A chemist has a supply of a 10% acid solution and a 30% acid solution. The chemist mixes a certain...

1. TRANSLATE the problem information

• Given information:
  • 10% acid solution (unknown amount)
  • 30% acid solution (unknown amount)
  • Final mixture: 50 liters at 22% acid concentration

• What this tells us: We need two equations - one for total volume and one for total acid content

2. TRANSLATE into mathematical equations

• Let \(\mathrm{x}\) = liters of 10% solution, \(\mathrm{y}\) = liters of 30% solution

• Volume equation: \(\mathrm{x + y = 50}\)
• Acid equation: \(\mathrm{0.10x + 0.30y = 0.22(50) = 11}\)

3. INFER the solution approach

• We have a system of two equations with two unknowns
• Substitution method will work well since the first equation easily gives us y in terms of x

4. SIMPLIFY using substitution

• From equation 1: \(\mathrm{y = 50 - x}\)
• Substitute into equation 2: \(\mathrm{0.10x + 0.30(50 - x) = 11}\)
• Distribute: \(\mathrm{0.10x + 15 - 0.30x = 11}\)
• Combine like terms: \(\mathrm{-0.20x + 15 = 11}\)
• Isolate x: \(\mathrm{-0.20x = -4}\), so \(\mathrm{x = 20}\)

5. Verify the answer

• If \(\mathrm{x = 20}\), then \(\mathrm{y = 30}\)
• Check: \(\mathrm{0.10(20) + 0.30(30) = 2 + 9 = 11}\) ✓
• Concentration: \(\mathrm{11/50 = 22\%}\) ✓

Answer: 20 liters

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to set up the acid content equation correctly. They might write something like \(\mathrm{10x + 30y = 22(50)}\) instead of \(\mathrm{0.10x + 0.30y = 0.22(50)}\), forgetting to convert percentages to decimals. This leads to completely wrong numbers and typically results in confusion and random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when distributing \(\mathrm{0.30(50 - x)}\) or when combining like terms. A common mistake is getting the signs wrong: writing \(\mathrm{+0.20x}\) instead of \(\mathrm{-0.20x}\), leading to \(\mathrm{x = -20}\). Since negative volume doesn't make sense, this causes confusion and may lead them to select Choice (D) (30) by incorrectly assuming they need the other solution.

The Bottom Line:

This problem tests whether students can bridge the gap between percentage language and decimal mathematics while maintaining algebraic accuracy throughout a multi-step solution.