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Boris and Michael are jogging around a lake. It takes Boris 6 minutes to complete a lap and
Michael takes 8 minutes. After completing each lap, the boys take a rest, Boris for one minute
and Michael for two minutes. Then they continue jogging. If they started jogging together,
how long, in minutes, does it take until they are next together?
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This problem is non-standard Travel and Distance problem.
The first move/thought is to analyse it as @ankor@dixie-net.com did in his post.
But the normal state of a person who solves it for the first time, is DO NOT BELIEVE and TO CHECK himself (or herself).
THEREFORE, I decided to make such a check. My solution is below. Read it attentively.
Let make timing Table, showing time/times, when the persons, Boris and Michael, are in the starting point again,
while they make their trips around the lake. In the Table, I placed timing, when they achieve this starting point and rest there.
T A B L E
# of lap 1 2 3 4 5 6 7 8 9 10
Boris' time 6-7 13-14 20-21 27-28 34-35 41-42 48-49 55-56 62-63 69-70
Michael's time 8-10 18-20 28-30 38-40 48-50 58-60 68-70
# of lap 1 2 3 4 5 6 7
We look to find non-zero intersection between the given time intervals, when they rest.
Such non-zero / (non-empty) intersection is time period from 48 to 49 minutes, when Boris completed his 7-th lap
and Michael completed his 5-th lap.
ANSWER. First time they are next together is 48 minutes after start.
Solved.
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At t = 28 minutes, they " are together " during the time interval which lasts 0 minutes and 0 seconds
(empty intersection of time intervals).
So, according to the context (and common sense), I do not consider this time moment as if " they are together ".