A community fundraiser sells wristbands for $8 each and T-shirts for $18 each. The fundraiser must raise at least $200...

1. TRANSLATE the problem information

• Given information:
  • Wristbands cost $8 each, with \(\mathrm{w}\) wristbands sold
  • T-shirts cost $18 each, with \(\mathrm{t}\) T-shirts sold
  • Must raise at least $200 total

• What this tells us: We need to find an inequality representing the minimum revenue requirement

2. INFER the revenue relationship

• Total revenue comes from two sources that we add together
• Revenue from wristbands = $8 × \(\mathrm{w}\) = \(\mathrm{8w}\) dollars
• Revenue from T-shirts = $18 × \(\mathrm{t}\) = \(\mathrm{18t}\) dollars
• Therefore: Total revenue = \(\mathrm{8w + 18t}\)

3. TRANSLATE the constraint into mathematical form

• "At least $200" means the total must be greater than or equal to $200
• This translates to: \(\mathrm{8w + 18t \geq 200}\)

4. Match with answer choices

• Looking at the options, this exactly matches choice (B): \(\mathrm{8w + 18t \geq 200}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "at least" with "at most" and use ≤ instead of ≥

Students think: "They can't exceed $200, so it must be less than or equal to $200"

This leads them to select Choice A (\(\mathrm{8w + 18t \leq 200}\))

Second Most Common Error:

Poor TRANSLATE execution: Students swap the price coefficients, thinking wristbands cost $18 and T-shirts cost $8

Students misread or mix up which item has which price, creating the expression \(\mathrm{18w + 8t}\) instead of \(\mathrm{8w + 18t}\)

This may lead them to select Choice C (\(\mathrm{18w + 8t \geq 200}\)) or Choice D (\(\mathrm{18w + 8t \leq 200}\))

The Bottom Line:

This problem tests careful reading and translation skills more than complex math. The key challenge is accurately converting everyday language ("at least") into mathematical symbols while keeping track of which price goes with which variable.