Mixture Problem: Blending Orange Juice

Skills This Problem Builds

• Setting up equations based on conservation of pure substances
• Translating percentage concentrations into algebraic expressions
• Understanding the difference between total volume and pure content
• Recognizing mixture problems as weighted average situations
• Connecting real-world scenarios to algebraic modeling

Visualizing the Problem

Let's organize what we know in a table to see the relationship between volumes, concentrations, and pure juice content:

The key insight: the pure juice in the final mixture must equal the sum of pure juice from both original components.

Solution: Method 1 — Pure Substance Balance

Step 1 — Define the variable

Let x = gallons of 70% orange juice we need to add.

Step 2 — Identify the pure juice amounts

From the 70% juice: 0.7x gallons of pure juice
From the 30% juice: 0.3 × 16 = 4.8 gallons of pure juice
In the final mixture: 0.5(x + 16) gallons of pure juice

Step 3 — Set up the equation

The pure juice from both components equals the pure juice in the final mixture:

Step 4 — Solve for x

Expand the right side:

Subtract 0.5x from both sides:

Subtract 4.8 from both sides:

Divide both sides by 0.2:

Solution: Method 2 — The Weighted Average Approach

Step 1 — Set up the weighted average

The final concentration is a weighted average of the two original concentrations, where the weights are the volumes:

Step 2 — Substitute known values

We want a 50% final concentration:

Step 3 — Cross multiply

Step 4 — Solve

This gives us the same equation as Method 1:

Verification

Let's check our answer by substituting back into the original problem:

• Pure juice from 70% component: 0.7 × 16 = 11.2 gallons
• Pure juice from 30% component: 0.3 × 16 = 4.8 gallons
• Total pure juice: 11.2 + 4.8 = 16 gallons
• Total volume: 16 + 16 = 32 gallons
• Final concentration: 16 ÷ 32 = 0.5 = 50% ✓

Perfect! The verification confirms our answer is correct.

Reality Check

Does our answer make intuitive sense? We're mixing equal amounts (16 gallons each) of 30% and 70% juice. The average of 30% and 70% is indeed 50%, so we'd expect equal volumes to give us the target concentration.

If we needed more of the stronger solution, that would mean our target concentration was closer to 70%. If we needed less, the target would be closer to 30%. Since our target of 50% is exactly halfway between 30% and 70%, equal volumes make perfect sense.

Common Pitfalls

The Underlying Pattern

All mixture problems follow this general structure:

Where "Amount × Concentration" represents the pure substance in each component. This pattern works for any mixture problem:

• Mixing solutions of different concentrations
• Combining alloys with different metal contents
• Blending coffee beans of different strengths
• Mixing investments with different interest rates

Where This Shows Up in Real Life

• Food industry: Blending different grades of milk to achieve specific fat content, or mixing fruit juices to create signature blends.
• Manufacturing: Creating metal alloys with precise compositions for aerospace and automotive applications—too much of one metal makes the alloy brittle, too little makes it weak.
• Pharmacy: Compounding medications by mixing solutions of different concentrations to create custom dosages that aren't available commercially.
• Agriculture: Mixing fertilizers with different nitrogen contents to achieve optimal soil nutrition for specific crops.

What If?