The Bike-Share Pricing Puzzle
A city bike-share program charges a flat $3 to unlock a bike plus $0.10 per minute of riding. A monthly pass costs $25 and makes each ride free for the first 30 minutes, with $0.10 per minute after that (no unlock fee). A commuter rides twice each workday (about 22 workdays per month), and each ride takes 18 minutes. She is deciding between paying per ride and buying the pass. A friend tells her the pass is 'obviously worth it because she rides so often.' The program also offers a weekly pass for $8 that covers unlimited 30-minute rides for 7 days.
1. Calculate her monthly cost under the pay-per-ride plan and under the monthly pass. Which is cheaper, and by how much?
Read the worked answer
Pay-per-ride: each ride costs $3 + (18 × $0.10) = $3 + $1.80 = $4.80. She takes 2 rides × 22 days = 44 rides per month. Total = 44 × $4.80 = $211.20. Monthly pass: $25 flat, and each 18-minute ride is under the 30-minute free window, so no per-minute charges. Total = $25. The pass saves $211.20 − $25 = $186.20 per month. Here the friend's conclusion happens to be right, but it should be verified by the numbers rather than assumed.
2. Compare the weekly pass ($8 for 7 days) with the monthly pass for a month of 22 workdays spread across roughly 4.3 weeks. Which is the better buy, and what assumption could change the answer?
Read the worked answer
Four weekly passes cost 4 × $8 = $32, which already exceeds the $25 monthly pass; covering the extra days could require a fifth week, pushing the cost to $40. The monthly pass is cheaper as long as the rider uses the bike on enough days that weekly passes would need to cover more than about 3 weeks (3 × $8 = $24 is just under $25, but leaves days uncovered). The answer could change if she rides only sporadically-say, fewer than about 9 rides per month, where pay-per-ride (9 × $4.80 ≈ $43) still loses to the pass but the gap narrows-or if she sometimes exceeds 30 minutes, which adds cost under every plan but affects the pass plans only after the free window.