Tom scored 85, 78, and 98 on his first three exams in history class. Solving which inequality gives the score,...

1. TRANSLATE the problem information

• Given information:
  • First three exam scores: 85, 78, 98
  • Fourth exam score: G (unknown)
  • Need mean of all four exams to be at least 90

• "At least 90" means \(\geq 90\)

2. INFER what mean formula to use

• Mean requires ALL four scores in the calculation
• Mean = (sum of all scores) ÷ (number of scores)
• For four exams: Mean = \(\frac{85 + 78 + 98 + \mathrm{G}}{4}\)

3. Set up the inequality

• We need: \(\frac{85 + 78 + 98 + \mathrm{G}}{4} \geq 90\)
• This directly matches Choice C

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget to include G in the mean calculation, thinking the mean should only involve the known scores.

They incorrectly reason: "The mean of the first three scores needs to be compared to 90 minus something involving G." This leads them to select Choice D: \(\frac{85 + 78 + 98}{4} \geq 90 - 4\mathrm{G}\), which only includes three scores in the mean instead of four.

Second Most Common Error:

Poor TRANSLATE execution: Students correctly identify that all four scores need to be in the mean, but make algebraic errors when setting up the inequality.

They might multiply through by 4 incorrectly or rearrange terms wrong, leading them toward Choice B, which has the right idea but incorrect algebraic structure.

The Bottom Line:

This problem tests whether students truly understand what "mean of four exams" means - it must include ALL four scores, including the unknown one. Many students get confused about whether to include G in the calculation or treat it separately.