A supplier charges $30 per unit for all units purchased, but provides a discount of $8 per unit for each...

1. TRANSLATE the problem information

• Given information:
  • Regular price: \(\$30\) per unit for all units
  • Discount: \(\$8\) per unit discount for each unit beyond the first 20 units
  • Need to find: Total cost function \(\mathrm{g(q)}\) for \(\mathrm{q \geq 20}\)

• What this tells us: We have a two-tier pricing structure where the first 20 units cost full price, and additional units get a discount.

2. INFER the approach

• Since the discount only applies to units beyond 20, we need to treat this as two separate cost components
• Strategy: Calculate cost of first 20 units + cost of additional discounted units

3. Calculate the cost of the first 20 units

• First 20 units at full price: \(\mathrm{20 \times \$30 = \$600}\)

4. Calculate the cost of additional units beyond 20

• Number of additional units: \(\mathrm{(q - 20)}\)
• Price per additional unit after discount: \(\mathrm{\$30 - \$8 = \$22}\)
• Cost of additional units: \(\mathrm{(q - 20) \times \$22 = 22(q - 20)}\)

5. SIMPLIFY the expression for additional units

• \(\mathrm{22(q - 20)}\)
• \(\mathrm{= 22q - 22(20)}\)
• \(\mathrm{= 22q - 440}\)

6. Combine all costs

• Total cost = Cost of first 20 units + Cost of additional units
• \(\mathrm{g(q) = \$600 + (22q - 440)}\)
• \(\mathrm{= 600 + 22q - 440}\)
• \(\mathrm{= 22q + 160}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "discount of \(\$8\) per unit for each unit beyond the first 20 units" and apply the \(\$8\) discount to ALL q units instead of just the additional \(\mathrm{(q - 20)}\) units.

Their reasoning: "Each unit gets \(\$8\) off, so the price per unit is \(\mathrm{\$30 - \$8 = \$22}\), and the total cost is \(\mathrm{22q}\)."

This leads them to look for an answer like \(\mathrm{g(q) = 22q}\), but since that's not available, they might select Choice B (\(\mathrm{g(q) = 22q + 600}\)), thinking they need to add something to account for a base cost.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the two-tier structure but make algebraic errors when combining terms, particularly when distributing \(\mathrm{22(q - 20)}\).

They might incorrectly expand \(\mathrm{22(q - 20)}\) as \(\mathrm{22q - 20}\) instead of \(\mathrm{22q - 440}\), leading to \(\mathrm{g(q) = 600 + 22q - 20 = 22q + 580}\), which doesn't match any answer choice. This causes confusion and guessing.

The Bottom Line:

This problem challenges students to carefully parse conditional pricing language and correctly apply mathematical operations to different portions of the purchase quantity. The key insight is recognizing that "beyond the first 20 units" creates a piecewise cost structure.