Optimizing a Transit Purchase Around an Expiring Credit
Mateo buys transit for 20 weekday commutes per month, one round trip per workday (40 rides). Fares: single ride $3.00; a 10-ride ticket costs $24 and expires after 90 days; a monthly pass costs $132. Off-peak singles are $2.10, but Mateo travels at 9:00 a.m. and 5:30 p.m., both peak. He also has a $15 promotional credit that can be used only on single rides and expires in 10 days. He will take 10 peak rides in those 10 days. Tickets are transferable; passes are not.
1. What is the lowest-cost combination for a normal month, and by how much does it beat the monthly pass?
Read the worked answer
Four 10-ride tickets cover 40 rides: 4 x $24 = $96, an effective $2.40 per ride. Singles would be 40 x $3 = $120; the pass is $132. So $96 is lowest and beats the pass by $36. The off-peak fare is irrelevant because both trips are peak.
2. How should Mateo use the expiring credit, and what total does that produce?
Read the worked answer
Use the $15 credit on five of the ten rides in the next 10 days, since it cannot buy tickets. That leaves 35 rides for the month: three tickets cover 30 rides ($72), and five peak singles cost $15. Total = $72 + $15 = $87. Buying four tickets would cost $96 and leave five unused rides, so the credit plan saves $9. The credit is worth using only because he has enough rides before it expires.
3. If the credit could be applied to any purchase, would the best plan change? Explain the adjustment.
Read the worked answer
Yes. Apply the credit to the tickets: four tickets cost $96 - $15 = $81, which beats $87. The rule change matters because the credit's value is no longer restricted to the more expensive per-ride option. Recompute from the constrained total, not from the sticker price.