15, 14, 18, 17, x The mean and the median of the five numbers above are equal. Which of the...

1. TRANSLATE the problem information

• Given information:
  • Five numbers: 15, 14, 18, 17, x
  • Constraint: \(\mathrm{mean = median}\)
  • Need to find: which value of x does NOT satisfy this constraint

2. INFER the solution strategy

• Since the problem asks "Which is NOT a possible value," we need to find the exception
• Strategy: Test each answer choice to see which one makes \(\mathrm{mean \neq median}\)
• For each choice, we'll calculate both mean and median, then compare

3. CONSIDER ALL CASES by testing each answer choice

Testing Choice A: x = 6

• Numbers become: 15, 14, 18, 17, 6
• SIMPLIFY by ordering: 6, 14, 15, 17, 18
• Median = 15 (middle of 5 numbers)
• Mean = \(\mathrm{\frac{15 + 14 + 18 + 17 + 6}{5}}\)
  \(\mathrm{= \frac{70}{5}}\)
  \(\mathrm{= 14}\)
• Since \(\mathrm{14 \neq 15}\), this does NOT satisfy our constraint ❌

Testing Choice B: x = 11

• Numbers become: 15, 14, 18, 17, 11
• SIMPLIFY by ordering: 11, 14, 15, 17, 18
• Median = 15
• Mean = \(\mathrm{\frac{15 + 14 + 18 + 17 + 11}{5}}\)
  \(\mathrm{= \frac{75}{5}}\)
  \(\mathrm{= 15}\)
• Since \(\mathrm{15 = 15}\), this WORKS ✓

Testing Choice C: x = 16

• Numbers become: 15, 14, 18, 17, 16
• SIMPLIFY by ordering: 14, 15, 16, 17, 18
• Median = 16
• Mean = \(\mathrm{\frac{15 + 14 + 18 + 17 + 16}{5}}\)
  \(\mathrm{= \frac{80}{5}}\)
  \(\mathrm{= 16}\)
• Since \(\mathrm{16 = 16}\), this WORKS ✓

Testing Choice D: x = 21

• Numbers become: 15, 14, 18, 17, 21
• SIMPLIFY by ordering: 14, 15, 17, 18, 21
• Median = 17
• Mean = \(\mathrm{\frac{15 + 14 + 18 + 17 + 21}{5}}\)
  \(\mathrm{= \frac{85}{5}}\)
  \(\mathrm{= 17}\)
• Since \(\mathrm{17 = 17}\), this WORKS ✓

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss that the question asks for what is NOT possible, so they look for a value that DOES make \(\mathrm{mean = median}\) instead of one that doesn't.

They might calculate correctly but then select a choice like B, C, or D because those values actually work. They essentially answer the opposite question.

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors when finding the mean or mistakes when ordering numbers to find the median.

For example, they might incorrectly order 15, 14, 18, 17, 6 or make arithmetic errors in calculating \(\mathrm{\frac{15 + 14 + 18 + 17 + 6}{5}}\). This leads to confusion and potentially random answer selection.

The Bottom Line:

This problem requires careful attention to what's being asked (the exception) and systematic checking of all cases with accurate calculations. The key insight is recognizing that you're looking for the value that breaks the constraint, not one that satisfies it.