A bicycle that originally cost $450 is on sale for $378. The sale price is what percentage of the original...

1. TRANSLATE the problem information

• Given information:
  • Original price: \(\$450\)
  • Sale price: \(\$378\)
• What we need: Sale price as a percentage of original price

2. TRANSLATE the question into mathematical form

• "What percentage of the original price" means: \((\mathrm{Sale~Price} \div \mathrm{Original~Price}) \times 100\%\)
• Set up: \((\$378 \div \$450) \times 100\%\)

3. SIMPLIFY the fraction before calculating

• Look for common factors in 378 and 450
• Both are divisible by 18:
  • \(378 \div 18 = 21\)
  • \(450 \div 18 = 25\)
• Simplified fraction: \(\frac{21}{25}\)

4. SIMPLIFY to convert to percentage

• To make denominator 100, multiply both parts by 4:
  • \(\frac{21 \times 4}{25 \times 4} = \frac{84}{100}\)
• \(\frac{84}{100} = 84\%\)

Answer: C) 84%

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students misinterpret the question and calculate the discount percentage instead of the sale price percentage.

They think: "How much of a discount is this?" and calculate:

\((\mathrm{Original~Price} - \mathrm{Sale~Price}) \div \mathrm{Original~Price}\)

\((\$450 - \$378) \div \$450\)

\(\$72 \div \$450 = 16\%\)

This leads them to select Choice A (16%)

Second Most Common Error:

Weak SIMPLIFY execution: Students set up the problem correctly but make arithmetic errors when reducing fractions or converting to percentages.

Common mistakes include incorrectly finding the GCD, making multiplication errors when converting \(\frac{21}{25}\) to a percentage, or miscalculating intermediate steps. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is correctly interpreting what "percentage of the original price" means - it's asking for the sale price as a fraction of the original, not the discount amount.