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:
  • Total cost = fixed fee (f) + cost per mile (m) × number of miles
  • 0-mile ride costs $7
  • 3-mile ride costs $19
  • Need to find m

• What this tells us: We have two data points that we can use to find the unknown values f and m.

2. INFER the solution strategy

• The key insight: The 0-mile ride eliminates the variable cost entirely, giving us the fixed fee directly
• Strategy: Use the 0-mile ride to find f, then substitute into the 3-mile ride equation to find m

3. TRANSLATE the first data point

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

4. TRANSLATE the second data point and SIMPLIFY

• For 3-mile ride: \(\mathrm{f + m(3) = 19}\)
• Substitute \(\mathrm{f = 7}\): \(\mathrm{7 + 3m = 19}\)
• SIMPLIFY the equation:
  • 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 might confuse which piece of information gives them which variable. They may try to use the 3-mile ride to find both f and m simultaneously, or incorrectly assume that f represents the cost per mile.

This leads to confusion about how to set up the equations, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{7 + 3m = 19}\) but make arithmetic errors. For example, they might incorrectly calculate \(\mathrm{19 - 7 = 11}\), leading to \(\mathrm{m = 11/3 ≈ 3.67}\), which would make them select Choice (B) 3 as the closest option.

The Bottom Line:

This problem tests whether students can recognize that a linear cost structure creates a simple system where one data point (0 miles) directly reveals one variable, making the rest straightforward. The mathematical work is basic, but the setup requires clear thinking about what each piece of information tells you.