Tilly earns p dollars for every w hours of work. Which expression represents the amount of money, in dollars, Tilly...

1. TRANSLATE the problem information

• Given information:
  • Tilly earns \(\mathrm{p}\) dollars for every \(\mathrm{w}\) hours of work
  • We need to find earnings for \(\mathrm{39w}\) hours

2. INFER the mathematical relationship

• Since earnings are proportional to hours worked, if hours increase by a factor, earnings increase by the same factor
• The time \(\mathrm{39w}\) is exactly 39 times the original time \(\mathrm{w}\)
• Therefore, earnings should be 39 times the original earnings \(\mathrm{p}\)

3. SIMPLIFY using rate reasoning

• Original rate: \(\mathrm{p}\) dollars per \(\mathrm{w}\) hours
• For \(\mathrm{39w}\) hours: \(\frac{\mathrm{p}}{\mathrm{w}} \times \mathrm{39w} = \mathrm{39p}\) dollars

Answer: A. \(\mathrm{39p}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize the proportional relationship between hours and earnings. Instead, they think about adding or subtracting the multiplier 39 to the earnings \(\mathrm{p}\).

They might reason: "She works \(\mathrm{39w}\) hours instead of \(\mathrm{w}\) hours, so that's 39 more, meaning she gets \(\mathrm{p + 39}\) dollars."

This leads them to select Choice C (\(\mathrm{p + 39}\)).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly identify that there's a rate involved but confuse what the rate represents. They think \(\frac{\mathrm{p}}{39}\) represents the earnings for the extended time period, misunderstanding the relationship between the multiplier and the rate.

This causes them to select Choice B (\(\frac{\mathrm{p}}{39}\)).

The Bottom Line:

This problem tests whether students understand that in proportional relationships, when one quantity is multiplied by a factor, the related quantity is multiplied by the same factor—not added to or divided by it.