Alcohol Mixture Problem: Finding Solution Volumes

What This Problem Teaches

• How to use one variable to represent two related quantities when there's a fixed difference
• Setting up equations based on pure substance content rather than total volume
• Converting percentage concentrations to decimal multipliers in calculations
• Recognizing when a mixture problem has constraints that link the variables
• Checking whether calculated concentrations fall within reasonable ranges

Solution: Method 1 — The Single Variable Approach

The key insight is that since Solution A is always 500 mL less than Solution B, we can express both volumes using just one variable.

Step 1 — Define the variable using the relationship

Let b = milliliters of Solution B that Juan uses.

Then Solution A = b - 500 milliliters (since A is 500 mL less than B).

Step 2 — Set up the pure alcohol equation

The total pure alcohol comes from both solutions:

Step 3 — Expand and collect like terms

Distribute the 0.17:

Step 4 — Solve for b

Add 85 to both sides:

Since we're dealing with milliliters, we can round to 1355 mL.

Solution: Method 2 — Setting Up a System

We can also approach this by explicitly writing two equations and solving the system.

Step 1 — Define two variables

Let a = milliliters of Solution A

Let b = milliliters of Solution B

Step 2 — Write the relationship equation

Since A is 500 mL less than B:

Step 3 — Write the pure alcohol equation

Step 4 — Substitute and solve

Substitute a = b - 500 into the pure alcohol equation:

This gives us the same result: approximately 1355 mL of Solution B.

Verification

Let's check our answer by substituting back:

If Solution B = 1355 mL, then Solution A = 1355 - 500 = 855 mL

Pure alcohol from each solution:

• From Solution A: 0.17 × 855 = 145.35 mL
• From Solution B: 0.14 × 1355 = 189.7 mL
• Total pure alcohol: 145.35 + 189.7 = 335.05 mL ✓

The slight difference (335.05 vs 335) is due to rounding, confirming our answer is correct.

Does This Seem Reasonable?

Let's do a quick sanity check on the final concentration:

• Total volume: 855 + 1355 = 2210 mL
• Final concentration: 335 ÷ 2210 ≈ 15.2%

This makes sense! The final concentration (15.2%) falls between the two original concentrations (14% and 17%). Since we used more of the weaker solution (1355 mL of 14%) than the stronger one (855 mL of 17%), the result is closer to 14%, which is exactly what we see.

Common Pitfalls

How to Spot This Problem Type

Look for these tell-tale phrases in mixture problems:

• "X milliliters less than" or "X milliliters more than" — signals a fixed difference relationship
• "The resulting mixture has X milliliters of pure [substance]" — tells you to focus on the substance, not total volume
• Two different percentage concentrations — means you'll be working with decimal multipliers
• "How many milliliters of [specific solution]" — usually the variable you should define first

Any time you see a fixed numerical relationship between two quantities in a mixture problem, you can almost always use a single variable approach to simplify the algebra.

The Pattern Behind This

This is a constrained mixture problem with the general form:

This simplifies to a single linear equation: (c₁ + c₂)V₂ - c₁k = Total pure substance

For our problem: (0.17 + 0.14)V₂ - 0.17(500) = 335, which gives 0.31V₂ = 420.

The constraint actually makes these problems easier than unrestricted mixture problems because it reduces the variables from two to one.

What If?