Two runners, A and B, start running simultaneously from the same point

Question

Two runners, A and B, start running simultaneously from the same point on a circular track of length 400 meters. They run in the same direction. Runner A's speed is 5 meters per second, and Runner B's speed is 3 meters per second. How many seconds after they start will they meet again at the starting point for the first time?

A) 50 
B) 80 
C) 200 
D) 400 
E) 1200

Lap Time for Runner A:  Time for A (TA) = 400 meters / 5 m/s = 80 seconds. Runner A returns to the start every 80 seconds. 
  Lap Time for Runner B:  Time for B (TB) = 400 meters / 3 m/s = 400/3 seconds. Runner B returns to the start every 400/3 seconds.
Least Common Multiple (LCM) of Lap Times: They will meet again at the starting point at the LCM of their individual lap times. We need LCM(80, 400/3). 
  
 Use the formula: LCM(a/b, c/d) = LCM(a, c) / HCF(b, d)  . 
 LCM(80, 400) = 400 (since 400 is a multiple of 80). 
 HCF(1, 3) = 1. 
 LCM = 400 / 1 = 400 seconds.  
The correct answer is D) 400. 
  Verify 
 
 In 400 seconds, Runner A completes 400s / 80s/lap = 5 laps. 
 In 400 seconds, Runner B completes 400s / (400/3)s/lap = 400 * (3/400) = 3 laps. Since both have completed an integer number of laps, they are both back at the starting point after 400 seconds.

(Common Trap: Calculating the time until they first meet anywhere on the track uses relative speed (5-3=2 m/s). Time = 400m / 2m/s = 200 seconds. This is when A first laps B, but not necessarily at the starting line).

Krunaal · Answer

Speed (S) * Time(T) = Distance (D)

Runner A:   5 * T_A = 400  =>  T_A = 80  seconds / lap

Therefore Runner A will be at starting position every 80 seconds, eg. 80s, 160s, 240s, 320s,  400s , 480s, 560s, etc.

Runner B:   3 * T_B = 400  =>  T_B = \frac{400}{3}  seconds / lap

Therefore Runner B will be at starting position every 400/3 seconds, eg. 400/3s, 800/3s,  1200/3s = 400s , 1600/3s, 2000/3s etc.

From above, we can see that both runners will be at the starting position at 400s

Answer D.

heyRaj · Answer

My Approach:

Time Taken by A: (400/5) -> 80 seconds 
 Time Taken by B: 400/3 seconds

Now the time at which they will    meet at the     starting point again    will be LCM(80, 400/3) which is 400 (Option D)

Two runners, A and B, start running simultaneously from the same point

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