Question:A paddleboard rental shop charges a mandatory equipment fee of $12 per visit.The hourly rental rate is $20, and a...

1. TRANSLATE the problem information

• Given information:
  • Equipment fee: \(\$12\) (mandatory, per visit)
  • Hourly rate: \(\$20\) with \(10\%\) sales tax applied to hourly charges only
  • Maya's budget: at most \(\$96\)
  • Must rent whole number of hours
• What this tells us: We need to find the maximum whole number of hours within her budget

2. TRANSLATE the cost structure into mathematical expressions

• Equipment fee = \(\$12\) (fixed cost, no tax)
• Hourly cost with tax = \(\$20 + (10\% \text{ of } \$20) = \$20 + \$2 = \$22\) per hour
• Total cost = \(\$12 + \$22\mathrm{h}\) (where h = number of hours)

3. TRANSLATE the budget constraint

• "At most \(\$96\)" means: Total cost \(\leq \$96\)
• Mathematical constraint: \(\$12 + \$22\mathrm{h} \leq \$96\)

4. SIMPLIFY by solving the inequality

• \(\$12 + \$22\mathrm{h} \leq \$96\)
• \(\$22\mathrm{h} \leq \$96 - \$12\)
• \(\$22\mathrm{h} \leq \$84\)
• \(\mathrm{h} \leq \$84 \div \$22\) (use calculator)
• \(\mathrm{h} \leq 3.818...\)

5. APPLY CONSTRAINTS to find the final answer

• Since hours must be whole numbers, we take the largest integer: \(\mathrm{h} = 3\)
• Verification check:
  • For \(\mathrm{h} = 3\): \(\$12 + \$22(3) = \$78 \leq \$96\) ✓ (within budget)
  • For \(\mathrm{h} = 4\): \(\$12 + \$22(4) = \$100 \gt \$96\) ✗ (exceeds budget)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students forget that tax only applies to hourly charges, not the equipment fee, and calculate \(10\%\) tax on the entire amount.

They might set up: Total cost = \((\$12 + \$20\mathrm{h}) \times 1.10\), leading to the inequality \((\$12 + \$20\mathrm{h}) \times 1.10 \leq \$96\). This gives approximately \(\mathrm{h} \leq 3.45\), and they might select 3, but their underlying cost calculation is wrong throughout.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that the hourly rate with tax is \(\$22\), but make arithmetic errors when solving \(\$22\mathrm{h} \leq \$84\).

Common mistakes include dividing incorrectly or forgetting to subtract \(\$12\) first. This leads to confusion about whether the answer should be 3 or 4, often resulting in guessing.

The Bottom Line:

This problem requires careful parsing of which costs have tax applied and systematic algebraic manipulation. The key insight is recognizing that the tax structure creates two separate cost components that must be handled differently.