Question 1184017
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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.



<pre>
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.


<U>ANSWER</U>. First time they are next together is 48 minutes after start.
</pre>

Solved.


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At &nbsp;&nbsp;t = 28 minutes, &nbsp;they &nbsp;&nbsp;" are together " &nbsp;during the time interval which lasts &nbsp;0 minutes and &nbsp;0 &nbsp;seconds 

(empty intersection of time intervals).


So, &nbsp;according to the context &nbsp;(and common sense), &nbsp;I do not consider this time moment as if &nbsp;" they are together ".