"GRANDPA, please tell me a story." "Dear child, I am old, so, I don't remember any story now," "My friend’s Grandpa is old as well, then, how does he remember so many stories?" "All right, what story would you like to hear?" "Father says that you participated in a great race once. Tell me about it?" "Yes… they called it a great race of the great Olympics. Your grandpa wasn't an old man then. He was the best runner in the country, but no matter how good you are, there is always someone who is better than you. Your grandfather, too, had such a competitor, fierce as a lion and good as a saint. The final race had a photo finish and even the camera could not tell who had won between your Grandpa and his rival, so, it was announced that there would be a challenge race to decide the winner."

"One thing more, dear child. Your grandpa wasn't a rich man then, but so was his worthy rival. Your father was a little boy then and so sick that only an expensive treatment in a foreign land could save his life… and the government had announced a huge reward for the winner of the race. This money could save your father's life, but I had no energy left for another race and my rival knew it. I was like a cheetah, quick in a short dash, but low on stamina for a long race, while he was like a Siberian crane, fast and untiring. As tears washed my cheeks, he came to me with an offer that I could not refuse; he called it a challenge, where he would run three laps, while I ran two."

"He was only 150 yards into his third lap when I won. The victory meant that your father's life could now be saved. Spectators were hugging me, trying to lift me up, but I was searching for my rival who was nowhere to be seen. I met him in the evening at the awards ceremony and we stayed on in the stadium after everyone else had left. He, then, offered to run four laps to my three on the spot and I had to accept. He quickened his pace from the first race in the ratio of 4:3. I also quickened my pace in the ratio of 9:8, but in the second lap fell off to my original pace. In the third lap, I covered only 9 yards for the 10 I went in the first race. My rival won the race by 180 yards… and I thanked him for saving my son's life. It was the greatest race that no one ever saw." "It was a good story, Grandpa." "Is it so? Then, tell me how long was each lap?" "You didn't tell me that." "But, I told you the story." "So, tell me the lap length as well."

"Well, I don't remember, but let X be the distance around the track (in yards), A be the rival's speed in the first race and B be Grandpa's speed in the first race. Since both runners stopped running at the same time, the distance they travelled divided by their speed will be equal, generating the following two equations: First race: (2*x+150)/A=(2*x)/B; second race: (4*x)/((4/3)*A)=x/((9/8)*B)+x/B+(x-180)/((9/10)*B). Looking at the first-race equation, it is evident that if you can express Grandpa's speed in terms of a ratio to the rival's speed, the lap size will fall out of the equation. To do this, we must remove the lap distance X from the picture. We solve each equation for X in terms of A and B. On one hand, X equals -150/[A*{ (2/A)-(2/B)} ] and on the other hand, X equals -200/[B*{ (3/A)-(3/B)} ]. Remove X and set the two equations equal to each other: -150/[A*{ (2/A)-(2/B)} ]= -200/[B*{ (3/A)-(3/B)} ]. Solving for B, we get B=(8/9)*A. Substituting this into either of the initial equations, we get (2X+150)/A=2X/[(8/9)*A]. Solving for X, we get X=600 yards."