P and Q run along a circular track in 4 minutes and 7 minutes respectively. They start from the same point at the same time. When Q has completed 12 rounds, how many times will they meet if they run in, (i) Same direction, (ii) Opposite directions?

Correct option is C

P completed in 4 minutes
Q completed in 7 minutes
LCM of 4 and 7 = 28 m
Speed of P = 28/4 = 7 m/min
Speed of Q = 28/7 = 4 m/min
Condition (i) when they running same direction
Relative speed = 7-4 = 3
So, they meet first time = 28/3 min.
Time taken to complete 12 round by Q = 7×12=84 min
Therefore, no of time they meet = 84/(28/3)=9 times
Condition (ii) when they running opposite direction
Relative speed = 7+4 = 11 m/min
They meet 1st time = 28/11 min
So, no of time they meet = 84/(28/11)=33 times
Therefore the correct option is (c).