In a 1000 m race, Arjun, Balaji and Charan are running. Arjun beats Balaji by 100m, and Balaji beats Charan by 100 m. In the next 1000 m race (the speeds are the same as in the previous), Balaji gives Charan a head start of 100 m, and Arjun gives Balaji a head start of 100m. Find the distance by which the winner is ahead of the person just behind him.

20 

In a 1000 m race, Arjun beats Balaji by 100m

ratio of                   Arjun     :     Balaji

Distance                 1000      :     900

Speed                       10       :       9

In a 1000 m race, Balaji beats Charan by 100 m

ratio of                   Balaji     :     Charan

Distance                 1000      :     900

Speed                       10       :       9

Overall,  Speed of Arjun  :  Balaji  : Charan = 100  :  90  :  81

In the next race,

Distance covered by Arjun = 1000 m

Time taken by Arjun to cover 1000m = \(\frac{1000}{100R}\) = \(\frac{10}{R}\)

Distance covered by Balaji = 900 m

Time taken by Balaji to cover 900m = \(\frac{900}{90R}\) = \(\frac{10}{R}\)

Distance covered by Charan = 800 m

Time taken by Charan to cover 800m = \(\frac{800}{90R}\) = \(\frac{9.8}{R}\)

Charan takes least time and winner of race.

Distance covered by Arjun in \(\frac{9.8}{R}\) sec = 100R x \(\frac{9.8}{R}\)

= 980 m

Distance covered by Balaji in \(\frac{9.8}{R}\) sec = 90R x \(\frac{9.8}{R}\)

= 880 m

Now, the distance by which the winner is ahead of the person just behind him

= 1000 - 980

= 20 m