A bicycle originally priced at $450 is on sale. The sale offers a 30% discount off the original price. What...

1. TRANSLATE the problem information

• Given information:
  • Original price: \(\$450\)
  • Discount rate: \(30\%\)
  • Find: Sale price (final price after discount)

2. INFER the solution approach

• Key insight: A discount means you pay less than the original price
• Two strategic approaches available:
  • Method 1: Calculate discount amount, then subtract from original price
  • Method 2: Calculate what percentage you actually pay (100% - 30% = 70%), then find that amount

3. SIMPLIFY using Method 1 (Discount Subtraction)

• Convert percentage to decimal: \(30\% = 0.30\)
• Calculate discount amount: \(\$450 \times 0.30 = \$135\)
• Calculate sale price: \(\$450 - \$135 = \$315\)

4. SIMPLIFY using Method 2 (Remaining Percentage) - Alternative approach

• INFER remaining percentage: \(100\% - 30\% = 70\%\)
• Convert to decimal: \(70\% = 0.70\)
• Calculate sale price directly: \(\$450 \times 0.70 = \$315\)

Answer: \(\$315\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly calculate the discount amount (\(\$135\)) but then add it to the original price instead of subtracting it.

Their reasoning: "The problem mentions \(\$450\) and \(\$135\), so I need to combine them somehow." This leads them to calculate \(\$450 + \$135 = \$585\), which represents paying more than the original price for a discounted item - clearly illogical but easy to do when rushing.

This leads to confusion and incorrect answer selection.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the concept but make arithmetic errors when converting percentages or performing calculations.

Common mistakes include: forgetting to convert \(30\%\) to \(0.30\) (trying to multiply by 30 instead), or making basic arithmetic errors in the final calculations. These computational errors lead to incorrect final answers even with correct reasoning.

This causes them to doubt their approach and potentially guess among remaining options.

The Bottom Line:

This problem tests whether students truly understand what a discount means (paying less, not more) and can accurately execute percentage calculations. The key insight is recognizing that discounts reduce the final price from the original price.