What remains constant during dilution?

The amount of pure solute remains constant; total volume changes.

Why does repeated replacement use multiplication?

Each step retains a fraction of the amount present immediately before that step.

RELATED COMPETENCIES

Continue your mathematics review.

Percent and Percent ChangeRatio and ProportionWeighted MeanLinear Equations and Modeling

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