Practice Problem - 5 : Seven children A, B, C, D, E, F and G started walking from the same point at the same time, with speeds in the ratio of 1 : 2 : 3 : 4 : 5 : 6 : 7 respectively and they are running around a circular park. Each of them carry flags of different colours and whenever two or more children meet, they place their respective flag at that point. However nobody places more than 1 flag at a same point. They are running in anti-clockwise direction. How many flags will be there in total, when there will be no scope of putting more flags?

Solution : When running in the same direction : If the ratio of speeds of two athletes (in the most reducible form) is a : b, the number of distinct meeting points on the track would be would be |a – b|

A and B will meet at |1 - 2| = 1 point.

A and C will meet at |1 - 3| = 2 points

A and D will meet at |1 - 4| = 3 points

A and E will meet at |1 - 5| = 4 points

A and F will meet at |1 - 6| = 5 points

A and G will meet at |1 - 7| = 6 points

So A will put 1 + 2 + 3 + 4 + 5 + 6 = 21 flags.

similarly B and C will meet at |2 - 3| = 1 point

B and D will meet at |2 - 4| = 2 points

B and E will meet at |2 - 5| = 3 points

B and F will meet at |2 - 6| = 4 points

B and G will meet at |2 - 7| = 5 points

So B will put 1 + 2 + 3 + 4 + 5 = 15 flags

Similarly find for C, D, E and F.

We will get 21 + 15 + 10 + 6 + 3 + 1 = 56 flags