Question 1184017: 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?
Found 3 solutions by ankor@dixie-net.com, greenestamps, ikleyn:
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! 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?
:
Include the rest times in the times to complete each lap except the final lap of each. B requires 7 min per lap, M requires 10 min
They should be together again at 69 min
Least common multiple of 7 and 10 is 70 min, B will complete 10 laps: 9(7)+6 = 69 min, M will complete 7 laps: 6(10) + 8 = 68 min, wait 1 min when B joins him at 69 min
Answer by greenestamps(13206)  (Show Source):
You can put this solution on YOUR website!
The next time the two are together will be when Boris has run one lap more than Michael. That might happen when they are both resting; or it might happen when both are jogging.
Since the ratio of the times it takes them to run a lap is 6:8 = 3:4, the first time they are together again should be near the time that Boris completes 4 laps and Michael completes 3.
So look at the times around the time when Boris completes 4 laps and Michael completes 3.
Boris' "cycle" is 7 minutes long. He completes the rest after his 4th lap at 28 minutes, which means he finished his 4th lap at 27 minutes.
Michael's cycle is 10 minutes long. He completes the rest after his 3rd lap at 30 minutes, which means he finished his 3rd lap at 28 minutes.
So the two boys are next together at 28 minutes, when Michael has just finished his 3rd lap and Boris is just finishing the rest after his 4th lap.
ANSWER: 28 minutes
Answer by ikleyn(52866)  (Show Source):
You can put this solution on YOUR website! .
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?
~~~~~~~~~~~~~~~~
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 ".
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