A ride-hailing service charges a fixed fee f dollars per ride and an additional m dollars per mile. A 0-mile...

1. TRANSLATE the problem information

• Given information:
  • Fixed fee \(\mathrm{f}\) dollars per ride, plus \(\mathrm{m}\) dollars per mile
  • 0-mile ride costs $7
  • 3-mile ride costs $19
  • Need to find \(\mathrm{m}\)

• What this tells us: We have a linear cost function where \(\mathrm{total\ cost = f + m \times (miles)}\)

2. INFER the solving strategy

• Key insight: The 0-mile ride gives us the fixed fee directly since there's no mileage charge
• Strategy: Use the first scenario to find \(\mathrm{f}\), then use the second scenario to find \(\mathrm{m}\)

3. TRANSLATE each scenario into equations

• For 0-mile ride: \(\mathrm{f + m(0) = 7}\), which simplifies to \(\mathrm{f = 7}\)
• For 3-mile ride: \(\mathrm{f + m(3) = 19}\)

4. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{f = 7}\) into the second equation: \(\mathrm{7 + 3m = 19}\)
• Subtract 7 from both sides: \(\mathrm{3m = 12}\)
• Divide by 3: \(\mathrm{m = 4}\)

Answer: C) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up incorrect equations by not clearly understanding what the "0-mile ride" scenario represents.

Some students might think they need to solve a complex system of equations instead of recognizing that \(\mathrm{f + m(0) = 7}\) simply means \(\mathrm{f = 7}\). They may write something like "\(\mathrm{f + 0m = 7}\) and \(\mathrm{f + 3m = 19}\)" but then get confused about how to proceed systematically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equations correctly but make arithmetic errors when solving.

They might correctly get to \(\mathrm{7 + 3m = 19}\), but then make errors like: \(\mathrm{3m = 19 + 7 = 26}\), so \(\mathrm{m = 26/3}\), or other calculation mistakes. Since none of the answer choices match these incorrect calculations, they end up guessing.

This may lead them to select Choice A (2) or Choice D (6) as "reasonable-looking" answers.

The Bottom Line:

This problem tests whether students can translate a real-world linear relationship into mathematics and recognize that one scenario can directly give them part of the solution, making the rest straightforward.