A taxi company charges a base fee plus a constant amount per mile. A 15-mile trip costs $11. The cost...

1. TRANSLATE the problem information

• Given information:
  • Taxi charges base fee + constant amount per mile
  • 15-mile trip costs $11
  • Cost increases by $0.40 for each additional mile

• What this tells us: The cost structure follows \(\mathrm{C = base\,fee + (rate\,per\,mile \times miles)}\), where the rate per mile is $0.40

2. INFER the mathematical approach

• Since we have a base fee plus a rate per mile, this is a linear function: \(\mathrm{C = b + 0.40m}\)
• We know one data point (15 miles, $11), so we can substitute to find the unknown base fee b
• Strategy: Substitute the known values and solve for b

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{m = 15}\) and \(\mathrm{C = 11}\) into \(\mathrm{C = b + 0.40m}\):

\(\mathrm{11 = b + 0.40(15)}\)
\(\mathrm{11 = b + 6}\)
\(\mathrm{b = 11 - 6 = 5}\)

Answer: C. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "cost increases by $0.40 for each additional mile" and think this means the base fee is $0.40, or they set up the equation incorrectly as \(\mathrm{C = 0.40 + bm}\).

This confusion about the cost structure leads them to set up wrong equations and get incorrect base fee values, potentially leading to guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(\mathrm{11 = b + 0.40(15)}\) but make arithmetic errors, such as calculating \(\mathrm{0.40 \times 15 = 0.60}\) instead of 6.00, leading to \(\mathrm{b = 11 - 0.60 = 10.40}\).

This may lead them to select Choice B (4) as the closest option, or causes confusion since 10.40 isn't among the choices.

The Bottom Line:

This problem tests whether students can properly translate a real-world linear relationship into mathematics and then execute basic substitution. The key insight is recognizing that "$0.40 per additional mile" means the rate per mile is $0.40, not that the base fee has anything to do with $0.40.