Two customers purchased the same kind of bread and eggs at a store. The first customer paid $12.45 for 1...

1. TRANSLATE the problem information

• Given information:
  • Customer 1: 1 loaf of bread + 2 dozen eggs = $12.45
  • Customer 2: 4 loaves of bread + 1 dozen eggs = $19.42
  • Find: cost of 1 dozen eggs

• Define variables:
  • Let b = cost of 1 loaf of bread
  • Let e = cost of 1 dozen eggs

• TRANSLATE to equations:
  • Customer 1: \(\mathrm{b + 2e = 12.45}\)
  • Customer 2: \(\mathrm{4b + e = 19.42}\)

2. INFER the solution approach

• We have a system of two linear equations with two unknowns
• Since we want to find e (eggs), we should eliminate b (bread)
• Elimination method works well here since we can multiply the first equation by -4

3. SIMPLIFY using elimination

• Multiply first equation by -4:
  \(\mathrm{-4(b + 2e) = -4(12.45)}\)
  \(\mathrm{-4b - 8e = -49.8}\)

• Add this to the second equation:
  \(\mathrm{(-4b - 8e) + (4b + e) = -49.8 + 19.42}\)
  \(\mathrm{-7e = -30.38}\)

• Solve for e:
  \(\mathrm{e = 30.38 \div 7 = 4.34}\) (use calculator)

Answer: D. 4.34

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may set up equations incorrectly, such as writing \(\mathrm{2b + e = 12.45}\) instead of \(\mathrm{b + 2e = 12.45}\), misunderstanding which coefficient goes with which variable.

This leads to a completely different system that produces wrong values for both variables, causing confusion and typically results in guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the system but make arithmetic errors during elimination, such as incorrectly multiplying \(\mathrm{-4 \times 12.45 = -49.8}\) or making sign errors when combining terms.

These calculation mistakes lead to incorrect values, potentially causing them to select Choice B (3.88) or Choice C (4.15).

Third Most Common Error:

Poor final interpretation: Students solve the system correctly but get confused about which variable represents what, solving for \(\mathrm{b = 3.77}\) (cost of bread) instead of \(\mathrm{e = 4.34}\) (cost of eggs).

This may lead them to select Choice A (3.77).

The Bottom Line:

This problem requires careful translation of English to mathematical equations, systematic algebraic manipulation, and clear tracking of what each variable represents throughout the solution process.