A print shop charges $0.07 for each black-and-white page and $0.55 for each color page. In addition, there is a...

1. TRANSLATE the problem information

• Given information:
  • Black-and-white pages: \(\$0.07\) each, with x pages printed
  • Color pages: \(\$0.55\) each, with y pages printed
  • Setup fee: \(\$3\) (one-time charge)
  • Total charge: \(\$92.14\)
• What this tells us: We need to build an equation where all costs add up to \(\$92.14\)

2. INFER the cost structure

• All costs must be added together to get the total
• Variable costs: price per page × number of pages
• Fixed costs: the setup fee applies once regardless of page count

3. TRANSLATE each cost component

• Black-and-white cost: \(\mathrm{0.07x}\) (since \(\$0.07\) per page × x pages)
• Color cost: \(\mathrm{0.55y}\) (since \(\$0.55\) per page × y pages)
• Setup fee: \(\mathrm{3}\) (fixed \(\$3\) charge)

4. Build the total cost equation

• Total cost = Variable costs + Fixed costs
• Total cost = \(\mathrm{0.07x + 0.55y + 3}\)
• Since total charge is \(\$92.14\): \(\mathrm{0.07x + 0.55y + 3 = 92.14}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which price goes with which variable, incorrectly thinking x represents color pages and y represents black-and-white pages.

Since the problem mentions black-and-white first, they might assume x goes with the first mentioned price (\(\$0.07\)), but then incorrectly pair y with color pages while using the wrong coefficient. This creates the equation \(\mathrm{0.55x + 0.07y + 3 = 92.14}\).

This may lead them to select Choice A (\(\mathrm{0.55x + 0.07y + 3 = 92.14}\))

Second Most Common Error:

Poor INFER reasoning about cost structure: Students incorrectly think the setup fee should be subtracted rather than added, perhaps misunderstanding it as a discount or credit rather than an additional charge.

This leads to the equation \(\mathrm{0.07x + 0.55y - 3 = 92.14}\).

This may lead them to select Choice B (\(\mathrm{0.07x + 0.55y - 3 = 92.14}\))

The Bottom Line:

The key challenge is carefully TRANSLATING the problem statement to match variables with their correct coefficients, while INFERRING that all costs (both variable and fixed) must be added together.