Question:A delivery service charges a flat handling fee of $5 per order and $8 per package delivered.The total cost, C...

1. TRANSLATE the problem information

• Given information:
  • Cost model: \(\mathrm{C = 8n + 5}\)
  • \(\mathrm{C}\) represents total cost in dollars
  • \(\mathrm{n}\) represents number of packages
  • We need to find \(\mathrm{C}\) when \(\mathrm{n = 7}\) packages

2. SIMPLIFY by substituting the given value

• Substitute \(\mathrm{n = 7}\) into the equation \(\mathrm{C = 8n + 5}\):
  \(\mathrm{C = 8(7) + 5}\)

• Calculate step by step:
  • First: \(\mathrm{8 \times 7 = 56}\)
  • Then: \(\mathrm{56 + 5 = 61}\)

Answer: 61

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students make arithmetic calculation errors, particularly in the multiplication step.

Common mistakes include calculating \(\mathrm{8 \times 7}\) incorrectly (getting 54 or 63) or making addition errors when combining \(\mathrm{56 + 5}\). These calculation errors lead to incorrect final answers and cause students to doubt their approach even when their method is correct.

Second Most Common Error:

Weak TRANSLATE reasoning: Students misunderstand what the problem is asking or misinterpret the variables.

Some students might think they need to solve for \(\mathrm{n}\) instead of \(\mathrm{C}\), or they might not recognize that they simply need to substitute the given value. This confusion leads to unnecessary complication of a straightforward substitution problem, causing them to get stuck and guess.

The Bottom Line:

This problem tests basic substitution skills in a real-world context. Success depends on careful reading of what's being asked and accurate arithmetic execution.