A sample of a certain alloy has a total mass of 50.0 grams and is 50.0% silicon by mass. The...

1. TRANSLATE the problem information

• Given information:
  • Total sample: 50.0 grams, 50.0% silicon
  • First piece: 30.0% silicon by mass
  • Second piece: 80.0% silicon by mass
  • Need: mass of silicon in second piece

• What this tells us: We need to find how much each piece weighs, then calculate the silicon content.

2. INFER the approach

• This is a mixture problem requiring a system of equations
• Let x = mass of first piece, y = mass of second piece
• We need two equations: one for total mass, one for silicon content

3. TRANSLATE into mathematical equations

• Total mass equation: \(\mathrm{x + y = 50.0}\)
• Silicon content equation: \(\mathrm{0.300x + 0.800y = 0.500(50.0) = 25.0}\)

4. INFER the solution method

• Use substitution: solve the first equation for x, then substitute into the second equation

5. SIMPLIFY through algebraic steps

• From \(\mathrm{x + y = 50.0}\), get \(\mathrm{x = 50.0 - y}\)
• Substitute: \(\mathrm{0.300(50.0 - y) + 0.800y = 25.0}\)
• Distribute: \(\mathrm{15.0 - 0.300y + 0.800y = 25.0}\)
• Combine like terms: \(\mathrm{15.0 + 0.500y = 25.0}\)
• Solve: \(\mathrm{0.500y = 10.0}\), so \(\mathrm{y = 20.0}\)

6. INFER the final calculation

• Second piece weighs 20.0 grams
• Silicon in second piece = \(\mathrm{0.800 \times 20.0 = 16.0}\) grams

Answer: B. 16.0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students set up incorrect equations, often confusing what quantities should add up to what totals. They might write something like "\(\mathrm{0.30 + 0.80 = 0.50}\)" thinking the percentages should add directly, rather than understanding that the weighted average of silicon content equals the final percentage.

This leads to confusion and abandoning systematic solution, resulting in guessing.

Second Most Common Error:

Incomplete solution execution: Students correctly find that the second piece has mass 20.0 grams but stop there, forgetting that the question asks for the mass of silicon IN the second piece, not the total mass of the second piece.

This may lead them to select Choice C (20.0).

Third Most Common Error:

Calculation mix-up during final step: Students correctly solve the system and find x = 30.0, y = 20.0, but then calculate the silicon content of the wrong piece, finding \(\mathrm{0.300 \times 30.0 = 9.0}\) instead of \(\mathrm{0.800 \times 20.0 = 16.0}\).

This may lead them to select Choice A (9.0).

The Bottom Line:

Mixture problems require careful translation of percentage relationships into algebraic equations, followed by systematic solution and attention to exactly what quantity the question is asking for.