To earn a B grade in Mr. Johnson's math class, a student's test average must be between 85 and 89...

1. TRANSLATE the problem information

• Given information:
  • Current test scores: 83, 87, 94
  • B grade requirement: average between 85 and 89 points, inclusive
  • Need to find: possible fourth test score
• What this tells us: We need to set up an inequality for the average of four tests.

2. TRANSLATE the mathematical setup

• Let \(\mathrm{x}\) = fourth test score
• Average of four tests = \(\mathrm{\frac{83 + 87 + 94 + x}{4} = \frac{264 + x}{4}}\)
• B grade requirement becomes: \(\mathrm{85 \leq \frac{264 + x}{4} \leq 89}\)

3. SIMPLIFY to solve for x

• Multiply all parts of the inequality by 4:
  \(\mathrm{85 \times 4 \leq \frac{264 + x}{4} \times 4 \leq 89 \times 4}\)
  \(\mathrm{340 \leq 264 + x \leq 356}\)
• Subtract 264 from all parts:
  \(\mathrm{340 - 264 \leq x \leq 356 - 264}\)
  \(\mathrm{76 \leq x \leq 92}\)

4. APPLY CONSTRAINTS to check answer choices

• Jessica's fourth test score must be between 76 and 92 points
• Check each option:
  • (A) 74: Too low (less than 76)
  • (B) 84: Perfect (between 76 and 92)
  • (C) 94: Too high (greater than 92)
  • (D) 98: Too high (greater than 92)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse "between 85 and 89, inclusive" and set up the inequality incorrectly, perhaps using strict inequalities (< and >) instead of inclusive ones (\(\mathrm{\leq}\) and \(\mathrm{\geq}\)), or they might think Jessica needs exactly 85 or 89 points rather than understanding the range concept.

This often leads to setting up wrong constraints or misunderstanding what scores are acceptable, causing them to select Choice A (74) or Choice C (94) without proper justification.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{85 \leq \frac{264 + x}{4} \leq 89}\) but make algebraic errors when solving, such as forgetting to multiply/subtract from all three parts of the compound inequality or making arithmetic mistakes with 264 + 83 + 87 + 94.

This leads to getting the wrong range for x, potentially causing them to accept Choice C (94) or Choice D (98) as valid when they exceed the actual upper bound.

The Bottom Line:

This problem challenges students to translate a real-world grading scenario into mathematical inequalities and then systematically work through compound inequality algebra. The key insight is recognizing that "between...inclusive" creates a range of acceptable values, not just specific target numbers.