Question : In a 1000 m race, Arjun, Balaji, and Charan are running. Arjun beats Balaji by 100 m, 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 100 m. Find the distance by which the winner is ahead of the person just behind him.

Option 1: 100 m

Option 2: 40 m

Option 3: 30 m

Option 4: 20 m

Correct Answer: 20 m

Solution :

Let us denote Arjun by A, Balaji by B, and Charan by C.
In a 1000 m race, Arjun beats Balaji by 100 m.
Hence, while Arjun covers 1000 m, Bajali covers 900 m.
So, $\small \text{Speed of Arjun}:\small \text{Speed of Balaji} = 1000 : 900 = 10 : 9 = 100 : 90$
In a 1000 m race, Balaji beats Charan by 100 m.
Hence, while Balaji covers 1000 m, Charan covers 900 m.
So, $\small \text{Speed of Balaji}: \small \text{Speed of Charan}= 1000 : 900 = 10 : 9 = 90 : 81$
Hence, $\small \text{Speed of Arjun} :\small \text{Speed of Balaji}: \small \text{Speed of Charan} = 100 : 90 : 81$
Let the speed of Arjun, Balaji, and Charan be $100s, 90s, \text{and}\ 81s$ respectively.
In the next race,
Arjun has to cover = 1000 m
Balaji has to cover = 1000 – 100 = 900 cm
Charan has to cover = 900 – 100 m = 800 m
Also,
Time taken by Arjun to reach the finish line = $\frac{1000}{100s} = \frac{10}{s}$
Time taken by Balaji to reach the finish line = $\frac{900}{90s} = \frac{10}{s}$
Time taken by Charan to reach the finish line = $\frac{800}{81s}$
It is visible that Charan will be the winner.
By that time,
Arjun will cover = $100s \times \frac{800}{81s} ≈ 980\ \text{m}$
Balaji will cover = $90s \times \frac{800}{81s} ≈ 880\ \text{m}$
Now, the distance by which the winner is ahead of the person just behind him = $1000 - 980 = 20\ \text{m}$
Hence, the correct answer is 20 m.