In a race of one kilometre. A gives B a start of 100 metres and still wins by 20 seconds. But if A gives B a start of 25 seconds, B wins by 50 metres. The time taken by A to run kilometre is :

Answer: Option B

Let the time taken by A to run 1000 metres = x seconds
Time taken by B to sun 900 metres = x + 20 seconds
Speed of A and B :
$$=\frac{1000}{x},\frac{900}{x+20}$$   respectively
According to the question,
$$\frac{950}{\frac{1000}{x}}-\frac{1000}{\frac{900}{x+20}}=25$$
(As B is the winner now)
$$\begin{array}{rl}& \Rightarrow \frac{950x}{1000}-\frac{1000\left(x+20\right)}{900}=25\\ & \Rightarrow \frac{19x}{20}-\frac{10x+200}{9}=25\\ & \Rightarrow 171x-200x-4000=25\times 20\times 9\\ & \Rightarrow 29x=500\\ & \Rightarrow x=\frac{500}{29}\text{sec}\end{array}$$