Tony spends $80 per month on public transportation. A 10-ride pass costs $12.50, and a single-ride pass costs $1.50. If...

1. TRANSLATE the problem information

• Given information:
  • Tony spends \(\$80\) total per month on public transportation
  • 10-ride pass costs \(\$12.50\) each
  • Single-ride pass costs \(\$1.50\) each
  • \(\mathrm{g}\) = number of 10-ride passes bought per month
  • \(\mathrm{t}\) = number of single-ride passes bought per month

• What this tells us: We need an equation that shows how Tony's spending adds up to \(\$80\)

2. TRANSLATE each cost component

• Cost for 10-ride passes: Since each costs \(\$12.50\) and he buys g of them:
  • Total cost for 10-ride passes = \(\mathrm{12.50g}\)

• Cost for single-ride passes: Since each costs \(\$1.50\) and he buys t of them:
  • Total cost for single-ride passes = \(\mathrm{1.50t}\)

3. INFER the relationship we need

• Tony's total monthly spending = Cost of 10-ride passes + Cost of single-ride passes
• Since he spends exactly \(\$80\) per month: \(\mathrm{12.50g + 1.50t = 80}\)

• This means we're looking for a cost equation, not a quantity equation

4. Check against answer choices

• Choice A \(\mathrm{(g + t = 80)}\): This would mean he buys 80 total passes, which doesn't match the \(\$80\) spending
• Choice B \(\mathrm{(g + t = 1.50 + 12.50)}\): This doesn't make mathematical sense
• Choice C \(\mathrm{(1.50g + 12.50t = 80)}\): This reverses the costs - puts the wrong price with each pass type
• Choice D \(\mathrm{(12.50g + 1.50t = 80)}\): This correctly matches our derived equation ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students confuse quantity relationships with cost relationships.

They might think "g + t should equal 80" because they see the number 80 and assume it represents the total number of passes Tony buys. This completely misses that \(\$80\) represents money spent, not quantity purchased.

This may lead them to select Choice A \(\mathrm{(g + t = 80)}\).

Second Most Common Error:

Poor TRANSLATE execution: Students correctly recognize they need a cost equation but mix up which coefficient goes with which variable.

They might think: "10-ride passes are more expensive, so they should have the bigger coefficient" but then incorrectly write \(\mathrm{1.50g + 12.50t = 80}\), putting \(\$1.50\) (the single-ride cost) with g (the 10-ride pass quantity).

This may lead them to select Choice C \(\mathrm{(1.50g + 12.50t = 80)}\).

The Bottom Line:

This problem tests whether students can distinguish between representing quantities versus costs, and whether they can accurately match cost coefficients with the correct variables. The key insight is recognizing that \(\$80\) represents total spending, not total items purchased.