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 i…

Question

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?

1. A. 35 (Correct answer)
2. B. 34
3. C. 31
4. D. 29

Correct Answer

Option A — 35

Detailed Solution & Explanation

The correct answer is Option A (35 meetings).

Key Points

• Speeds are in ratio X : Y = 7 : 5 (X does 7 rounds while Y does 5). Same-direction meetings occur on each unit of relative rounds.
• After Y's first 5 rounds it reverses and doubles its speed, so meetings then occur on the sum of rounds covered (opposite direction), which happens far more frequently.
• Counting the same-direction meetings in the first phase plus the head-on meetings in the second phase (until X finishes 21 rounds) totals 35.

Additional Information

• The key idea: same-direction meetings depend on the difference of rounds, head-on meetings on the sum; adding both phases gives Option A.