A local transit company sells a monthly pass for $95 that allows an unlimited number of trips of any length....

1. TRANSLATE the problem setup

• Given information:
  • Monthly pass costs \(\$95\)
  • Individual tickets cost \(\$1.50\), \(\$2.50\), or \(\$3.50\)
  • Want minimum trips where pass is cheaper

• What this means: We need individual ticket costs \(\gt \$95\)

2. INFER the strategic approach

• To find the minimum number of trips, use the maximum ticket price
• Why? Higher price per trip means fewer trips needed to exceed \(\$95\)
• Use \(\$3.50\) per trip for our calculation

3. TRANSLATE into mathematical inequality

• "Monthly pass costs less than individual tickets" becomes:

\(\$95 \lt \text{(price per trip)} \times \text{(number of trips)}\)

• With maximum price: \(95 \lt 3.50\mathrm{n}\)

4. SIMPLIFY to solve the inequality

• Divide both sides by 3.50:

\(95 \div 3.50 \lt \mathrm{n}\)

• Calculate: \(\mathrm{n} \gt 27.14...\) (use calculator)

5. APPLY CONSTRAINTS to find the final answer

• Since you can only take whole trips: \(\mathrm{n} \geq 28\)
• Check: \(28 \times \$3.50 = \$98.00 \gt \$95\) ✓

Answer: 28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Using the wrong ticket price in the calculation

Students might use \(\$1.50\) (cheapest) or \(\$2.50\) (middle price) instead of \(\$3.50\). This leads to calculating when the pass becomes cheaper for the wrong scenario. For example, using \(\$2.50\) gives \(\mathrm{n} \gt 38\), which doesn't answer the question about the minimum number of trips.

This leads to confusion and incorrect reasoning about what the problem is actually asking.

Second Most Common Error:

Poor TRANSLATE reasoning: Setting up the inequality backwards

Students might write \(3.50\mathrm{n} \gt 95\) (thinking "when do individual trips cost more?") instead of recognizing we want \(95 \lt 3.50\mathrm{n}\) (when does the pass cost less?). While these are mathematically equivalent, the confusion can lead to misunderstanding what the solution represents.

The Bottom Line:

This problem requires recognizing that "minimum number of trips for which the pass is cheaper" means using the maximum individual price to find when that threshold is crossed. Students must think strategically about which price scenario gives the smallest trip count.