A ride-share company charges a fixed base fee plus a constant rate per mile. A 2-mile trip costs $11, and...

1. TRANSLATE the problem information

• Given information:
  • Ride-share pricing: base fee + rate per mile
  • 2-mile trip costs $11
  • 8-mile trip costs $29
  • Need to find: cost of 10-mile trip

• This describes a linear function: \(\mathrm{Cost = (rate\,per\,mile)(miles) + base\,fee}\)

2. INFER the mathematical approach

• Set up the linear function: \(\mathrm{C(m) = rm + b}\)
• We have two data points: \(\mathrm{(2, 11)}\) and \(\mathrm{(8, 29)}\)
• Strategy: Find rate r using slope formula, then find base fee b

3. SIMPLIFY to find the rate per mile

• Using slope formula: \(\mathrm{r = \frac{29 - 11}{8 - 2}}\)
  \(\mathrm{= \frac{18}{6}}\)
  \(\mathrm{= 3}\)
• The rate is $3 per mile

4. SIMPLIFY to find the base fee

• Substitute into \(\mathrm{C(2) = 11}\):
• \(\mathrm{2(3) + b = 11}\)
• \(\mathrm{6 + b = 11}\)
• \(\mathrm{b = 5}\)
• The base fee is $5

5. SIMPLIFY to calculate the 10-mile cost

• \(\mathrm{C(10) = 3(10) + 5}\)
  \(\mathrm{= 30 + 5}\)
  \(\mathrm{= 35}\)

Answer: C ($35)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize this as a linear function problem requiring both a base fee and rate component.

Some students try to find a simple rate by dividing cost by miles (like \(\mathrm{\$11 \div 2 = \$5.50}\) per mile), then multiply by 10 to get $55. When this doesn't match any answer choice, they become confused and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the approach but make arithmetic errors in the slope calculation.

For example, they might incorrectly calculate the slope as \(\mathrm{r = \frac{18}{6} = 2}\) instead of 3. Then finding \(\mathrm{b = 11 - 2(2) = 7}\), they get \(\mathrm{C(10) = 2(10) + 7 = 27}\). This may lead them to select Choice A ($27).

The Bottom Line:

This problem requires recognizing that ride-share pricing follows a linear model with both fixed costs (base fee) and variable costs (per-mile rate). Students who miss this two-component structure often struggle with the setup, while calculation errors in the slope formula can lead to plausible wrong answers.