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: (b) 34

Let X's speed be v_x and Y's speed be v_y. When X completed 7 rounds, Y completed 5 rounds in the same time. Thus, v_x / v_y = 7 / 5. Let v_x = 7 units and v_y = 5 units per unit time. After X completed 5 rounds, Y changed direction and doubled his speed. Wait, the problem states: 'After completing 5 rounds, Y changed his direction...' Since Y completes 5 rounds when X completes 7 rounds, let's calculate time. Time taken for Y to complete 5 rounds is t1 = 5 / v_y. During this time, X ran 7 rounds. Then X continues to run at speed 7, and Y runs in the opposite direction with speed 2 x 5 = 10 units. X runs until he completes 21 rounds in total. Total rounds of X = 21. Time taken for X to complete 21 rounds in total: first 7 rounds took some time, remaining 14 rounds at speed 7 take 2 time units. Relative speed when moving in opposite directions = 7 + 10 = 17. By calculating meeting points using relative speeds and total time intervals, we find the total number of meetings is 34. Hence, option (b) is correct.