Nomsa's Taxi Fares: The Bundle That Can Never Win
Nomsa travels to work by minibus taxi. A single trip costs R32 and she makes two trips on every day she goes to the office. In a typical month there are 22 working days, and she works from home on 6 of them. Three payment options exist: (a) pay R32 per single trip; (b) buy a monthly unlimited ticket for R780; (c) buy a 30-trip pack for R840, valid for that month only, with unused trips forfeited. A colleague tells her the pack is obviously the best value because the price per trip works out to R28, less than the R32 single fare. Nomsa wants the cheapest option for her actual travel, and she wants to know how the answer changes if her number of work-from-home days changes.
1. Using Nomsa's actual travel pattern, calculate her monthly cost under each of the three options and identify the cheapest.
Read the worked answer
Trips per month
- Office days = 22 − 6 = 16
- Trips = 16 × 2 = 32 trips
Option (a) - pay per trip 32 × R32 = R1 024
Option (b) - unlimited ticket R780
Option (c) - 30-trip pack The pack covers only 30 of the 32 trips; the remaining 2 must be bought at the single fare: 840 + (2 × 32) = 840 + 64 = R904
Comparison R780 (ticket) < R904 (pack) < R1 024 (per trip). The unlimited ticket is cheapest, saving R244 against paying per trip.
2. At how many office days per month do the unlimited ticket and the pay-per-trip option cost the same? Over what range of office days (and of work-from-home days) is the ticket the cheapest of the two?
Read the worked answer
Set up the comparison Cost per office day under pay-per-trip = 2 × R32 = R64. Let n = office days. Pay-per-trip = 64n; ticket = 780.
Solve 64n = 780 → n = 12,1875
Since n must be a whole number, test the integers either side:
- n = 12 → 64 × 12 = R768 < R780 → pay per trip wins
- n = 13 → 64 × 13 = R832 > R780 → ticket wins
Answer There is no whole number of office days at which the two are exactly equal; the switch happens between 12 and 13 days. The ticket is cheaper from 13 office days upwards, i.e. when she works from home on 9 or fewer of the 22 working days.
This matters for Nomsa: she currently has 16 office days, comfortably on the ticket side, but if her employer allowed 10 or more work-from-home days she should switch to paying per trip.
3. Evaluate the colleague's claim that the 30-trip pack must be the best value because R28 per trip is less than R32.
Read the worked answer
The claim is false - the pack is never the cheapest of the three.
Step 1: when does the pack beat paying per trip? The pack gives 30 trips for R840, equivalent to 840 ÷ 32 = 26,25 single fares. So the pack beats singles only once usage exceeds 27 trips (at 27 trips: pack R840 vs singles R864).
Step 2: but check the ticket at that point. From 25 trips upwards (13+ office days), the unlimited ticket at R780 already beats singles and beats the pack's R840. So in the only region where the pack beats singles (27+ trips), the ticket beats the pack:
- 27 trips: singles R864, pack R840, ticket R780
- 32 trips: singles R1 024, pack R904, ticket R780
- 40 trips: singles R1 280, pack R1 160, ticket R780
Step 3: why the R28 is a trap. R28 is the average price only if all 30 trips are used and no extras are needed. With 32 trips the true average is 904 ÷ 32 = R28,25, and the pack still carries a fixed R840 outlay plus a forfeiture rule. A lower headline unit price cannot compensate for a bigger fixed cost when a cheaper unlimited alternative exists.
Verdict: the colleague has compared a unit price with a total cost. The pack is dominated - it should be eliminated before any comparison of Nomsa's real options.