Lesson 7.12

Word Problems: Mixtures

The dreaded mixture problem! "Mix a 20% solution with a 50% solution to get a 30% solution." Don't worry—it's just a Value Problem in disguise.

Introduction

Whether you are mixing acid in a lab, interest rates at a bank, or nuts at a grocery store, the math is the same. We track the Total Amount and the Pure Ingredient.

Past Knowledge

Lesson 7.11 (Value Problems). Instead of "Price", we use "Percentage" as the value.

Today's Goal

Use the "Bucket Method" to visualize and solve mixture systems.

Future Success

This skill is heavily used in chemistry (stoichiometry solutions) and nursing (dosage calc).

Key Concepts

The "Bucket Method"

Draw 3 buckets: Bucket A + Bucket B = Final Bucket.

20%

50%

10 Liters

30%

Bucket Equation (Volume)

Ingredient Equation (Stuff)

Worked Examples

Example 1: Chemical Solutions

Basic

How many liters of 10% acid and 40% acid must be mixed to get 12 liters of 20% acid?

Step 1: Set Up Equations

Volume:

Acid:

Clean up Acid Eq:

(Multiply by 100)

Step 2: Solve

Target . Multiply Volume by -10.

So .

8L of 10%, 4L of 40%

Example 2: Investment Interest

Intermediate

You invest $5000 total. Part is in a safe account (2% interest) and part in risky stocks (8% interest). You made $220 total interest. How much in each?

Step 1: Buckets

Total Money:

Interest:

Step 2: Solve

Clear decimals (x100):

Multiply Top by -2:

Add:

$2000 in Risky, $3000 in Safe

Example 3: Dry Mix (Nuts)

Advanced

Cashews cost $10/lb. Peanuts cost $4/lb. We want 20lbs of mixed nuts worth $5.50/lb.

Step 1: The "Final Value"

This is tricky! The final value isn't given directly.

Final Value = total.

Step 2: System

Weight:

Cost:

Step 3: Solve

Multiply top by -4:

Add:

5lbs Cashews, 15lbs Peanuts

Common Pitfalls

Forgetting the Right Side

In , don't forget to multiply the final percentage by the total liters! Often students just write which is wrong.

Pure Ingredients

Watch out for "Pure Acid" (100% or 1.00) or "Pure Water" (0% or 0.00). If you add water, the percent is 0.

Real-Life Applications

Pharmacy:

IV drips must be exact. If a patient needs a 15% solution but the hospital only has 10% and 50% bags, the nurse must do this calculation.
Being wrong could injure the patient. Math saves lives!