Solution: ① It takes X minutes for technicians to complete all tasks and Y minutes for apprentices to complete all tasks, and all tasks are regarded as a whole "1"; Then the speed of the mechanic is 1/X, and the speed of the apprentice is 1/y, and the time is calculated in minutes.
(2) According to "it takes 6 hours and 40 minutes for the apprentice to complete the remaining tasks after the mechanic completes two thirds of the tasks", we can list the equivalence relation about time (the time for the mechanic to complete two thirds of the workload+the time for the apprentice to complete the remaining workload =6 hours and 40 minutes):
(2/3)/(1/x)+(1-2/3)/(1/y) = 400 (6 hours and 40 minutes);
Similarly, according to "it takes * * * seven and a half hours for the apprentice to complete the rest after the mechanic completes one-third of the tasks":
(1/3)/(1/x)+(1-1/3)/(1/y) = 450 (seven and a half hours);
Two simultaneous equations can be solved: X=350 (minutes) and Y=500 (minutes).