Question:A city bike-share program offers a monthly membership for $87 that allows an unlimited number of rides of any length....

1. TRANSLATE the problem information

• Given information:
  • Monthly membership: \(\$87\) (unlimited rides)
  • Single-ride costs: \(\$2.25\), \(\$3.50\), or \(\$4.50\) depending on duration
  • Find: minimum rides where membership costs less

2. INFER the optimal strategy

• To find the MINIMUM number of rides where membership is better, consider when individual rides are most expensive
• Use \(\$4.50\) per ride (highest cost) - this gives the smallest number of rides needed to justify membership

3. TRANSLATE into mathematical inequality

• Set up: Cost of individual rides \(\gt\) Cost of membership
• Inequality: \(\$4.50 \times n \gt \$87\)

4. SIMPLIFY to solve for n

• Divide both sides by 4.50:

\(n \gt 87 \div 4.50\)
\(n \gt 19.333...\)

5. APPLY CONSTRAINTS to find the final answer

• Since you can't take a fraction of a ride, n must be a whole number
• The smallest whole number greater than 19.333... is 20

6. Verify the answer

• 19 rides: \(\$4.50 \times 19 = \$85.50 \lt \$87\) (membership not cheaper)
• 20 rides: \(\$4.50 \times 20 = \$90 \gt \$87\) (membership is cheaper)

Answer: 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students might use the cheapest ride cost (\(\$2.25\)) instead of the most expensive (\(\$4.50\)), thinking they need to find when membership beats the 'typical' cost.

Using \(\$2.25\): \(n \gt 87 \div 2.25 = 38.67\), so \(n = 39\)

This reasoning misses that we want the MINIMUM number of rides, which occurs when individual rides are most expensive. This leads to an answer of 39 rides instead of 20.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students correctly find \(n \gt 19.333...\) but incorrectly round down to 19 instead of up to 20.

They might think '19.333 rides' means 19 rides is enough, not realizing that at exactly 19 rides, membership actually costs MORE than individual passes (\(\$85.50 \lt \$87\)).

The Bottom Line:

This problem requires strategic thinking about optimization - recognizing that to minimize something in one direction, you often need to maximize something else. The key insight is that minimum rides needed happens when individual rides cost maximum amount.