Two persons, A and B are running on a circular track. At the start, B is ahead of A and their positions make an angle of 30° at the centre of the circle. When A reaches the point diametrically opposite to his starting point, he meets B. What is the ratio of speeds of A and B, if they are running with uniform speeds?
- (a) 6 : 5 ✓ UPSC's answer
- (b) 4 : 3
- (c) 6 : 1
- (d) 4 : 2
Why the answer is (a)
• Let the circumference of the circular track be $C$. The distance corresponding to a central angle of $30^\circ$ is $\frac{30}{360}C = \frac{1}{12}C$.
• Since B is ahead of A by $30^\circ$, the distance B needs to cover to reach the point diametrically opposite to A's starting point is $\frac{C}{2} - \frac{C}{12} = \frac{6C - C}{12} = \frac{5C}{12}$.
• The distance A needs to cover to reach the point diametrically opposite to his starting point is exactly half the circumference, which is $\frac{C}{2} = \frac{6C}{12}$.
• Since they meet at this point at the same time, the ratio of their speeds is equal to the ratio of the distances covered by them.
• Therefore, the ratio of speed of A to speed of B is $\frac{6C/12}{5C/12} = \frac{6}{5}$, which is 6 : 5.
Why the other options are wrong
- (b) 4 : 3
- Option (b) 4:3 implies a distance ratio of 4:3, which does not match the calculated distance ratio of 6:5.
- (c) 6 : 1
- Option (c) 6:1 implies B covers negligible distance, but B actually covers 5/12 of the track.
- (d) 4 : 2
- Option (d) 4:2 simplifies to 2:1, which is incorrect as the actual ratio is 6:5.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.