Solve Mixture Problems: Combining Two Solutions

What This Problem Teaches

• Setting up equations based on the conservation of pure medication
• Converting percentages to decimals for calculations
• Understanding that mixing solutions requires balancing substance amounts, not just volumes
• Solving linear equations with practical constraints (volumes must be positive)
• Verifying solutions by checking final concentrations make physical sense

Visualizing the Problem

Let's create a clear picture of what we're mixing:

Solution: Method 1 — Medication Mass Balance

The key insight is that pure medication is conserved when we mix solutions. The total amount of medication before mixing equals the total amount after mixing.

Step 1 — Define the variable

Let x = milliliters of 16% solution needed.

Step 2 — Calculate pure medication in each component

Pure medication is found by multiplying volume by concentration:

• From 16% solution: x × 0.16 = 0.16x ml of medication
• From 5% solution: 6 × 0.05 = 0.30 ml of medication
• In final mixture: (x + 6) × 0.10 = 0.10(x + 6) ml of medication

Step 3 — Set up the medication balance equation

The medication from both original solutions equals the medication in the final mixture:

Step 4 — Solve the equation

Expand the right side and collect like terms:

Step 5 — State the complete answer

The veterinarian needs 5 ml of the 16% solution.

Solution: Method 2 — The Concentration Difference Approach

This method uses the concept that we're moving the 5% solution "up" to 10%, while the 16% solution gets "diluted down" to 10%.

Step 1 — Calculate concentration differences

Find how far each solution is from the target concentration:

• 5% solution needs to increase by: 10% - 5% = 5%
• 16% solution will decrease by: 16% - 10% = 6%

Step 2 — Set up the ratio

The volume ratio should be inversely related to the concentration differences:

Step 3 — Solve the proportion

This confirms our answer: 5 ml of the 16% solution.

Verification

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

Calculate total medication

• From 5 ml of 16% solution: 5 × 0.16 = 0.80 ml
• From 6 ml of 5% solution: 6 × 0.05 = 0.30 ml
• Total medication: 0.80 + 0.30 = 1.10 ml

Calculate total volume and final concentration

• Total volume: 5 + 6 = 11 ml
• Final concentration: 1.10 ÷ 11 = 0.10 = 10% ✓

Perfect! Our answer gives exactly the required 10% concentration.

Common Pitfalls

Does This Seem Reasonable?

Let's do a sanity check on our answer of 5 ml:

• Volume check: We're adding 5 ml of strong solution to 6 ml of weak solution. Since 5 < 6, we expect the final concentration to be closer to the weak solution, which it is (10% is closer to 5% than to 16%).
• Concentration logic: The 16% solution is 6 percentage points above our target, while the 5% solution is 5 points below. We need slightly less of the stronger solution, which matches our 5:6 ratio.
• Practical context: For treating a small bird, 11 ml total volume (5 + 6) is a reasonable amount for a veterinarian to work with.

The answer passes all reasonableness tests.

The Pattern Behind This

All mixture problems follow the same fundamental principle: the amount of pure substance is conserved.

This same structure applies whether you're mixing:

• Medication solutions (like this problem)
• Salt water of different concentrations
• Alcohol solutions for chemistry
• Fertilizer mixtures for agriculture

The key is identifying what substance is being conserved (medication, salt, alcohol, etc.) and setting up the balance equation correctly.

Why This Matters

Mixture calculations appear throughout healthcare and science:

• Pharmacy: Compounding custom medication strengths for patients with specific needs
• Veterinary medicine: Adjusting medication concentrations for different animal sizes and species
• Laboratory work: Preparing reagent solutions with precise concentrations for experiments
• Food industry: Creating products with specific nutrient or additive concentrations

In clinical settings, these calculations are often double-checked by multiple professionals because concentration errors can have serious consequences for patient safety.