A construction project requires 127 screws total. The contractor already has 23 screws in stock. Additional screws are sold in...

1. TRANSLATE the problem information

• Given information:
  • Total screws needed: 127
  • Screws already in stock: 23
  • Package size: 20 screws per package
  • Find: minimum number of packages to purchase

2. TRANSLATE to find how many more screws are needed

• Screws still needed = Total needed - Already have
• \(\mathrm{127 - 23 = 104}\) screws needed

3. TRANSLATE to determine theoretical packages needed

• Packages needed = Screws needed ÷ Package size
• \(\mathrm{104 \div 20 = 5.2}\) packages

4. INFER the purchasing constraint and APPLY CONSTRAINTS

• Since you cannot buy 0.2 of a package, you must buy whole packages only
• To meet the requirement of at least 127 screws, round UP to 6 packages
• (Rounding down to 5 would leave you short of screws)

5. INFER the need to verify your answer

• Check: \(\mathrm{6 \times 20 = 120}\) new screws
• Total screws: \(\mathrm{120 + 23 = 143}\) screws
• Since \(\mathrm{143 \geq 127}\), this works ✓

Answer: C) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students calculate 5.2 packages but round DOWN to 5 packages, thinking this minimizes cost.

They calculate \(\mathrm{5 \times 20 + 23 = 123}\) screws total, which is less than the 127 needed, but don't recognize this creates a shortage. This may lead them to select Choice A) 4 or Choice B) 5.

Second Most Common Error:

Poor TRANSLATE execution: Students forget to subtract the screws already in stock and divide \(\mathrm{127 \div 20 = 6.35}\), then round up to 7.

This leads them to purchase more packages than necessary and select Choice D) 7.

The Bottom Line:

This problem tests whether students understand that "minimum packages" means the smallest number that still meets the requirement - requiring them to round UP when dealing with discrete packaging units, even though this seems counterintuitive to minimizing purchases.