SOLUTION: In a circular track of 63 meters two runners run with speeds 9m/s and 7 m/s. At how many distinct meeting points will they meet? i figured they meet every 31.5 second and i work

Question 984312: In a circular track of 63 meters two runners run with speeds 9m/s and 7 m/s. At how many distinct meeting points will they meet?
i figured they meet every 31.5 second and i worked out that the number of distinct meeting points is 2. however, what i do not understand is this solution that you can take the ratio of the speeds in simplest form, subtract, and then get the number of distinct meeting points.
What I mean is 9-7 =2 which is your answer
Can you please derive this method.
I AM LITERALLY LOOSING it, i cannot understand how this works even though it is correct.
please help me i've been asking since yesterday.
Also if they were traveling in opposite directions the method says to add 9 and 7 which gives you 16 distinct meeting points. I don't understand this either.

I AM LITERALLY LOOSING it, I cannot understand how this works even though it is correct.
please help me i've been asking since yesterday!
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Opposite, they are separating at 16 m/sec.
Therefore, every 63/16 seconds they meet, and after 16 such places, they start the same cycle over again. (periodicity is 63/16 sec)
Together, they are separating at 2 m/sec. 63/2=31.5 seconds, and the periodicity is 2, the number of distinct places they meet.
The subtracting gives you the denominator to the circumference of the track, and that gives you the periodicity.
Going in opposite directions, the denominator is larger, and there are more places they can meet.
It's why you see more cars come towards you than ones you pass. It is the same issue of overtaking vs. oncoming.
I'm not sure that answers your question, but drawing it and seeing where the meeting points occur might help.
Think of a clock. One hand moves around at 12 times the speed of the other. The difference is 11. How many places do the hands cross? Eleven, and they are the same places.
9 vs 7: A goes a full circle while B goes 7/9. Second rep for A, B is now 5/9. Third rep for A, B is now 3/9, 4th rep for A, B is 1/9, 5th rep, A passes, and B 8/9, 6th Rep, 6/9. 9th rep, passes as A has done 9 and B 7, difference two.
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