Rates – Time, Speed and Race

In a 2000 m race, Gates gives Steve a headstart of 200 m and beats him by 30 sec. When Gates gives Steve a headstart of 3 mins he gets beaten by 1000 m. How long would Gates and Steve take to run the race individually?

Answer: Gates – 4 mins and Steve – 5 mins.

Let the speeds of Gates and Steve be ‘g’ and ‘s’ respectively.

In the first case, since Gates gives Steve a headstart of 200 m Steve has to run only 1800 m. It is also known that Gates finishes the race 30 sec before Steve.
Hence, the time that Steve takes to run 1800 m is the time that Gates takes to run 2000 m plus 30 sec (0.5 min).

1800/s = 2000/g + ½ – – – – – – – – – – – – – –  (1)

In the second case, Gates gives Steve a headstart of 3 mins and gets beaten by 1000 m.
Hence, while Steve runs 2000 m Gates runs only 1000 m. We also see that the time taken by Steve to run a distance of (2000 – 3s) is equal to the time taken by Gates to run a distance of 1000 m.

2000/s – 3 = 1000/g – – – – – – – – – – – – – – – (2)

Solving (1) and (2), we get

s = 400 m/min and g = 500 m/min.

So, to run a distance of 2000 m Steve and Gates would take 5 min and 4 min respectively.