A startup company needs cloud storage for their data backup system. The cloud service provider charges a monthly base fee...

1. TRANSLATE the problem information

• Given information:
  • Monthly base fee: $45
  • Cost per gigabyte: $8.75
  • Monthly budget limit: $180
  • Need: Maximum whole gigabytes without exceeding budget

• What this tells us: We need to find the largest whole number of gigabytes where total cost ≤ $180

2. TRANSLATE into mathematical inequality

• Let \(\mathrm{x}\) = number of gigabytes used
• Total monthly cost = Base fee + (Cost per GB × Number of GB)
• Total monthly cost = \(\$45 + \$8.75\mathrm{x}\)
• Constraint: Total cost cannot exceed budget
• Mathematical inequality: \(\$45 + \$8.75\mathrm{x} \leq \$180\)

3. SIMPLIFY by solving the linear inequality

• Subtract $45 from both sides: \(\$8.75\mathrm{x} \leq \$135\)
• Divide both sides by $8.75: \(\mathrm{x} \leq 15.428...\) (use calculator for \(\$135 \div \$8.75\))

4. APPLY CONSTRAINTS to find final answer

• Since storage is purchased in whole gigabytes only: \(\mathrm{x}\) must be an integer
• Since we want maximum storage without exceeding budget: \(\mathrm{x} = 15\) gigabytes
• Verification: \(\$45 + \$8.75(15) = \$176.25 \leq \$180\) ✓ (use calculator for \(\$8.75 \times 15\))

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "without exceeding budget" and set up an equation instead of an inequality (\(\$45 + \$8.75\mathrm{x} = \$180\)), or they confuse which direction the inequality should face.

When they solve \(\$45 + \$8.75\mathrm{x} = \$180\), they get exactly \(\mathrm{x} = 15.428...\), but then they might round up to 16 gigabytes thinking "maximum means round up." However, 16 gigabytes would cost \(\$45 + \$8.75(16) = \$185\), which exceeds their $180 budget. This leads to confusion and potentially guessing when their check doesn't work.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS reasoning: Students correctly solve the inequality to get \(\mathrm{x} \leq 15.428...\) but fail to recognize they need the largest whole number that satisfies this constraint. They might give the decimal answer 15.428 or incorrectly round up to 16.

This causes them to either provide a non-integer answer when the problem specifically requests a whole number, or select 16 which violates the budget constraint.

The Bottom Line:

This problem combines inequality setup, algebraic solving, and real-world constraints. Success requires carefully reading "without exceeding" (inequality, not equation) and understanding that "maximum" means the largest valid integer, not rounding up the decimal result.