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# A can run $500\;m$ in $30\;\sec$ and B in $35\;\sec$ . How many meters can A give to B in a km race so that race may end in a dead heat?A. $139\dfrac{5}{7}$ B. $138\dfrac{5}{7}$ C. $142\dfrac{6}{7}$ D. $140\dfrac{5}{7}$

Last updated date: 09th Sep 2024
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Hint: Assume that A gives a start of $x$ meters to B in the race. As the race end in a dead heat it means they end the race at the same time. Equate their times and get the value of $x$ .

A can run $500\;m$ in $30\;\sec$ and B in $35\;\sec$ .
As per given that A can run $500\;m$ in $30\;\sec$ . The formula for the speed of a person is equal to the distance over time. So, the speed of A is equal to
$\dfrac{{500}}{{30}} = \dfrac{{50}}{3}\;m/s$ .
It is also given that B can run $500\;m$ in $35\;\sec$ . So, the speed of B is equal to $\dfrac{{500}}{{35}} = \dfrac{{100}}{7}\;m/s$ .
Now as given that the race is of $1\;km$ i.e. $1000\;m$ and as assumed that A gets a start of $x$ meters in the race. So, B needs to travel only $1000 - x$ meters of distance.
The speed of the B is equal to $\dfrac{{100}}{7}\;m/s$ and the distance is $1000 - x$ meters. The formula for the time is equal to distance over speed. So, the time taken by B in the race is equal to
$\dfrac{{1000 - x}}{{\dfrac{{100}}{7}}} = \dfrac{{7000 - 7x}}{{100}}\;s$ .
The speed of the A is equal to
$\dfrac{{50}}{3}\;m/s$ and the distance is $1000$ meters. The formula for the time is equal to distance over speed. So, the time taken by A in the race is equal to $\dfrac{{1000}}{{\dfrac{{50}}{3}}} = 60\;s$ .
As the race ended in dead heat i.e. both the players took equal time to complete the race. Now equate the time of both the players:
$60 = \dfrac{{7000 - 7x}}{{100}} \\ 6000 = 7000 - 7x \\ 7x = 1000 \\ x = \dfrac{{1000}}{7} \\ \Rightarrow x = 142\dfrac{6}{7}\;m \;$
So, the start which A gives to B is equal to $142\dfrac{6}{7}\;m$ .
So, the correct answer is “Option C”.

Note: It is given in the question that the race ends in dead heat that means that both the players took nearly equal time to complete the race. The formula for the speed is distance over time and the formula for the time is distance over speed.