A smartphone has a total storage capacity of 64 gigabytes (GB). The operating system and pre-installed applications use 12 GB...

1. TRANSLATE the problem information

• Given information:
  • Total smartphone storage: 64 GB
  • Storage used by OS and pre-installed apps: 12 GB
  • Size of each new app: 1.6 GB
  • Need to find: maximum number of new apps

• What this tells us: Available storage = \(64 - 12 = 52\) GB for new apps

2. TRANSLATE the constraint into mathematics

• The key insight: Total storage used by new apps cannot exceed available storage
• If \(\mathrm{x}\) = number of new apps, then storage used = \(1.6\mathrm{x}\) GB
• Mathematical constraint: \(1.6\mathrm{x} \leq 52\)

3. SIMPLIFY by solving the inequality

• Divide both sides by 1.6: \(\mathrm{x} \leq 52 \div 1.6\)
• Calculate: \(\mathrm{x} \leq 32.5\) (use calculator for this division)

4. APPLY CONSTRAINTS to select final answer

• Since you cannot install a fraction of an app, \(\mathrm{x}\) must be a whole number
• The maximum whole number \(\leq 32.5\) is 32

Answer: B. 32

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS skill: Students correctly solve to get \(\mathrm{x} \leq 32.5\) but then round up to 33 instead of down to 32.

Their reasoning: "32.5 is closer to 33 than 32, so the answer should be 33."

However, this ignores the constraint that you cannot install 0.5 of an app. Since 32.5 apps would require \(32.5 \times 1.6 = 52\) GB exactly, but 33 apps would need \(33 \times 1.6 = 52.8\) GB (exceeding the 52 GB available), only 32 apps can actually be installed.

This may lead them to select Choice C (33).

Second Most Common Error:

Weak TRANSLATE skill: Students set up the problem incorrectly by not accounting for the 12 GB already used.

They solve \(1.6\mathrm{x} \leq 64\) directly, getting \(\mathrm{x} \leq 40\). Since none of the answer choices are 40, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students understand that mathematical solutions must make sense in real-world contexts - you can't install partial apps, and you can't exceed storage capacity.