Question:A taxi company charges a flat fee of $3 and an additional constant charge per mile. The total cost C,...

1. TRANSLATE the problem information

• Given information:
  • Taxi equation: \(\mathrm{C = 2.75m + 3}\)
  • \(\mathrm{C}\) = total cost in dollars
  • \(\mathrm{m}\) = miles traveled
  • Question asks: What does 2.75 represent?
• What this tells us: We have a linear equation relating cost to distance

2. INFER the mathematical structure

• This follows the linear form \(\mathrm{y = mx + b}\) where:
  • The coefficient (2.75) multiplies the variable
  • The constant (3) stands alone
• In any linear relationship, the coefficient tells us the rate of change
• Here: 2.75 tells us how much cost increases per additional mile

3. INFER the real-world meaning

• The coefficient 2.75 represents the cost per mile
• The constant 3 represents the flat starting fee
• This makes sense: you pay $3 to get in the taxi, then $2.75 for each mile you travel

4. Verify against answer choices

• (A) Starting fee = constant term = 3 ✗
• (B) Cost per mile = coefficient = 2.75 ✓
• (C) Miles traveled = variable m ✗
• (D) Cost for 1-mile trip = \(\mathrm{2.75(1) + 3 = \$5.75}\) ✗

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the roles of coefficient vs constant in linear equations

They might think "2.75 comes first in the equation, so it must be the starting fee" or fail to understand that coefficients represent rates of change. This leads them to incorrectly associate 2.75 with the flat fee concept.

This may lead them to select Choice A ($3 starting fee) by incorrectly swapping the roles of 2.75 and 3.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread what the question is asking for

They might calculate the total cost for specific scenarios instead of interpreting what 2.75 represents. For instance, they calculate what a 1-mile trip costs \(\mathrm{(2.75 + 3 = 5.75)}\) and think this relates to the meaning of 2.75.

This may lead them to select Choice D or causes confusion leading to guessing.

The Bottom Line:

Success requires understanding that in linear models like \(\mathrm{y = mx + b}\), the coefficient (m) always represents the rate of change - how much the output changes per unit increase in the input. The real-world context helps verify this interpretation makes sense.