Maria plans to rent a boat. The boat rental costs $60 per hour, and she will also have to pay...

1. TRANSLATE the problem information

• Given information:
  • Boat rental: \(\$60\) per hour
  • Water safety course: \(\$10\) (one-time fee)
  • Maximum budget: \(\$280\)
  • Hours must be whole numbers

• What this tells us: We need to find the largest whole number of hours where total cost \(\leq \$280\)

2. TRANSLATE the constraint into math

• Total cost = (hourly rate × hours) + one-time course fee
• Let h = number of hours
• Total cost = \(60\mathrm{h} + 10\)
• Constraint: \(60\mathrm{h} + 10 \leq 280\)

3. SIMPLIFY by solving the inequality

• Start with: \(60\mathrm{h} + 10 \leq 280\)
• Subtract 10 from both sides: \(60\mathrm{h} \leq 270\)
• Divide both sides by 60: \(\mathrm{h} \leq 4.5\)

4. APPLY CONSTRAINTS to select final answer

• Since hours must be whole numbers, we can't rent for 4.5 hours
• The largest whole number \(\leq 4.5\) is 4
• Check: \(60(4) + 10 = 250\), which is \(\leq 280\) ✓

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might set up the inequality incorrectly, such as writing \(60\mathrm{h} \leq 280\) and forgetting to include the \(\$10\) course fee.

This leads them to solve \(60\mathrm{h} \leq 280\), getting \(\mathrm{h} \leq 4.67\), and potentially answering 4 hours while missing that they could actually afford less time due to the overlooked course fee. In reality, this would mean Maria has \(\$280 - \$10 = \$270\) for boat rental, allowing only \(\mathrm{h} \leq 4.5\), so the maximum is still 4 hours, but the reasoning would be flawed.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students correctly solve to get \(\mathrm{h} \leq 4.5\) but then round up to 5 hours instead of down to 4 hours.

This happens because they might think "4.5 rounds to 5" without recognizing that we need the largest value that satisfies the constraint, not standard rounding rules. This leads to confusion about whether the answer should be 4 or 5.

The Bottom Line:

This problem tests your ability to translate real-world spending limits into mathematical inequalities and then properly apply the constraint that time must be measured in whole hours. The key insight is recognizing that "no more than" means \(\leq\), and that fractional hours aren't allowed in this context.