The regular price of a shirt at a store is $11.70. The sale price of the shirt is 80% less...

1. TRANSLATE the problem information

• Given information:
  • Regular price = $11.70
  • Sale price is 80% less than regular price
  • Sale price is 30% greater than store's cost
  • Need to find: store's cost
• What this tells us: We need to work through the sale price to connect regular price and cost

2. INFER the solution approach

• Key insight: The sale price is the connecting link between regular price and cost
• Strategy: Find sale price first, then use it to find cost
• This is a two-step process where each percentage relationship gives us one equation

3. TRANSLATE and calculate the sale price

• "80% less than regular price" means:
  \(\mathrm{Sale\,price = Regular\,price - 80\%\,of\,regular\,price}\)
  \(\mathrm{Sale\,price = \$11.70 - 0.80 \times \$11.70}\)
  \(\mathrm{Sale\,price = \$11.70 - \$9.36 = \$2.34}\)

4. TRANSLATE the cost relationship and SIMPLIFY

• "Sale price is 30% greater than store's cost" means:
  \(\mathrm{Sale\,price = Store's\,cost + 30\%\,of\,store's\,cost}\)
  \(\mathrm{\$2.34 = Cost + 0.30 \times Cost}\)
  \(\mathrm{\$2.34 = 1.30 \times Cost}\)

5. SIMPLIFY to find the final answer

• Solve for cost:
  \(\mathrm{Cost = \$2.34 \div 1.30 = \$1.80}\) (use calculator)

Answer: 1.80

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misunderstanding "80% less than"

Students often think "80% less than $11.70" means "80% of $11.70" = $9.36, rather than "$11.70 - 80% of $11.70" = $2.34. This fundamental translation error means they use $9.36 as the sale price instead of $2.34, leading to a completely incorrect cost calculation of approximately $7.38. This leads to confusion and guessing since this value won't match any reasonable answer format.

Second Most Common Error:

Poor INFER reasoning: Trying to set up one equation instead of recognizing the two-step process

Some students attempt to directly relate regular price to cost without finding the sale price first. They might try something like \(\mathrm{Cost = \$11.70 \div 1.30 \div 0.20}\) which creates a meaningless calculation. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests your ability to carefully translate percentage language into math and recognize when you need intermediate steps. The sale price isn't just a detail—it's the crucial bridge between the given information and what you need to find.