A cloud storage company offers an unlimited storage plan for $120 per month. Alternatively, customers can choose a pay-per-use plan...

1. TRANSLATE the pricing information

• Unlimited plan: \(\$120\) per month
• Pay-per-use plan:
  • First 20 GB: \(\$2.00/GB\)
  • Next 20 GB (21-40 GB): \(\$3.00/GB\)
  • Beyond 40 GB: \(\$5.00/GB\)
• Goal: Find minimum GB where unlimited costs LESS than pay-per-use

2. INFER the approach strategy

• Since we're looking for when unlimited becomes cheaper, we need to:
  • Calculate pay-per-use costs at different usage levels
  • Find where pay-per-use exceeds \(\$120\)
• Start by calculating the "breakpoints" in the pricing tiers

3. SIMPLIFY the piecewise calculation

Calculate cumulative costs:

• First 20 GB: \(\mathrm{20 \times \$2.00 = \$40}\)
• Next 20 GB: \(\mathrm{20 \times \$3.00 = \$60}\)
• Total for first 40 GB: \(\mathrm{\$40 + \$60 = \$100}\)

Since \(\mathrm{\$100 \lt \$120}\), we need to go beyond 40 GB.

4. TRANSLATE the general cost formula

For usage beyond 40 GB:

• Base cost for first 40 GB: \(\$100\)
• Additional cost per GB: \(\$5.00\)
• Total cost for n GB (where n > 40): \(\mathrm{\$100 + \$5.00(n - 40)}\)

5. INFER the inequality setup

We want unlimited (\(\$120\)) to cost LESS than pay-per-use:

\(\mathrm{\$120 \lt \$100 + \$5.00(n - 40)}\)

6. SIMPLIFY the inequality

\(\mathrm{\$120 \lt \$100 + \$5.00(n - 40)}\)

\(\mathrm{\$20 \lt \$5.00(n - 40)}\)

\(\mathrm{4 \lt n - 40}\)

\(\mathrm{n \gt 44}\)

7. APPLY CONSTRAINTS to find the minimum integer

Since we need \(\mathrm{n \gt 44}\) and the answer must be a whole number of gigabytes:

• At \(\mathrm{n = 44}\): Pay-per-use = \(\mathrm{\$100 + \$5.00(4) = \$120}\) (equal, not less)
• At \(\mathrm{n = 45}\): Pay-per-use = \(\mathrm{\$100 + \$5.00(5) = \$125 \gt \$120}\) ✓

Answer: 45

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Setting up the wrong inequality direction, making it \(\mathrm{\$120 \gt \text{pay-per-use}}\) instead of \(\mathrm{\$120 \lt \text{pay-per-use}}\).

Students often think "I want to find when unlimited is better" and incorrectly write \(\mathrm{\$120 \gt \$100 + \$5.00(n - 40)}\). This gives them \(\mathrm{n \gt 40}\), leading them to think 41 GB is the answer. Since 41 isn't among typical answer choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors in the piecewise calculation, especially forgetting that the middle tier is only for GB 21-40 (20 GB total), not 40 GB.

Some students calculate: First 20 GB = \(\$40\), next 40 GB = \(\$120\), getting \(\$160\) total for 60 GB. This makes them think much higher usage is needed, leading to incorrect answer selection.

The Bottom Line:

This problem challenges students to correctly interpret "less than" in the context of finding a minimum threshold, while accurately handling piecewise pricing calculations.