A class of 12 students took a quiz, and 11 of their scores are recorded below:78, 82, 85, 78, 81,...

1. TRANSLATE the problem information

• Given information:
  • 11 known scores: 78, 82, 85, 78, 81, 88, 79, 74, 82, 86, 78
  • Mean of these 11 scores is 81
  • Mean of all 12 scores must be an integer greater than 81
  • No student scored higher than 95
  • Find the highest possible score

2. SIMPLIFY to find the sum and set up equations

• First, let's find the sum of the 11 known scores:
  \(78 + 82 + 85 + 78 + 81 + 88 + 79 + 74 + 82 + 86 + 78 = 891\) (use calculator)

• Let x be the missing 12th score
• Total sum of all 12 scores = \(891 + \mathrm{x}\)
• Mean of all 12 scores = \(\frac{891 + \mathrm{x}}{12}\)

3. TRANSLATE the constraint conditions

• For mean \(\gt 81\): \(\frac{891 + \mathrm{x}}{12} \gt 81\)
• SIMPLIFY: \(891 + \mathrm{x} \gt 972\), so \(\mathrm{x} \gt 81\)
• For mean to be an integer: \((891 + \mathrm{x})\) must be divisible by 12
• No score \(\gt 95\): \(\mathrm{x} \leq 95\)

4. INFER the modular arithmetic requirement

• Since \(891 = 74 \times 12 + 3\), we have \(891 \equiv 3 \pmod{12}\)
• For \((891 + \mathrm{x})\) to be divisible by 12: \(891 + \mathrm{x} \equiv 0 \pmod{12}\)
• This means: \(3 + \mathrm{x} \equiv 0 \pmod{12}\)
• Therefore: \(\mathrm{x} \equiv 9 \pmod{12}\)

5. APPLY CONSTRAINTS to find the valid solution

• x must satisfy three conditions:
  • \(\mathrm{x} \gt 81\)
  • \(\mathrm{x} \leq 95\)
  • \(\mathrm{x} \equiv 9 \pmod{12}\)

• Values of form \(12\mathrm{k} + 9\): ..., 81, 93, 105, ...
• Only \(\mathrm{x} = 93\) satisfies all three conditions

6. SIMPLIFY to verify the answer

• Check: \(\frac{891 + 93}{12} = \frac{984}{12} = 82\) (use calculator)
• 82 is indeed an integer greater than 81 ✓

Answer: 93

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often miss that for the mean to be an integer, the total sum must be divisible by 12. They might try to find the mean by testing random values or just find the minimum value where \(\mathrm{x} \gt 81\), leading to answers like 82 or 84. This approach ignores the crucial 'integer mean' constraint and leads to confusion when their calculated mean isn't an integer.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students may correctly identify that \(\mathrm{x} \equiv 9 \pmod{12}\) but fail to systematically check all three conditions. They might select \(\mathrm{x} = 81\) (forgetting it must be greater than 81) or \(\mathrm{x} = 105\) (ignoring the upper limit of 95). This leads them to select incorrect answers or get stuck trying to verify impossible solutions.

The Bottom Line:

This problem requires students to juggle multiple constraints simultaneously while using modular arithmetic—a combination that challenges both algebraic reasoning and systematic constraint checking. The key insight is recognizing that 'integer mean' creates a very specific mathematical requirement that dramatically limits the possible answers.