A shipping company charges customers based on total package weight. Each shipment has a base processing weight of 2 pounds...

1. TRANSLATE the problem information

• Given information:
  • Base processing weight: \(2\) pounds (added to every shipment)
  • Contents weight: \(\mathrm{w}\) pounds
  • Total weight charged: \(2 + \mathrm{w}\) pounds
  • Rate: $3 per pound for total weight
  • Budget limit: $45 maximum

2. INFER the cost calculation approach

• Since the company charges $3 per pound for the TOTAL weight (not just contents), we need to:
  • Find total weight first: \(2 + \mathrm{w}\) pounds
  • Then multiply by the rate: $3 per pound

3. SIMPLIFY the cost expression

• Total cost = \($3 \times (\text{total weight})\)

Total cost = \($3 \times (2 + \mathrm{w})\)

Using distributive property: \($3 \times (2 + \mathrm{w}) = $3 \times 2 + $3 \times \mathrm{w}\)

Total cost = \($6 + $3\mathrm{w}\)

4. INFER the constraint relationship

• Maria has a maximum budget of $45
• This means her cost must be less than or equal to $45
• Set up inequality: \($6 + $3\mathrm{w} \leq $45\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what gets multiplied by the $3 rate, thinking it only applies to the contents weight rather than the total weight.

They might calculate: Cost = \($2 + $3\mathrm{w}\) (thinking the 2 pounds costs $2 and the w pounds costs $3w), leading them to select Choice A (\(2 + 3\mathrm{w} \leq 45\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that total cost = \($3 \times (2 + \mathrm{w})\) but fail to use the distributive property correctly.

They might incorrectly calculate \($3 \times (2 + \mathrm{w})\) as \($3 + $2\mathrm{w}\) instead of \($6 + $3\mathrm{w}\), leading them to select Choice B (\(3 + 2\mathrm{w} \leq 45\)).

The Bottom Line:

This problem tests whether students can correctly translate a multi-component cost structure (base weight + contents weight) and apply the rate to the total, not the individual parts. The key insight is recognizing that the $3 per pound applies to the entire combined weight of \(2 + \mathrm{w}\) pounds.