Six tasks, two teams, one shared review step. The arithmetic is the whole point.
task description duration depends on
---- -------------------------- -------- -----------
A schema agreed 5d -
B API built 15d A
C mobile client built 20d A
D integration test 8d B, C
E security review 5d B
F launch readiness 3d D, E
paths through the plan
A -> B -> D -> F = 5 + 15 + 8 + 3 = 31d
A -> C -> D -> F = 5 + 20 + 8 + 3 = 36d <- critical path
A -> B -> E -> F = 5 + 15 + 5 + 3 = 28d
float
C 0d on the critical path
B 5d B finishes day 20, D cannot start until C finishes day 25
E 8d E can finish as late as day 33 without delaying F
The program is 36 days, not 56, and not the sum of anything. Candidates who add durations are describing effort, not duration, and the two differ by however much runs in parallel.
The useful finding is in B's five days of float. Speeding up the API buys nothing at all, because integration test waits on the mobile client either way. If a director offers you two engineers, putting them on B is a visible burst of activity that does not move the date by one day, and saying so is exactly the value a TPM adds at that moment.
The second finding is what happens when C slips. Six days of slip on C moves the program date by six days and simultaneously consumes B's entire float, so a plan that had two independent paths now has one, and B becomes critical too. That is the point at which reporting has to change, because the number of things that can now hurt you has doubled while the date moved by less than a week.
Float on E is worth reading as an opportunity rather than a comfort. Eight days of slack on the security review means the review can be scheduled early without cost, which converts a late-program unknown into an early finding. Deliberately spending float to pull risk forward is one of the more sophisticated moves available in program planning.