A café lists a specialty drink at a price that is 320% of its ingredient cost. During a promotion, the...

1. TRANSLATE the problem information

• Given information:
  • Listed price = 320% of ingredient cost
  • Promotional discount = 65% off listed price
  • Promotional price = $28.00
  • Find: ingredient cost

• What this tells us: We need to work backwards from the final promotional price through two pricing adjustments.

2. INFER the solution approach

• Key insight: This involves two percentage applications in sequence:
  1. Markup: ingredient cost → listed price (320% means multiply by 3.2)
  2. Discount: listed price → promotional price (65% off means multiply by 0.35)

• Strategy: Set up one equation that connects ingredient cost directly to promotional price.

3. TRANSLATE the pricing relationships into algebra

Let \(\mathrm{C}\) = ingredient cost

• Listed price = 320% of C = \(\mathrm{3.2C}\)
• Promotional price = 35% of listed price (since 65% off means paying 35%)
• Promotional price = \(\mathrm{0.35 \times (3.2C) = 1.12C}\)

4. SIMPLIFY to solve for ingredient cost

• We know promotional price = $28.00, so:
  \(\mathrm{1.12C = 28.00}\)

• Solve for C:
  \(\mathrm{C = 28.00 \div 1.12}\) (use calculator)
  \(\mathrm{C = 25.00}\)

Answer: D) $25.00

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misunderstanding what "65% off" means - thinking customers pay 65% instead of 35% of the listed price.

Students might set up: Promotional price = \(\mathrm{0.65 \times 3.2C = 2.08C}\), leading to \(\mathrm{C = 28.00 \div 2.08 \approx 13.46}\). This may lead them to select Choice B ($13.46).

Second Most Common Error:

Poor INFER reasoning: Not recognizing this as a sequential percentage problem and instead treating the percentages as additive or trying to apply them directly to the promotional price.

Some students might think: "If it's 320% markup and 65% discount, maybe I subtract: 320% - 65% = 255%, so ingredient cost = \(\mathrm{28.00 \div 2.55 \approx 10.98}\)." This leads to confusion and guessing since 10.98 isn't among the choices.

The Bottom Line:

This problem tests your ability to work backwards through sequential percentage applications. The key insight is recognizing that "65% off" means paying 35%, and that you need to trace the money flow in reverse: promotional price → listed price → ingredient cost.