A taxi ride has a base fare of $3.50. The meter adds $2.25 for each mile traveled. Which equation best...

1. TRANSLATE the problem information

• Given information:
  • Base fare: \(\$3.50\) (paid no matter how far you travel)
  • Additional cost: \(\$2.25\) for each mile traveled
  • Need to find: equation for total cost C in terms of miles m

2. INFER the cost structure

• This is a linear cost function with two components:
  • Fixed cost (doesn't change): \(\$3.50\) base fare
  • Variable cost (changes with distance): \(\$2.25\) per mile

• The pattern is: Total Cost = Fixed Cost + (Rate per unit × Number of units)

3. TRANSLATE into mathematical form

• Substitute our values:
  \(\mathrm{C = 3.50 + 2.25m}\)

• Rearrange to standard form (coefficient first):
  \(\mathrm{C = 2.25m + 3.50}\)

• Compare with answer choices: This matches choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which value goes where in the linear equation

They might think "base fare of \(\$3.50\)" means \(\$3.50\) per mile, leading them to write \(\mathrm{C = 3.50m + 2.25}\). This reasoning puts the larger number as the coefficient because it seems more important.

This may lead them to select Choice (B) (\(\mathrm{C = 3.50m + 2.25}\))

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the difference between fixed and variable costs

They might think both costs should multiply by miles, reasoning that "everything depends on how far you go." This leads to equations like \(\mathrm{C = 3.50m}\) or forgetting one component entirely.

This may lead them to select Choice (A) (\(\mathrm{C = 3.50m}\)) or causes confusion and guessing

The Bottom Line:

The key insight is recognizing that taxi fares have a fixed starting cost (base fare) plus a variable cost that grows with distance. Students who miss this structure will struggle to set up the correct linear equation.