The correct option is (a) - 35.
[as per provisional answerkey]Solution
Step 1: Determine initial speeds
Let the length of the circular track be L.
X and Y run for the same duration. In the first phase, X completes 7 rounds while Y completes 5 rounds.
Ratio of speeds Vx:Vy=7:5.
Let Vx=7 units/sec and Vy=5 units/sec. Let the track length L=1 unit.
Step 2: Phase 1 (Same direction)
This phase ends when Y completes 5 rounds. Since Vx:Vy=7:5, when Y has done 5 rounds, X has done 7 rounds.
Relative speed in same direction = Vx−Vy=7−5=2 units/sec.
Number of meetings in same direction = (Relative distance covered) / (Track length).
Relative distance = (Vx−Vy)×Time. Since Vy×Time=5 rounds, Time=5/5=1 unit.
Relative distance = 2×1=2 units. This means X overlapped Y exactly 2 times.
However, they started at the same point. In "same direction" races, the start is usually not counted as a "meeting" unless specified. They meet at the end of this phase (at the start line) because 7 and 5 are integers.
Meetings in Phase 1: At relative distances 1 and 2. (Total 2 times).
Step 3: Phase 2 (Opposite direction)
X has 14 rounds left to complete (from round 7 to 21).
Y doubles his speed and reverses direction. New Vy=5×2=10 units/sec.
Relative speed in opposite direction = Vx+Vy=7+10=17 units/sec.
Time taken for X to finish remaining 14 rounds = Distance / Vx=14/7=2 units of time.
Relative distance covered in Phase 2 = (Relative Speed) × Time = 17×2=34 units.
In opposite direction movement, the number of meetings = Relative distance / Track length = 34/1=34 times.
Step 4: Total Meetings
Total meetings = Meetings in Phase 1 + Meetings in Phase 2.
Phase 1: 2 times (at the end of X's 3.5th and 7th round).
Phase 2: 34 times.
Wait, we must check the transition point. At the end of Phase 1, they are at the same spot (the start line). This is the 2nd meeting of Phase 1 and the 0th point of Phase 2. To avoid double counting:
Meetings = (Relative distance in Phase 1) + (Relative distance in Phase 2) - (overlap at transition if any).
Actually, the question asks for meetings "after they started" and "before they stopped".
Phase 1 relative distance = 2. Phase 2 relative distance = 34.
Total = 2+34=36.
Since the very last meeting occurs exactly when X completes the 21st round (the moment they stop), and the question says "before they finally stopped", we exclude the final meeting.
Total = 36−1=35.
Why the other options are incorrect
- Option (b) - 34: This result is obtained if one fails to account for the meetings during the first phase or incorrectly calculates the relative speed in the second phase.
- Option (c) - 31: This is a common error if the speed of Y is not doubled or if the relative speed calculation for opposite directions is handled as a subtraction instead of an addition.
- Option (d) - 29: This value does not correlate with the relative distances covered in either phase and likely stems from a miscalculation of the time duration for the second phase.
Key Concept
The number of meetings on a circular track is determined by the total relative distance covered divided by the length of the track, using (V1−V2) for the same direction and (V1+V2) for opposite directions.