The race is of one kilometre, that is 1000 metres.
Tarun starts running 20 seconds earlier than Satish. Suppose Tarun covers a distance of "x" metres during this 20 seconds. At this point, Satish starts running and covers the 1000 metres in say "t" seconds. He beats Tarun by 40 metres. Therefore Tarun has covered a distance of 960 metres at this point of time. For this, he has taken 20 seconds plus "t" seconds. Thus, he has covered 960 metres in t+20 seconds.
Now, speed = distance / time. Therefore, Tarun's speed is S1 = 960/(t+20). Satish's speed is S2 = 1000 / t.
If Satish had given Tarun a 25-second start, both of them would have finished the race at the same instant. Okay, we have Tarun starting off first and covering some distance "y" in 25 seconds. Satish starts now and covers the distance in "t" seconds. Tarun also finishes the race at the same instant. But he has taken 25+t seconds. In this case, we have Satish's speed S2 = 1000 / t and Tarun's speed S1 = 1000 / (t+25).
Tarun's speed is S1 = 960 / (t +20) and S1 = 1000 / (t + 25).
So, 960 / (t + 20) = 1000 / (t + 25).
Thus, 960 (t + 25) = 1000 (t + 20).
Therefore 960t + 24000 = 1000t + 20000.
So, 40t = 4000. Therefore t = 100 seconds.
Substituting the value for "t" in any of the equations for S1 and S2, we get the speeds of the brothers.
S1 = 960 / (t + 20). So, S1 = 960 / 120. Therefore, Tarun's speed is S1 = 8 metres/second.
S2 = 1000 / t. So, S2 = 1000 / 100. Therefore, Satish's speed is S2 = 10 metres/second.
A much shorter method would have been to consider Tarun's speed. When Satish gives him a 20-second start, Tarun loses by 40 metres. But when Satish gives him a 25-second start, both finish at the same instant. This means that Tarun requires 5 seconds to cover the distance of 40 metres. So, his speed is 40/5, that is, 8 metres/second.
Tarun takes 5 seconds to run a distance of 40 metres. Therefore to run a distance of 1000 metres, he will take: 1000 x 5 / 40 seconds, or 125 seconds. Both of them finish at the same instant when Satish gives Tarun a 25-second start. This means that Satish requires (125 - 25) seconds to run the distance of 1000 metres. So, he requires 100 seconds to run 1000 metres, which means his speed is 1000 / 100 or 10 metres/second.