$X$ 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?
Detailed Explanation:
Phase 1 (Same direction): When X completes 7 rounds, Y completes 5 rounds in time T. Speed ratio is 7:5. Relative speed = 7 − 5 = 2 rounds/T. In time T, they meet 2 times (at t = 0.5T and t = T).
Phase 2 (Opposite directions): After Y completes 5 rounds, he reverses and doubles his speed to 10 rounds/T. X continues at 7 rounds/T. Relative speed = 7 + 10 = 17 rounds/T. X runs 21 − 7 = 14 more rounds in time 2T. Meetings in Phase 2 = 17 × 2 = 34 times.
Therefore, total meetings after start and before final stop = 2 + 33 = 35 (excluding the meeting at final stop), making Option 1 the correct answer.
Question 3 of 5 Time, Speed & Distance
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