A ride-share service charges according to the function \(\mathrm{T(d) = 1.8(d - 2) + 2.50}\), where \(\mathrm{T(d)}\) is the total...

1. TRANSLATE the problem information

• Given information:
  • Fare function: \(\mathrm{T(d) = 1.8(d - 2) + 2.50}\)
  • One trip is 3.5 miles longer than another
  • Need to find the difference in their fares

• What this tells us: We need to compare \(\mathrm{T(d + 3.5)}\) and \(\mathrm{T(d)}\) for some distance d

2. INFER the approach

• Since we're looking at the difference between function values, we need: \(\mathrm{T(d + 3.5) - T(d)}\)
• Key insight: For linear functions like this one, the difference depends only on how the coefficient of d affects the change in input

3. SIMPLIFY to find the difference

• Method 1 (Direct property): Since \(\mathrm{T(d) = 1.8(d - 2) + 2.50}\) is linear with coefficient 1.8, an increase of 3.5 miles increases the fare by \(\mathrm{1.8 \times 3.5}\)

• Method 2 (Full calculation):
  • \(\mathrm{T(d + 3.5) = 1.8((d + 3.5) - 2) + 2.50}\)
    \(\mathrm{= 1.8(d + 1.5) + 2.50}\)
  • \(\mathrm{T(d + 3.5) - T(d) = [1.8(d + 1.5) + 2.50] - [1.8(d - 2) + 2.50]}\)
  • \(\mathrm{= 1.8(d + 1.5) - 1.8(d - 2)}\)
    \(\mathrm{= 1.8[3.5]}\)
    \(\mathrm{= 6.3}\)

4. Calculate the final result

• \(\mathrm{1.8 \times 3.5 = 6.3}\)

Answer: D (6.3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to plug in specific values for d instead of recognizing the general property of linear functions. They might calculate \(\mathrm{T(5)}\) and \(\mathrm{T(8.5)}\), or \(\mathrm{T(10)}\) and \(\mathrm{T(13.5)}\), getting the right answer through more work, but missing the elegant insight about linear functions.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might misinterpret the problem and try to find \(\mathrm{T(3.5)}\) instead of finding the difference between fares. This leads to:

\(\mathrm{T(3.5) = 1.8(3.5 - 2) + 2.50}\)
\(\mathrm{= 1.8(1.5) + 2.50}\)
\(\mathrm{= 2.7 + 2.50}\)
\(\mathrm{= 5.2}\)

which isn't among the answer choices, causing confusion and guessing.

The Bottom Line:

This problem tests whether students recognize that linear functions have constant rates of change - the coefficient tells you exactly how much the output changes per unit change in input, regardless of the specific starting point.