A printing machine uses ink at an average rate of 3.5 milliliters per page. Which function f models the number...

1. TRANSLATE the problem information

• Given information:
  • Rate: 3.5 milliliters per page
  • Variable: p pages
• What this tells us: We need total milliliters for p pages

2. INFER the mathematical relationship

• "Per page" means "for each page"
• This is a rate problem: \(\mathrm{Total = Rate \times Quantity}\)
• We multiply the rate by the number of pages

3. Set up the function

• Total ink needed = \(\mathrm{3.5 \, milliliters/page \times p \, pages}\)
• \(\mathrm{f(p) = 3.5p}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what "per page" means mathematically and think they need to divide pages by the rate.

They might reason: "If it's 3.5 milliliters per page, then for p pages I need \(\mathrm{p \div 3.5}\)." This backwards thinking about rates leads them to select Choice A (\(\frac{\mathrm{p}}{\mathrm{3.5}}\)).

Second Most Common Error:

Poor INFER skill: Students don't recognize this as a multiplication scenario and instead think about adding the rate to the number of pages.

They might think: "I have p pages plus 3.5 milliliters per page, so \(\mathrm{p + 3.5}\)." This leads them to select Choice B (\(\mathrm{p + 3.5}\)).

The Bottom Line:

Rate problems require understanding that "per unit" means you multiply the rate by the number of units to get the total. The key insight is recognizing that 3.5 milliliters per page means 3.5 milliliters for every single page.