Last week, an interior designer earned a total of $1,258 from consulting for x hours and drawing up plans for...

1. TRANSLATE the problem information

• Given information:
  • Interior designer earned \(\$1,258\) total
  • \(\mathrm{x}\) = hours spent consulting
  • \(\mathrm{y}\) = hours spent drawing up plans
  • Equation: \(68\mathrm{x} + 85\mathrm{y} = 1,258\)

2. INFER what each part of the equation represents

• In equations of the form (coefficient × variable), the coefficient typically represents a rate
• Since \(\mathrm{x}\) represents hours consulting, then \(68\mathrm{x}\) represents:
  • 68 (rate per hour) × x (hours) = total earned from consulting
• Since \(\mathrm{y}\) represents hours for plans, then \(85\mathrm{y}\) represents:
  • 85 (rate per hour) × y (hours) = total earned from drawing plans

3. TRANSLATE the question back to the equation

• The question asks: "What does 68 represent?"
• From our analysis: 68 is the coefficient of \(\mathrm{x}\) (consulting hours)
• Therefore: 68 represents the hourly rate for consulting work

Answer: A. The interior designer earned \(\$68\) per hour consulting last week.

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students confuse which activity corresponds to which variable in the equation.

They might think \(\mathrm{x}\) represents drawing plans and \(\mathrm{y}\) represents consulting, leading them to believe 68 is the rate for drawing plans instead of consulting. This confusion happens because they don't carefully track which variable represents which activity from the problem statement.

This may lead them to select Choice C (\(\$68\) per hour drawing up plans)

Second Most Common Error:

Weak INFER skill about equation structure: Students don't understand that coefficients represent rates in this context.

They might think 68 represents the actual number of hours worked rather than the hourly rate. This happens when students don't recognize the standard "rate × time = total" structure of the equation.

This may lead them to select Choice B (68 hours drawing up plans) or Choice D (68 hours consulting)

The Bottom Line:

This problem requires careful attention to variable definitions and understanding how coefficients function as rates in linear equations representing real-world scenarios.