A and B started from the same point on a circular track. The ratio of their speed is n:1 (n is a whole number). If 5th and 17th time they meet at the same point, what can not be the value of n?
Options : A) 2 b) 3 c) 4 d) 6
Answer is : D
Solution :
If speed of two bodies moving in same direction in circular path are in the simplest ratio of a:b, then no. of pts they will meet = |a-b|Let circumference of circle be 'L'.
Time for 1st meeting at any of the |a-b| pts = L/(a-b).
Time for 1st meeting at Starting Pt = LCM of (L/a, L/b).
Time for nth meeting at Starting Pt = n*LCM of (L/a, L/b).
Checking with the options :-
1 . Check Option A :
When n = 2 :
a=2,b=1.
No:of meeting pts = (2-1)=1pt.
Only meeting pt is L.
Time for first meeting = L/1.
Time for first meeting at starting pt = LCM of (L/2, L/1) = L.
Time for 5th meeting = 5L.
Time for 17th meeting = 17L.
Both tyms, they meet at pt L.
So, n=2 is valid.
Check Option B :
When n = 3 :
a=3,b=1.
No:of meeting pts = (3-1)=2pts.
Meeting pts are : L/2, L .
Time for first meeting = L/2.
Time for first meeting at starting pt = LCM of (L/3, L/1) = L.
Time for 5th meeting = 5*L/2.
Time for 17th meeting = 17*L/2.
Both tyms, they meet at pt L/2.
So, n=3 is valid.
Check Option C :
When n = 4 :
a=4,b=1.
No.of meeting pts = (4-1)=3pts.
Meeting pts are : L/3, L/2, L .
Time for first meeting = L/3.
Time for first meeting at starting pt = LCM of (L/4, L/1) = L.
Time for 5th meeting = 5*L/3.
Time for 17th meeting = 17*L/3.
Both times, they meet at pt L/3.
So, n=4 is valid.
When n = 6 :
a=6, b=1.
No:of meeting pts = (6-1)=5pts.
Meeting pts are : L/5, L/4, L/3, L/2, L .
Time for first meeting = L/5.
Time for first meeting at starting pt = LCM of (L/5, L/1) = L.
Time for 5th meeting = 5*L/5 = L.
Time for 17th meeting = 17*L/5.
Both tyms, they meet at different pts namely L and L/5.
So, n=6 is not valid.