Sean rents a tent at a cost of $11 per day plus a one­time insurance fee of $10. Which equation...

1. TRANSLATE the problem information

• Given information:
  • Tent rental: \(\$11\) per day
  • Insurance fee: \(\$10\) one-time
  • Need total cost c for d days

• What this tells us: We have both a variable cost (depends on days) and a fixed cost (one-time only)

2. INFER the cost structure

• Total cost = Variable cost + Fixed cost
• Variable cost changes with number of days
• Fixed cost stays the same regardless of days
• We need to build an equation combining both components

3. TRANSLATE each cost component into algebra

• Daily rental for d days: \(\$11 \times \mathrm{d} = \$11\mathrm{d}\)
• One-time insurance fee: \(\$10\) (doesn't change with d)

4. Combine the components

• Total cost c = Daily rental cost + Insurance fee
• \(\mathrm{c} = \$11\mathrm{d} + \$10\)

Answer: C. \(\mathrm{c} = 11\mathrm{d} + 10\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret which part of the cost applies "per day" versus "one-time"

They might think both costs apply daily, translating to: "\(\$(11 + 10)\) per day for d days" which gives \(\mathrm{c} = 21\mathrm{d}\). Or they might reverse which number goes with which component, thinking the \(\$10\) is per day and \(\$11\) is one-time.

This may lead them to select Choice D (\(\mathrm{c} = 10\mathrm{d} + 11\)) when they reverse the components.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand the phrase "per day plus a one-time fee" and incorrectly group the components

They might interpret this as "\(\$11\) per (day plus \(\$10\))" meaning the \(\$10\) gets added to the number of days first, then multiplied by \(\$11\).

This may lead them to select Choice A (\(\mathrm{c} = 11(\mathrm{d} + 10)\)).

The Bottom Line:

This problem tests whether students can correctly distinguish between variable costs (that change with quantity) and fixed costs (that don't change with quantity) when translating a word problem into algebra.