A contract for a certain service requires a onetime activation cost of $35 and a monthly cost of $23. Which...

1. TRANSLATE the problem information

• Given information:
  • One-time activation cost: \(\$35\)
  • Monthly cost: \(\$23\)
  • Total cost c for t months
• What this tells us: We have one fixed cost and one cost that varies with time

2. INFER the cost structure

• The monthly cost of \(\$23\) gets multiplied by the number of months t
• The activation cost of \(\$35\) happens only once, regardless of duration
• Total cost = (variable cost) + (fixed cost)

3. TRANSLATE into mathematical form

• Variable cost: \(23 \times \mathrm{t} = 23\mathrm{t}\)
• Fixed cost: 35
• Combined equation: \(\mathrm{c} = 23\mathrm{t} + 35\)

4. Verify against answer choices

• Choice C matches our equation: \(\mathrm{c} = 23\mathrm{t} + 35\)

Answer: C. \(\mathrm{c} = 23\mathrm{t} + 35\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students mix up which cost is variable versus fixed, leading them to multiply the wrong value by t.

They might think "the activation cost happens over time" and create \(\mathrm{c} = 35\mathrm{t} + 23\), where they incorrectly multiply the one-time fee by the number of months. This fundamental misunderstanding of the cost structure leads them to select Choice D (\(\mathrm{c} = 35\mathrm{t} + 23\)).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly identify the cost components but create fractional relationships instead of multiplicative ones.

They might think "cost per month means divide" and write expressions like \(\mathrm{t}/23\), not recognizing that "monthly cost of \(\$23\)" means "\(\$23\) times the number of months." This confusion about mathematical language leads them to select Choice A (\(\mathrm{c} = \mathrm{t}/23 + 35\)) or Choice B (\(\mathrm{c} = \mathrm{t}/35 + 23\)).

The Bottom Line:

This problem tests whether students can distinguish between costs that scale with time (variable) versus costs that occur once (fixed), and then correctly express these relationships mathematically. The key insight is recognizing that "monthly cost" means multiplication, not division.