Two friends A and B run around a circular track ... - with answer

Two friends A and B run around a circular track of length 510 meters, starting from the same point, simultaneously and in the same direction. A who runs faster laps B in the middle of the 5th round. If A and B were to run a 3 km race long race, how much start, in terms of distance, should A give B so that they finish the race in a dead heat?

Answer:  666.67 metres.

A and B run around a circular track. A laps B in the middle of the $5^{th}$ lap. i.e. when A has run four and a half laps he has covered a distance which is 1 lap greater than that covered by B's.

Therefore, when A runs $\frac{9}{2}$ laps, B runs $\frac{7}{2}$ laps.

This is same as saying when A runs 9 laps, B runs 7 laps.

i.e in a race that is 9 laps long, A can give B a start of 2 laps.

So, if the race is of 3000 metres long, then A can give B a start of  $\left(\frac{2}{9}\right)\times 3000=$ 666.67 metres.