A community kitchen starts the month with 3{,600} paper napkins in stock. The kitchen uses 45 napkins each day and...

1. TRANSLATE the problem information

• Given information:
  • Starting napkins: 3,600
  • Daily usage rate: 45 napkins per day
  • No restocking occurs
  • Need to find: when remaining napkins = used napkins

• What this tells us: We need expressions for both "remaining" and "used" napkins after \(\mathrm{d}\) days

2. INFER the mathematical setup

• Since we want remaining = used, we'll set up an equation
• Let \(\mathrm{d}\) = number of days
• After \(\mathrm{d}\) days: Used napkins = \(\mathrm{45d}\)
• After \(\mathrm{d}\) days: Remaining napkins = \(\mathrm{3{,}600 - 45d}\)

3. TRANSLATE the key condition into an equation

• "Remaining equals used" means: \(\mathrm{3{,}600 - 45d = 45d}\)

4. SIMPLIFY to solve for d

• Add \(\mathrm{45d}\) to both sides: \(\mathrm{3{,}600 = 90d}\)
• Divide both sides by 90: \(\mathrm{d = 3{,}600 \div 90 = 40}\)

5. Verify with alternative approach

• INFER: If remaining = used, each portion is exactly half the total
• Half of 3,600 = 1,800 napkins
• Time to use 1,800 napkins: \(\mathrm{1{,}800 \div 45 = 40}\) days ✓

Answer: C (40 days)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students often struggle to correctly set up the "remaining napkins" expression, writing it as just \(\mathrm{45d}\) instead of \(\mathrm{3{,}600 - 45d}\).

Their thinking: "After \(\mathrm{d}\) days, there are \(\mathrm{45d}\) napkins remaining"

This fundamental misunderstanding of what "remaining" means leads to the wrong equation: \(\mathrm{45d = 45d}\), which gives no useful information. This causes confusion and typically leads to guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{3{,}600 - 45d = 45d}\) but make arithmetic errors when solving.

Common mistakes include:

• Getting \(\mathrm{3{,}600 = 45d}\) (forgetting to add \(\mathrm{45d}\) to both sides)
• Calculating \(\mathrm{3{,}600 \div 45 = 80}\) instead of \(\mathrm{3{,}600 \div 90 = 40}\)

This may lead them to select Choice D (80) due to the incorrect division.

The Bottom Line:

This problem tests whether students can correctly interpret "remaining quantity" as "starting amount minus used amount" and then set up a proper equation. The key insight is recognizing that when two quantities are equal, you can solve by setting their algebraic expressions equal to each other.