A class has 125 students with a mean test score of 88 points. If 50 students with a mean test...

1. TRANSLATE the problem information

• Given information:
  • Original class: 125 students with mean score of 88 points
  • Removed group: 50 students with mean score of 94 points
  • Need to find: Mean score of remaining students

2. INFER the solution strategy

• Key insight: To find a new mean after removing students, we need to:
  • Work backwards from the given means to find total points
  • Subtract the removed points from the total
  • Divide by the new student count

• This approach works because \(\mathrm{mean} \times \mathrm{count} = \mathrm{total}\), so we can reconstruct totals from means

3. Calculate total points for original class

Total points = \(125 \times 88 = 11{,}000\) points

4. Calculate total points being removed

Points removed = \(50 \times 94 = 4{,}700\) points

5. SIMPLIFY to find remaining totals

Remaining points = \(11{,}000 - 4{,}700 = 6{,}300\) points

Remaining students = \(125 - 50 = 75\) students

6. Calculate the new mean

New mean = \(6{,}300 \div 75 = 84\) points

Answer: B) 84

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to work directly with the means instead of converting to totals first.

They might think: "The overall mean is 88, and we're removing students with mean 94, so the remaining mean should be less than 88." Then they look for an answer less than 88 and pick A) 82 or B) 84 without proper calculation. While this intuitive reasoning happens to point toward the right direction, it doesn't give the exact answer and could easily mislead on similar problems.

This leads to guessing among the lower choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what's being removed and calculate incorrectly.

Some students might subtract the means directly (\(88 - 94 = -6\)) or try to find some kind of weighted average using just the means (88 and 94) without considering the different group sizes properly. This conceptual confusion about how means combine leads to abandoning systematic solution and guessing.

The Bottom Line:

This problem tests whether students understand that means are summaries of totals, and to work with changing groups, you need to "unpack" the means back into totals first. The key insight is recognizing that mean problems often require this backwards-then-forwards approach.