The average (arithmetic mean) weight of a group of 12 packages is 15 pounds. Let this be Mean P. One...

1. TRANSLATE the problem information

• Given information:
  • 12 packages with average weight of 15 pounds (\(\mathrm{Mean\ P}\))
  • Remove one package weighing 26 pounds
  • Find average of remaining 11 packages (\(\mathrm{Mean\ R}\))
  • Compare \(\mathrm{Mean\ P}\) and \(\mathrm{Mean\ R}\)

2. INFER the key relationship

• The crucial insight: We're removing a 26-pound package from a group with a 15-pound average
• Since \(\mathrm{26 \gt 15}\), we're removing something heavier than the current average
• When you remove a value above the mean, the remaining values will have a lower average

3. Apply the conceptual understanding

• \(\mathrm{Mean\ P = 15}\) pounds (original average)
• Since we removed something above average, \(\mathrm{Mean\ R \lt Mean\ P}\)
• Therefore: \(\mathrm{Mean\ P \gt Mean\ R}\)

4. SIMPLIFY with calculations (verification)

• Original total weight: \(\mathrm{12 \times 15 = 180}\) pounds
• New total weight: \(\mathrm{180 - 26 = 154}\) pounds
• New average: \(\mathrm{154 \div 11 = 14}\) pounds
• Confirms our reasoning: \(\mathrm{15 \gt 14}\)

Answer: A. Mean P is greater than Mean R.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students focus only on calculations without grasping the conceptual relationship between removed values and resulting means. They may correctly calculate both averages but fail to recognize the logical pattern: removing values above the mean always decreases the average, while removing values below the mean increases it. This conceptual gap leads to confusion about which direction the comparison should go, causing them to guess randomly among the choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt the computational approach but make arithmetic errors in calculating totals or the final division (\(\mathrm{154 \div 11}\)). Common mistakes include miscalculating the original total (\(\mathrm{12 \times 15}\)) or the division step. These calculation errors can produce incorrect values that don't match any answer choice clearly, leading to confusion and potentially selecting Choice D (not enough information) when they can't make sense of their results.

The Bottom Line:

This problem rewards conceptual thinking over pure computation. Students who recognize the fundamental principle—that removing above-average values decreases the mean—can solve it instantly, while those who rely solely on calculations risk arithmetic errors and miss the elegant logical relationship at the problem's heart.