Data set C consists of 7 positive integers less than 28. The list shown gives 6 of the integers from...

1. TRANSLATE the problem information

• Given information:
  • 6 known integers: 10, 12, 14, 16, 18, 20
  • Their mean is 15
  • Total dataset has 7 integers
  • Mean of all 7 must be an integer > 15
  • All integers are positive and < 28

2. TRANSLATE the sum relationships

• Sum of 6 known integers: 10 + 12 + 14 + 16 + 18 + 20 = 90
• This confirms: \(6 \times 15 = 90\) ✓
• If seventh integer = x, total sum = 90 + x
• Mean of all 7 = \(\frac{90 + x}{7}\)

3. INFER the constraint strategy

• Since mean must be an integer greater than 15, we can call it k
• Where \(k \geq 16\) (since \(k \gt 15\) and k is an integer)
• This gives us: \(\frac{90 + x}{7} = k\)
• Solving: \(x = 7k - 90\)

4. APPLY CONSTRAINTS to find valid k values

• Since x must be positive: \(7k - 90 \gt 0\) → \(k \gt 12.86\) → \(k \geq 13\)
• Since x must be less than 28: \(7k - 90 \lt 28\) → \(k \lt 16.86\) → \(k \leq 16\)
• Combined with \(k \geq 16\): \(k = 16\) is the only possibility

5. SIMPLIFY to find the seventh integer

• \(x = 7(16) - 90\)
• \(x = 112 - 90\)
• \(x = 22\)
• Check: \(22 \lt 28\) ✓ and \(22 \gt 0\) ✓

6. INFER the largest value

• Given integers: 10, 12, 14, 16, 18, 20 (all ≤ 20)
• Seventh integer: 22
• Therefore, largest integer = 22

Answer: C) 22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that requiring the mean to be an integer severely limits possible values. They might set up \(\frac{90 + x}{7} \gt 15\) and solve \(x \gt 15\), then pick any value like x = 21 or x = 27 without checking if it produces an integer mean.

This may lead them to select Choice B (21) by assuming any value greater than 15 works.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students correctly find \(x = 7k - 90\) but don't systematically check which integer values of \(k \geq 16\) keep \(x \lt 28\). They might try k = 17, get x = 29, but not realize this violates the constraint, leading to confusion about which answer to select.

This leads to confusion and guessing among the larger answer choices.

The Bottom Line:

This problem requires recognizing that seemingly small constraints (mean being an integer, x < 28) create very specific solution requirements - there's often only one valid answer once all constraints are properly applied.