A bakery sells croissants and muffins. Yesterday, the bakery sold C croissants and M muffins for a total profit represented...

1. TRANSLATE the equation information

• Given equation: \(\mathrm{6C + 3M = 180}\)
• This represents: (profit per croissant) × (croissants sold) + (profit per muffin) × (muffins sold) = total profit
• What this tells us:
  • Coefficient 6 = profit per croissant
  • Coefficient 3 = profit per muffin

2. INFER what the question is asking

• The question asks for the difference in profit per item
• We don't need to find how many croissants or muffins were sold
• We just need the difference: (profit per croissant) - (profit per muffin)

3. Calculate the difference

• Profit per croissant = $6
• Profit per muffin = $3
• Difference = \(\mathrm{6 - 3 = 3}\) dollars

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that coefficients represent profit per item, instead thinking they need to solve for C and M first.

They might try to find specific values like "\(\mathrm{C = 20, M = 20}\)" by solving the equation, then get confused about what to do next. This leads to unnecessary complexity and potential calculation errors, causing them to abandon the systematic approach and guess.

Second Most Common Error:

Poor INFER reasoning: Students understand the coefficients but misinterpret the question, thinking it asks for total profit difference rather than profit difference per item.

They might calculate something like "If \(\mathrm{C = 10}\) and \(\mathrm{M = 40}\), then total croissant profit is 60 and total muffin profit is 120, so the difference is 60." This conceptual confusion about the question leads them to work with totals instead of per-unit rates.

The Bottom Line:

This problem tests whether students understand that coefficients in linear equations often represent rates or prices per unit, not just arbitrary numbers to manipulate algebraically.