The cost of renting a carpet cleaner is $52 for the first day and $26 for each additional day. Which...

1. TRANSLATE the pricing structure

• Given information:
  • First day costs $52
  • Each additional day costs $26
  • Need cost function \(\mathrm{C(d)}\) for \(\mathrm{d}\) total days

2. INFER what "additional days" means

• If renting for \(\mathrm{d}\) days total, then:
  • Day 1 is the "first day"
  • Days 2, 3, 4, ..., \(\mathrm{d}\) are the "additional days"
  • Number of additional days = \(\mathrm{d - 1}\)

3. TRANSLATE to mathematical expression

• Total cost = First day cost + Additional days cost
• \(\mathrm{C(d) = 52 + 26(d - 1)}\)

4. SIMPLIFY to match answer format

• \(\mathrm{C(d) = 52 + 26(d - 1)}\)
• \(\mathrm{C(d) = 52 + 26d - 26}\)
• \(\mathrm{C(d) = 26d + 26}\)

5. Select the matching choice

Looking at the options, this matches Choice A: \(\mathrm{C(d) = 26d + 26}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students miss that "additional days" means \(\mathrm{(d-1)}\) days, not \(\mathrm{d}\) days

They think: "It costs $52 for the first day and $26 for each additional day, so for \(\mathrm{d}\) days it's $52 + $26\(\mathrm{d}\)"

This gives \(\mathrm{C(d) = 26d + 52}\), leading them to select Choice B (\(\mathrm{26d + 52}\))

Second Most Common Error:

Poor TRANSLATE execution: Students confuse which rate applies to which days

They might think the first day costs $26 and additional days cost $52, or get confused about the pricing structure entirely.

This leads to confusion and random guessing among the remaining choices.

The Bottom Line:

The key challenge is carefully parsing "each additional day" to realize this means \(\mathrm{(d-1)}\) days when renting for \(\mathrm{d}\) total days. Students who rush through the translation miss this crucial distinction.