X and Y are two runners who run for the same duration of time on the same circular track. They started running at the same time in the same direction with uniform speeds. When X completed 7 rounds, Y did exactly 5. After completing 5 rounds, Y changed his direction and started running in the opposite direction with speed which is double of his earlier speed. On the other hand, X continued to run with the same speed. They stopped running when X completed exactly 21 rounds. How many times did X and Y meet after they had started and before they finally stopped?

Correct Option: A (35)

• Phase 1 (Same direction): Let the time taken for X to complete 7 rounds and Y to complete 5 rounds be T. Their uniform speeds are proportional to the distance covered: V_x = 7/T and V_y = 5/T rounds per unit time. When running in the same direction, their relative speed is V_x - V_y = 2/T. The number of times they meet during this phase is Relative Speed × Time = (2/T) × T = 2 times (meeting exactly at T/2 and T).
• Phase 2 (Opposite directions): Y reverses direction and doubles his speed, making V_y′ = 10/T. X continues at V_x = 7/T. Because they now run in opposite directions, the new relative speed is V_x + V_y′ = 17/T.
• X stops when completing exactly 21 rounds. Having already completed 7 rounds, X must run 14 more. At a speed of 7/T, this takes 14 / (7/T) = 2T time.
• The number of meetings in Phase 2 is Relative Speed × Time = (17/T) × 2T = 34 times.
• Total Meetings: The mathematical total of meetings is 2 + 34 = 36. However, exactly when they stop, X completes 21 rounds (returning to the start point) and Y completes 5 - 20 = -15 rounds (also exactly at the start point). Thus, their 36th meeting happens exactly at the moment they stop. Because the question explicitly specifies calculating meetings "before they finally stopped," we must strictly exclude this final instance. The correct count is 36 - 1 = 35.

Why Wrong Options are Incorrect:

• Option B (34): Incorrectly accounts only for the 34 meetings that occurred in Phase 2, completely missing the Phase 1 meetings.
• Option C (31): Incorrect distractor, likely chosen if one miscalculates the remaining distance or incorrectly proportions Y's altered speed.
• Option D (29): Incorrectly reached by making arithmetic errors in relative speed or misunderstanding the number of rounds left for X.

Concluding Takeaway: When solving circular track problems, find the relative speed for each distinct phase (subtract for the same direction; add for opposite directions). Always note phrasing constraints like "before they stopped" to determine if final boundary meetings should be excluded.