A utility company charges a fixed monthly fee of $18, plus $0.15 per kilowatt-hour (kWh) used during peak hours and...

1. TRANSLATE the equation components

• Given equation: \(\mathrm{C = 18 + 0.15p + 0.09q}\)
• What each variable means:
  • \(\mathrm{C}\) = total monthly charge (dollars)
  • \(\mathrm{p}\) = peak hours usage (kWh)
  • \(\mathrm{q}\) = off-peak hours usage (kWh)

2. INFER what mathematical operations represent in context

• In cost equations, when we multiply rate × quantity, we get total cost for that category
• The term \(\mathrm{0.09q}\) breaks down as:
  • \(\mathrm{0.09}\) = cost per off-peak kWh (the rate)
  • \(\mathrm{q}\) = number of off-peak kWh used (the quantity)
  • \(\mathrm{0.09q}\) = total dollars spent on off-peak electricity

3. TRANSLATE back to answer choices

• \(\mathrm{0.09q}\) represents the total cost charged for off-peak electricity used that month
• This is only part of the total bill, not the whole thing

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the individual components of an expression with the expression itself.

They might see \(\mathrm{0.09q}\) and focus on just one part - thinking \(\mathrm{0.09q}\) means "the number of off-peak kilowatt-hours" (focusing on the \(\mathrm{q}\)) or "the cost per off-peak kilowatt-hour" (focusing on the \(\mathrm{0.09}\)). They don't recognize that multiplying \(\mathrm{rate \times quantity}\) gives a total cost amount.

This may lead them to select Choice A (The number of off-peak kilowatt-hours used) or Choice B (The cost per off-peak kilowatt-hour).

Second Most Common Error:

Poor INFER reasoning: Students don't distinguish between a component cost and the total bill.

They recognize that \(\mathrm{0.09q}\) involves cost, but they think any cost term in the equation represents the total cost. They don't realize that \(\mathrm{C}\) (the left side) represents the complete monthly charge, while \(\mathrm{0.09q}\) is just the off-peak portion.

This may lead them to select Choice C (The total cost of the customer's electricity for the month).

The Bottom Line:

Success requires recognizing that in linear cost models, each term represents a specific cost component, and multiplying \(\mathrm{rate \times quantity}\) always gives the total cost for that category alone.