Mixture Problem: Antifreeze Solution

What This Problem Teaches

• How to track the amount of a substance (not just percentages) through mixture changes
• Setting up equations where the same unknown appears in both "removal" and "addition" terms
• Understanding that draining removes substance proportionally to current concentration
• Recognizing drain-and-replace as a fundamental mixture operation in real applications
• Converting between percentage concentrations and actual quantities

Solution: Method 1 — The Antifreeze Balance Approach

The key insight is to track the actual amount of antifreeze, not just the percentages. We'll follow where the antifreeze goes: some gets removed when we drain, and some gets added when we replace.

Step 1 — Calculate the current antifreeze amount

The radiator currently holds:

Step 2 — Set up what happens when we drain and replace

Let x = quarts of fluid drained and replaced.

When we drain x quarts of the 30% mixture, we remove:

When we replace those x quarts with pure antifreeze, we add:

Step 3 — Write the equation for final antifreeze amount

The final mixture should be 60% antifreeze out of 13 quarts total:

The balance equation is:

Step 4 — Solve for x

Simplify the left side:

Let's be more precise:

Solution: Method 2 — The Final Composition Setup

Instead of tracking the changes, we can set up an equation based on what the final mixture contains.

Step 1 — Identify what's in the final mixture

After draining x quarts and replacing with pure antifreeze, we still have 13 quarts total. This final mixture contains antifreeze from two sources:

• Antifreeze remaining from the original mixture: (13-x) × 0.30 quarts
• Pure antifreeze added: x quarts

Step 2 — Set up the final concentration equation

The total antifreeze in the final 13 quarts should equal 60% of 13:

Step 3 — Solve the equation

Expand and simplify:

This confirms our answer using a different setup approach.

Verification

Let's check our answer by calculating the final concentration:

Starting mixture after draining 5.57 quarts:

After adding 5.57 quarts of pure antifreeze:

Using the exact fraction 39/7:

Common Pitfalls

The General Formula

For any drain-and-replace mixture problem, if we drain and replace x units:

Where:

• Concentration₁ = current concentration (what's being drained)
• Concentration₂ = concentration of replacement fluid
• Target amount = desired concentration × total volume

This simplifies to:

Real Applications

This exact calculation appears frequently in practical situations:

• Automotive maintenance: Adjusting antifreeze concentration for different climate zones or seasonal changes
• Swimming pool chemistry: Adjusting chlorine or pH buffer concentrations by partial water changes
• Industrial manufacturing: Adjusting solution concentrations in chemical processes, plating baths, or cleaning solutions
• Medical applications: Diluting or concentrating IV solutions, preparing topical treatments with specific active ingredient percentages

Does the Answer Make Sense?

Let's check whether 5.57 quarts seems reasonable: