After 20% of the original number of marbles in a group were removed from the group, 360 marbles remained in...

1. TRANSLATE the problem information

• Given information:
  • 20% of original marbles were removed
  • 360 marbles remained after removal
  • Need to find: how many marbles were removed

2. INFER the relationship between removed and remaining

• If 20% were removed, then 80% remained
• This means: remaining marbles = 80% of original marbles
• Strategy: Find original number first, then calculate removed amount

3. TRANSLATE into mathematical equation

• Let \(\mathrm{x}\) = original number of marbles
• Remaining marbles: \(\mathrm{0.80x = 360}\)

4. SIMPLIFY to find the original number

• Solve: \(\mathrm{0.80x = 360}\)
• Divide both sides by 0.80: \(\mathrm{x = 360 \div 0.80}\) (use calculator)
• \(\mathrm{x = 450}\) original marbles

5. SIMPLIFY to find removed marbles

• Removed marbles = 20% of original = \(\mathrm{0.20 \times 450 = 90}\)

Answer: B. 90

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misidentify what to take 20% of, thinking they should take 20% of the remaining 360 marbles instead of the original unknown amount.

They calculate: \(\mathrm{0.20 \times 360 = 72}\), reasoning "if 360 remained and 20% were removed, then 20% of 360 were removed."

This may lead them to select Choice A (72).

Second Most Common Error:

Incomplete INFER reasoning: Students correctly find the original number (450) but stop there, forgetting that the question asks for the number removed, not the original number.

This may lead them to select Choice C (450).

The Bottom Line:

This problem challenges students to work backwards from the remaining amount to find the original, then forward again to find the removed amount. The key insight is recognizing that percentages must be taken from the original unknown quantity, not the given remaining amount.