Two friends, P and Q simultaneously start running from the same point around a circular track. They run in the same direction. P runs at 6 m/sec, and Q runs at b m/sec. If they cross each other at exactly two points on the circular track, and b is a natural number less than 6, then how many values can b take?
Show answer & explanation
Correct answer: B
Explanation
The correct answer is option 2 i.e. 1
Given that
The speed of P = 6 m/s
The speed of Q = b m/s
If they cross each other at exactly two points on the circular track, and b is a natural number less than 6
The relative speed when running in same direction = 6 - b
Let the length of the track be L
According to the question
TIme taken to meet for the first time = L/(6 - b)
Time taken to Lap for P = L/6
Time taken to Lap for Q = L/b
The first meeting at starting point again = LCM (L/6, L/b) = L/HCF(6, b)
As we know,
Number of meeting points on track = The first meeting at starting point again/TIme taken to meet for the first time
Number of meeting points on track = [L/HCF(6, b)]/[L/(6 - b)] = (6 - b)/HCF(6, b)
So, basically, we have to find the value of b as = (6 - b)/HCF(6, b) = 2
Hence, the only value less than 6 satisfies this equation is 2 i.e. only 1 value