A manager is responsible for ordering supplies for a shaved ice shop. The shop's inventory starts with 4,500 paper cups,...

1. TRANSLATE the problem information

• Given information:
  • Starting inventory: 4,500 paper cups
  • Daily usage rate: 70 cups per day
  • Target inventory level: 1,700 cups
• Find: Number of days (x) to reach target

2. INFER the mathematical relationship

• Key insight: Remaining inventory decreases linearly over time
• After x days: Remaining = Starting - (Daily rate × Days)
• This gives us: \(4,500 - 70\mathrm{x} = 1,700\)

3. SIMPLIFY by solving the linear equation

• Start with: \(4,500 - 70\mathrm{x} = 1,700\)

\(-70\mathrm{x} = 1,700 - 4,500\)

• Simplify the right side:

\(-70\mathrm{x} = -2,800\)

• Divide both sides by -70:

\(\mathrm{x} = -2,800 ÷ (-70) = 40\)

Answer: B. 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may set up the equation incorrectly by adding instead of subtracting daily usage, thinking \(4,500 + 70\mathrm{x} = 1,700\) because they focus on the cups being "used" rather than "remaining."

This backward setup leads to \(-70\mathrm{x} = -2,800\), but then \(\mathrm{x} = -40\), which makes no sense. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(4,500 - 70\mathrm{x} = 1,700\) but make arithmetic errors when handling negative numbers, particularly getting confused by \((-2,800) ÷ (-70)\).

Common arithmetic mistakes include getting \(\mathrm{x} = -40\) or \(\mathrm{x} = 2,800/70 ≈ 40\). If they get \(\mathrm{x} = -40\), this may lead them to select Choice A (20) after taking the absolute value.

The Bottom Line:

This problem tests whether students can correctly model a decreasing linear relationship and handle the resulting negative number arithmetic without getting confused.