Mixture Problems

Learn to solve word problems involving mixing solutions, alloys, and other substances with different concentrations.

Advanced30 minLesson
Printable worksheet

Definition

A mixture problem involves combining two or more substances with different properties (concentrations, prices, rates) to create a mixture with a new property value.
The key principle is:
Amount of substance in part 1+Amount of substance in part 2=Amount in mixture\text{Amount of substance in part 1} + \text{Amount of substance in part 2} = \text{Amount in mixture}
For concentration problems:
C1⋅V1+C2⋅V2=Cmix⋅VmixC_1 \cdot V_1 + C_2 \cdot V_2 = C_{mix} \cdot V_{mix}
Where:
  • CC = concentration (as decimal)
  • VV = volume
  • Subscripts 1, 2 = the two solutions being mixed
  • Subscript mixmix = the resulting mixture

Try it now

If you mix 10 liters of 20% salt solution with 10 liters of 40% salt solution, what is the concentration of the mixture?

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?
1

Identify what we know

Solution 1: 10L at 30% = 10×0.3010 \times 0.30 Solution 2: 6L at 50% = 6×0.506 \times 0.50 → Pure fruit amounts
2

Calculate pure fruit in each

10×0.30=310 \times 0.30 = 3 L of pure fruit 6×0.50=36 \times 0.50 = 3 L of pure fruit → 3L + 3L = 6L total pure fruit
3

Find total mixture volume

10+6=1610 + 6 = 16 liters total → 16 liters
4

Calculate mixture concentration

616=0.375=37.5%\frac{6}{16} = 0.375 = 37.5\% → 37.5% fruit content

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: 0.00×Vwater=00.00 \times V_{water} = 0

Using the wrong total volume

Why it's wrong: The mixture volume is the sum of all parts, not just one solution.

Correct: Vtotal=V1+V2V_{total} = V_1 + V_2 (or V1+V2+V3V_1 + V_2 + V_3 for three solutions)

Converting percentages incorrectly

Why it's wrong: 30% must be written as 0.30 (not 30) in calculations.

Correct: Always convert: 30%=30100=0.3030\% = \frac{30}{100} = 0.30

Interactive Visual

Ratio Tape Diagram

5:3
Juice 1, 30%
2
2
2
2
2
= 10
Juice 2, 50%
2
2
2
= 6
Mixture (L)
= 8 parts (16)
Juice 1, 30%:5
Juice 2, 50%:3
If total is:
Ratio:5:3
Fraction form:5/8 and 3/8
Juice 1, 30% value:10
Juice 2, 50% value:6

Adjust the ratio parts using + and - buttons.

Bar Chart

Juice 1 (10 L)(30)
Juice 2 (6 L)(50)
Mixture (16 L)(37.5)

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems
Problem 1 of 18
Easy
If 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

Mixture problems appear everywhere in real life:
  • 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
Understanding mixtures helps you think systematically about how quantities combine!

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.

1Try It Yourself

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 0.10×x=0.04×1000.10 \times x = 0.04 \times 100 and solve: how many mL of the 10% solution?

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.

2Try It Yourself

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. Solve 0.75x+0.417(50−x)=0.583×500.75x + 0.417(50-x) = 0.583 \times 50 for xx, rounded to whole grams:

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.

3Try It Yourself

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. Solve 18x+12(30−x)=14×3018x + 12(30-x) = 14 \times 30 for xx:

Key Takeaways

  • 1Mixture problems combine substances with different properties to find the resulting mixture's property
  • 2The key formula: C1V1+C2V2=CmixVmixC_1V_1 + C_2V_2 = C_{mix}V_{mix}
  • 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

Yes! The same principle applies: C1V1+C2V2+C3V3=CmixVmixC_1V_1 + C_2V_2 + C_3V_3 = C_{mix}V_{mix}. Just add more terms for each substance.
Yes! The same principle applies: C1V1+C2V2+C3V3=CmixVmixC_1V_1 + C_2V_2 + C_3V_3 = C_{mix}V_{mix}. Just add more terms for each substance.
The same weighted average approach works! For prices: P1Q1+P2Q2=PmixQmixP_1Q_1 + P_2Q_2 = P_{mix}Q_{mix} where P is price and Q is quantity.
Read carefully: if the problem asks 'how much of solution A' - that's your variable. The unknown quantity is what you solve for.

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

C1V1+C2V2=CmixVmixC_1V_1 + C_2V_2 = C_{mix}V_{mix}

Sum of amounts in parts equals amount in mixture

Find Concentration

Cmix=C1V1+C2V2V1+V2C_{mix} = \frac{C_1V_1 + C_2V_2}{V_1 + V_2}

Calculate resulting concentration from known values

Dilution

C1V1=C2V2C_1V_1 = C_2V_2

When diluting: concentration times volume stays constant

Today's challengeOne problem, three tries

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