In a 500 metres race, B starts 45 metres ahead of A, but A wins the race while B is still 35 metres behind. What is the ratio of the speeds of A to B assuming that both start at the same time ?

(a) 25 : 21 ✓ UPSC's answer
(b) 25 : 20
(c) 5 : 3
(d) 5 : 7

Why the answer is (a)

• In a 500 m race, A starts at the 0 m mark and must cover the full 500 m to win.

• B starts 45 m ahead, so B's starting position is at 45 m, meaning B only needs to cover 500 - 45 = 455 m to reach the finish line.

• The problem states A wins while B is still 35 m behind the finish line, so B has covered 455 - 35 = 420 m when A finishes.

• Since both start at the same time and A finishes at the same moment B is 35 m short, the time taken by both is equal.

• The ratio of their speeds is therefore equal to the ratio of the distances covered in that same time: Speed of A : Speed of B = 500 : 420.

• Simplifying the ratio 500 : 420 by dividing both terms by 20 gives 25 : 21, which matches option (a).

Why the other options are wrong

(b) 25 : 20: Option (b) 25:20 implies B covered 400 m, which would mean B was 55 m behind the finish line, not 35 m.
(c) 5 : 3: Option (c) 5:3 implies a distance ratio of 500:300, meaning B covered only 300 m, which is inconsistent with B starting 45 m ahead and being 35 m behind.
(d) 5 : 7: Option (d) 5:7 implies B is faster than A, which contradicts the fact that A wins the race.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.