Data set A consists of 10 positive integers less than 60. The list shown gives 9 of the integers from...

1. TRANSLATE the problem information

• Given information:
  • Data set A has 10 positive integers all less than 60
  • 9 known values: 43, 45, 44, 43, 38, 39, 40, 46, 40
  • Mean of these 9 values is 42
  • Mean of all 10 values is an integer greater than 42
  • Need to find the largest value in data set A

2. TRANSLATE the mean constraint into algebra

• Let \(\mathrm{x}\) = the missing 10th value
• Sum of 9 known values = \(43 + 45 + 44 + 43 + 38 + 39 + 40 + 46 + 40 = 378\) (use calculator)
• Mean of all 10 values = \(\frac{378 + \mathrm{x}}{10}\)
• This mean is an integer \(\gt 42\)

3. INFER what constraints this creates

• From mean \(\gt 42\): \(\frac{378 + \mathrm{x}}{10} \gt 42\), so \(\mathrm{x} \gt 42\)
• From "mean is an integer": \(\frac{378 + \mathrm{x}}{10}\) must be a whole number
• This means \(378 + \mathrm{x}\) must be divisible by 10

4. INFER the divisibility requirement

• For a number to be divisible by 10, it must end in 0
• 378 ends in 8
• For \(378 + \mathrm{x}\) to end in 0, \(\mathrm{x}\) must end in 2

5. APPLY CONSTRAINTS to find x

• We need: \(42 \lt \mathrm{x} \lt 60\) and \(\mathrm{x}\) ends in 2
• The only integer satisfying both conditions is \(\mathrm{x} = 52\)

6. INFER the final answer

• Compare all values: 43, 45, 44, 43, 38, 39, 40, 46, 40, 52
• The largest value is 52

Answer: 52

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss that "the mean is an integer" creates a divisibility constraint. They correctly find \(\mathrm{x} \gt 42\) but then guess among values like 43, 45, or 50 without using the fact that \(378 + \mathrm{x}\) must be divisible by 10. This leads to confusion and guessing among seemingly reasonable values.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students find that \(\mathrm{x}\) must end in 2 but fail to systematically check which values between 42 and 60 actually end in 2. They might incorrectly select \(\mathrm{x} = 50\) or \(\mathrm{x} = 45\), thinking these are "close enough" to satisfy the constraints. This may lead them to select an incorrect final answer.

The Bottom Line:

This problem requires students to connect multiple mathematical ideas: the mean formula, inequality constraints, and divisibility rules. The key insight is recognizing that "integer mean" isn't just descriptive—it's a mathematical constraint that severely limits possible values.