Mixture Problems
Learn to solve word problems involving mixing solutions, alloys, and other substances with different concentrations.
Definition
- = concentration (as decimal)
- = volume
- Subscripts 1, 2 = the two solutions being mixed
- Subscript = the resulting mixture
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Worked Examples
A juice company mixes 10 liters of juice with 30% fruit content with 6 liters of juice with 50% fruit content. What is the fruit concentration in the mixture?
Identify what we know
Solution 1: 10L at 30% = Solution 2: 6L at 50% = → Pure fruit amounts
Calculate pure fruit in each
L of pure fruit L of pure fruit → 3L + 3L = 6L total pure fruit
Find total mixture volume
liters total → 16 liters
Calculate mixture concentration
→ 37.5% fruit content
Answer: The mixture has a 37.5% fruit concentration
Common Mistakes
Adding percentages directly: 30% + 50% = 80%
Why it's wrong: Percentages don't add when mixing different volumes. You must weight them by volume.
Correct: Calculate the actual amounts, add those, then find the percentage of the total.
Forgetting that water has 0% concentration
Why it's wrong: When diluting, students often forget to account for water as a solution with 0% solute.
Correct: Water contributes to volume but not to solute amount:
Using the wrong total volume
Why it's wrong: The mixture volume is the sum of all parts, not just one solution.
Correct: (or for three solutions)
Converting percentages incorrectly
Why it's wrong: 30% must be written as 0.30 (not 30) in calculations.
Correct: Always convert:
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Practice Problems
18 problemsIf you mix 10 liters of 20% salt solution with 10 liters of 40% salt solution, what is the concentration of the mixture?
Why It Matters
- Chemistry labs: Scientists mix solutions to achieve specific concentrations
- Pharmacy: Pharmacists dilute medications to safe dosages
- Food industry: Creating coffee blends, juice mixtures, or metal alloys
- Finance: Calculating weighted average returns on investments
- Manufacturing: Combining materials to meet specifications
Real World Applications
Pharmacy Compounding
Pharmacists mix medications to create specific concentrations for patients.
Example:
A pharmacy needs to prepare 200 mL of a 2% saline solution from a 5% stock solution and distilled water.
A pharmacist has a 10% iodine solution and needs to create 100 mL of a 4% iodine solution.
How much of the 10% solution and how much water are needed?
Step 1: Write the mathematical expression
Set up:
Metal Alloy Production
Metallurgists create alloys by mixing metals with different compositions.
Example:
Bronze is made by mixing copper (90%) and tin (10%). A factory mixes pure copper with a 70% copper alloy.
A jeweler wants to make 50 grams of 14-karat gold (58.3% gold) by mixing 18-karat gold (75% gold) with 10-karat gold (41.7% gold).
How many grams of each type are needed?
Step 1: Write the mathematical expression
Let x = grams of 18K gold. Then:
Coffee Roasting Business
Coffee roasters blend beans of different prices to create signature blends at target price points.
Example:
A roaster creates a house blend by mixing Ethiopian beans (18 euros/kg) with Brazilian beans (10 euros/kg) to sell at 13 euros/kg.
A coffee shop needs 30 kg of a blend priced at 14 euros/kg. They have Kenyan beans at 18 euros/kg and Colombian beans at 12 euros/kg.
How much of each type should they use?
Step 1: Write the mathematical expression
Let x = kg of Kenyan. Then:
Key Takeaways
- 1Mixture problems combine substances with different properties to find the resulting mixture's property
- 2The key formula:
- 3Always convert percentages to decimals before calculating
- 4Set up an equation where the amount of substance in the parts equals the amount in the mixture
- 5Verify your answer by checking that the mixture equation holds true
Frequently Asked Questions
Glossary
- Concentration
- The amount of solute in a solution, usually expressed as a percentage
- Solute
- The substance dissolved in a solution (e.g., salt in saltwater)
- Solvent
- The substance that dissolves the solute (e.g., water in saltwater)
- Dilution
- Adding solvent (usually water) to decrease concentration
- Weighted average
- An average where different values contribute proportionally to their quantities
Formula Card
Mixture Formula
Sum of amounts in parts equals amount in mixture
Find Concentration
Calculate resulting concentration from known values
Dilution
When diluting: concentration times volume stays constant