A student needs to maintain an average of at least 78 points across 5 quizzes to pass a course. The...

1. TRANSLATE the problem information

• Given information:
  • Target average: at least 78 points across 5 quizzes
  • Current quiz scores: 72, 81, 75, 77 points
  • Need: minimum score for 5th quiz

• What this tells us: If \(\mathrm{average = \frac{total}{number\ of\ quizzes}}\), then \(\mathrm{total = average \times number\ of\ quizzes}\)

2. INFER the approach

• Strategic insight: To find the minimum 5th quiz score, work backwards from the total points needed
• First find total points required, then subtract current points to get the minimum additional points needed

3. SIMPLIFY to find total points needed

• Total points needed = \(\mathrm{78 \times 5 = 390}\) points

4. SIMPLIFY to find current total

• Current total = \(\mathrm{72 + 81 + 75 + 77 = 305}\) points

5. SIMPLIFY to find minimum 5th quiz score

• Minimum 5th quiz score = \(\mathrm{390 - 305 = 85}\) points

Answer: C (85)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "average of at least 78" and think they just need to score 78 on the fifth quiz, not realizing they need to compensate for below-average earlier scores.

This reasoning leads them to select Choice A (78) without doing any calculations.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students understand the correct approach but make arithmetic errors when calculating totals or performing the final subtraction.

Common calculation mistakes include:

• Miscalculating \(\mathrm{78 \times 5}\) (getting 385 instead of 390)
• Adding the current scores incorrectly
• Subtracting incorrectly in the final step

These computational errors may lead them to select Choice B (80) or Choice D (87).

The Bottom Line:

This problem tests whether students can work backwards from an average requirement to find a missing value. The key insight is recognizing that because some quiz scores are below the target average, the final quiz must be above average to compensate.