The mean score on a biology test for a class of 25 students was 82. After the test, the 5...

1. TRANSLATE the problem information

• Given information:
  • Original class: 25 students with mean score of 82
  • Moved students: 5 students with mean score of 98
  • 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 with total sums first
• Strategy: Find total sum → subtract removed sum → divide by remaining count

3. TRANSLATE and calculate the total sum for all original students

• Using the mean formula: \(\mathrm{sum = mean \times count}\)
• Total sum = \(\mathrm{82 \times 25 = 2,050}\)

4. Calculate the sum for the students who were moved

• Sum of moved students = \(\mathrm{98 \times 5 = 490}\)

5. SIMPLIFY to find the remaining students' information

• Sum of remaining students = \(\mathrm{2,050 - 490 = 1,560}\)
• Number of remaining students = \(\mathrm{25 - 5 = 20}\)

6. Calculate the new mean

• New mean = \(\mathrm{1,560 \div 20 = 78}\)

Answer: C) 78

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to work directly with the means instead of understanding they need to work with sums first.

They might think: "The overall mean was 82, and we removed students with mean 98, so the remaining mean should be lower than 82." They may then guess that it should be around 80 or try to calculate something like \(\mathrm{(82 \times 25 - 98 \times 5) \div 25}\), which gives them \(\mathrm{70.4}\). This approach is fundamentally flawed because they're not adjusting for the changed number of students.

This leads to confusion and guessing among the given choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the correct approach but make arithmetic errors.

For example, they might correctly calculate \(\mathrm{2,050 - 490 = 1,560}\) but then divide by 25 instead of 20 (forgetting that 5 students left), giving them \(\mathrm{1,560 \div 25 = 62.4}\). Since this isn't an answer choice, they get confused and guess.

This may lead them to select Choice D (80) as the closest "reasonable" answer.

The Bottom Line:

This problem tests whether students understand that finding a mean after removing a subgroup requires working with the underlying sums, not just manipulating the means directly. The key insight is recognizing that both the sum and the count change when students are removed.