Time, Speed & Distance

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Two friends $A$ and $B$ run around a circular track of length 510 metres, starting from the same point, simultaneously and in the same direction. $A$ who runs faster laps $B$ in the middle of the $5^{th}$ 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? 


545.45 metres


666.67 metres


857.14 metres


Cannot be determined

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Option(B) is correct

$A$ 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  \(\dfrac{9}{2}\) laps, $B$ runs \(\dfrac{7}{2}\) 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 

\(\left(\dfrac{2}{9}\right)\times 3000=\)666.67 metres.

The information with regard to the length of the circular track is redundant information.

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