In a data analysis project, three test scores a, b, and c have a = 24 and b = 24....

1. TRANSLATE the problem condition

• Given information:
  • \(\mathrm{a = 24, b = 24}\)
  • We need the arithmetic mean to equal each of the three scores

• What this means mathematically: \(\mathrm{Mean = a = b = c}\)

2. INFER what this condition requires

• If the mean equals each individual score, then all scores must be equal
• Since \(\mathrm{a = 24}\) and \(\mathrm{b = 24}\) already, we need \(\mathrm{c = 24}\) as well
• But let's verify this algebraically

3. Set up the mean equation

• Mean = \(\mathrm{\frac{a + b + c}{3} = \frac{24 + 24 + c}{3} = \frac{48 + c}{3}}\)
• For this to equal each score, it must equal 24

4. SIMPLIFY to solve for c

• Set up equation: \(\mathrm{\frac{48 + c}{3} = 24}\)
• Multiply both sides by 3: \(\mathrm{48 + c = 72}\)
• Subtract 48: \(\mathrm{c = 24}\)

5. Verify the solution

• With \(\mathrm{c = 24}\): Mean = \(\mathrm{\frac{24 + 24 + 24}{3} = \frac{72}{3} = 24}\) ✓
• This equals each individual score

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "arithmetic mean equals each of the three scores" and think it means the mean should equal the sum of the scores, or they don't recognize that this requires all three scores to be equal.

This conceptual confusion leads them to set up incorrect equations or try different approaches that don't address the core requirement. They might calculate means for each answer choice without understanding what condition needs to be satisfied, leading to random guessing among the choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{\frac{48 + c}{3} = 24}\) but make algebraic errors when solving, such as forgetting to multiply both sides by 3 or making arithmetic mistakes.

This may lead them to select Choice C (12) or Choice D (8) based on incorrect calculations, or they abandon the systematic approach and guess.

The Bottom Line:

The key insight is recognizing that when a mean equals each individual value, all values must be identical. Students who miss this fundamental relationship about means struggle to set up the problem correctly.