Atul gives Vishu a head-start of 20 seconds in a 900 m race and beats him by 135
m. While running the same race again Atul gives a start of 189 m and beats him by 8 seconds. In how much time can Vishu complete the full race of 900 m? 

Let Atul's speed = A m/s and Vishu's speed = V m/s From the first race: Vishu gets 20 seconds head start When Atul finishes 900m, Vishu has covered only 765m (beaten by 135m) Time for Atul to finish = 900/A seconds Total time Vishu ran = 20 + 900/A seconds Distance covered by Vishu = V × (20 + 900/A) = 765 So: 20V + 900V/A = 765 ... (1) From the second race: Vishu gets 189m head start (so needs to cover only 711m) Atul beats Vishu by 8 seconds Time for Atul to finish 900m = 900/A seconds Time for Vishu to cover 711m = 900/A + 8 seconds So: 711/V = 900/A + 8 ... (2) From equation (2): 711A = 900V + 8AV Therefore: A = 900V/(711 - 8V) ... (3) Substituting (3) into (1): 20V + 900V × (711 - 8V)/(900V) = 765 20V + (711 - 8V) = 765 20V + 711 - 8V = 765 12V = 54 V = 4.5 m/s Time for Vishu to complete 900m = 900/4.5 = 200 seconds or 3 minutes 20 seconds.