Correct Option
When A reaches the point diametrically opposite to his starting position, A has covered an angular distance equivalent to half the circle, which is 180°. B was initially 30° ahead of A. Since A and B meet at the point diametrically opposite to A's starting position, B must have covered an angular distance that is 30° less than A's total travel. Therefore, the angular distance covered by B is 180° - 30° = 150°. As both A and B run for the same duration until they meet, the ratio of their speeds is directly proportional to the ratio of the distances they covered. The ratio of distances covered by A to B is 180° : 150°. Simplifying this ratio by dividing both sides by their greatest common divisor, 30: 180/30 : 150/30 = 6 : 5. Thus, the ratio of speeds of A and B is 6 : 5.
Incorrect Options
Options 2 (4:3), 3 (6:1), and 4 (4:2) are incorrect because they do not correspond to the calculated ratio of angular distances covered by A and B. The problem states A covers 180° and B, starting 30° ahead, covers 150° to meet A at the diametrically opposite point. The correct ratio of these distances, and consequently their speeds, is 180:150, which simplifies to 6:5. Any other ratio would imply different distances covered, which contradicts the conditions provided in the question.