The notice taped to the glass at the community pool has been there since June, and it is the clearest piece of writing in the building.
ENTRY WITHOUT A PASS: $8 each visit.
SUMMER PASS: $40, then $3 each visit.
Underneath, somebody on the staff has added a line in handwriting: «Ask us which one is better for you. It depends how often you swim.»
That sentence is doing real mathematical work. The pass has two parts, and they behave completely differently. The $40 is a one-time fee of forty dollars: you pay it in June whether you swim once or fifty times, and it never comes back. The $3 is a rate — it arrives again each visit, and it is the only part of the pass that grows.
So the arithmetic is not one sum but two. A swimmer who comes twelve times over the summer pays forty dollars once and three dollars twelve times: $76 altogether. Without the pass, twelve visits cost $96. Someone who comes only five times pays $55 with the pass and $40 without it, and has overpaid for the privilege.
The turning point is exactly eight visits, where both routes cost $64 and the choice stops mattering. Below eight, pay at the door. Above eight, the pass works out cheaper, and every visit after that saves five dollars.
The reception desk gets the same question forty times a week. The answer is never a number. It is another question: how often?