## 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.

**Check Option D :**

**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.