A bakery sells trays of cookies. Each tray contains at least 50 cookies but no more than 60. Which of...

1. TRANSLATE the problem constraints

• Given information:
  • Each tray: at least 50 cookies but no more than 60 cookies
  • Total trays: 4
  • Need: possible total number of cookies

• What this tells us: For each tray, \(\mathrm{50 \leq cookies \leq 60}\)

2. INFER the solution strategy

• To find what's possible for 4 trays, we need to find the range of possible totals
• Strategy: Calculate the minimum possible total and maximum possible total
• Any valid answer must fall between these bounds

3. Calculate the bounds

• Minimum total: If each tray has the least possible (50 cookies)
  • Minimum = \(\mathrm{50 \times 4 = 200}\) cookies
• Maximum total: If each tray has the most possible (60 cookies)
  • Maximum = \(\mathrm{60 \times 4 = 240}\) cookies

4. APPLY CONSTRAINTS to check answer choices

• Valid range: \(\mathrm{200 \leq total\ cookies \leq 240}\)
• Check each option:
  • A. 165: Too small (\(\mathrm{165 \lt 200}\)) ✗
  • B. 205: In range (\(\mathrm{200 \leq 205 \leq 240}\)) ✓
  • C. 245: Too large (\(\mathrm{245 \gt 240}\)) ✗
  • D. 285: Too large (\(\mathrm{285 \gt 240}\)) ✗

Answer: B. 205

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand constraint language like "at least" and "no more than," leading to incorrect bounds.

For example, they might think "at least 50" means exactly 50, or "no more than 60" means less than 60. This leads to wrong calculations like using 49 as minimum or 59 as maximum, throwing off their entire range calculation and causing them to select an incorrect answer or get confused and guess.

Second Most Common Error:

Poor INFER reasoning: Students calculate individual tray bounds correctly but fail to recognize they need to find the range for the total.

Instead of finding minimum and maximum totals, they might try to work backwards from answer choices or use the average (55 cookies per tray). This scattered approach leads to confusion and often results in selecting Choice A (165) if they somehow calculate \(\mathrm{55 \times 3}\) instead of \(\mathrm{55 \times 4}\), or causes them to abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can systematically work with ranges and constraints rather than getting caught up in the specific numbers. The key insight is recognizing that constraint problems require finding bounds first, then checking which answers fall within those bounds.