A student club sells wristbands for $3 each and pins for $5 each at a fundraiser. The club sells a...

1. TRANSLATE the problem information

• Given information:
  • Wristbands cost $3 each, pins cost $5 each
  • Total items sold: 120
  • Total recorded revenue: $540 (includes $60 from donations)
  • w = number of wristbands, p = number of pins

• What this tells us: We need two equations to solve for two unknowns

2. INFER what equations we need

• We need one equation for the total number of items
• We need one equation for the revenue from sales (not including donations)
• Key insight: The $60 donation is NOT from item sales, so we must subtract it from total revenue

3. Set up the total items equation

• Total items sold = wristbands + pins
• TRANSLATE: \(\mathrm{w + p = 120}\)

4. Set up the revenue equation

• INFER: Sales revenue = Total revenue - Donations
• Sales revenue = \(\$540 - \$60 = \$480\)
• TRANSLATE: Revenue from sales = (price of wristbands × number of wristbands) + (price of pins × number of pins)
• \(\mathrm{3w + 5p = 480}\)

5. Identify the correct system

• Our system is:
  • \(\mathrm{w + p = 120}\)
  • \(\mathrm{3w + 5p = 480}\)
• This matches choice (B)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students miss that the $60 donation must be subtracted from total revenue because it's not from item sales.

They incorrectly think all $540 comes from selling wristbands and pins, so they set up the equation \(\mathrm{3w + 5p = 540}\) instead of \(\mathrm{3w + 5p = 480}\). Combined with the correct total items equation \(\mathrm{w + p = 120}\), this leads them to select Choice A (3w + 5p = 540, w + p = 120).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which price goes with which variable, setting up \(\mathrm{5w + 3p = 480}\) instead of \(\mathrm{3w + 5p = 480}\).

They correctly subtract the donation but mix up the coefficients, thinking wristbands cost $5 and pins cost $3. This may lead them to select Choice D (5w + 3p = 480, w + p = 120).

The Bottom Line:

This problem tests your ability to carefully extract the relevant information and distinguish between total recorded revenue and actual sales revenue. The key is recognizing that donations don't come from item sales, so they must be excluded from the revenue equation.