A reseller buys certain books for a purchase price of $5.00 each and then marks them each for sale at...

1. TRANSLATE the problem information

• Given information:
  • Purchase price: \(\$5.00\) per book
  • Consumer price: \(270\%\) of purchase price
  • After 4 months: remaining books discounted \(70\%\) off consumer price
  • Need to find: discounted price
• What this tells us: We need to work through two percentage calculations in sequence.

2. INFER the solution approach

• This is a multi-step problem requiring us to:
  1. First find the consumer price using the \(270\%\) markup
  2. Then find the discount amount using \(70\%\) off that consumer price
  3. Finally subtract the discount from the consumer price
• We must complete these steps in order since each depends on the previous result.

3. SIMPLIFY to find the consumer price

• Consumer price = \(270\%\) of \(\$5.00\)
• Convert percentage: \(270\% = 2.70\)
• Calculate: \(2.70 \times \$5.00 = \$13.50\) (use calculator)

4. SIMPLIFY to find the discount amount

• Discount amount = \(70\%\) of consumer price
• Convert percentage: \(70\% = 0.70\)
• Calculate: \(0.70 \times \$13.50 = \$9.45\) (use calculator)

5. SIMPLIFY to find the final discounted price

• Discounted price = Consumer price - Discount amount
• Calculate: \(\$13.50 - \$9.45 = \$4.05\) (use calculator)

Answer: \(\$4.05\) (which can also be written as \(\frac{81}{20}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "\(70\%\) off" as meaning the discounted price is \(70\%\) of the consumer price, rather than understanding it means subtracting \(70\%\) from the consumer price.

Instead of calculating:

\(\$13.50 - (70\% \times \$13.50)\)
\(= \$13.50 - \$9.45\)
\(= \$4.05\)

They incorrectly calculate:

\(70\% \times \$13.50\)
\(= 0.70 \times \$13.50\)
\(= \$9.45\)

This leads to confusion and an incorrect final answer of \(\$9.45\).

Second Most Common Error:

Poor INFER reasoning: Students attempt to combine both percentage operations in one step, trying to find "\(70\%\) off of \(270\%\) of \(\$5.00\)" without recognizing the need to calculate the consumer price first.

They might incorrectly calculate something like:

\((270\% - 70\%) \times \$5.00\)
\(= 200\% \times \$5.00\)
\(= \$10.00\)

This causes them to get stuck with an incorrect intermediate step and guess at the final answer.

The Bottom Line:

This problem challenges students' ability to interpret percentage language correctly and maintain a clear sequence through multi-step percentage calculations. The key insight is recognizing that "% off" means subtraction, not multiplication alone.