Question

Please do it fast...
It's urgent
A person travelled at a speed of 50kmph and missed the bus by 40 minutes.Had he travelled at a speed of 60 kmph he would have still missed the bus by 20 min.At what minimum speed should he travel to catch the bus.

Answer 1

$\textsf{Let the Distance between the Person's House and the Bus Stop be : D km}$

$\textsf{Given : Person traveled at a Speed of 50 kmph and Missed the Bus by 40 Min}$

$\bigstar\;\;\textsf{We know that : \boxed{\mathsf{Time = \dfrac{Distance}{Speed}}}}$

$\implies \mathsf{Time\;taken\;by\;person\;to\;reach\;Bus\;Stop\;with\;Speed\;50\;kmph : \dfrac{D}{50}\;hour}$

$\textsf{Given : Person traveled at a Speed of 60 kmph and Missed the Bus by 20 Min}$

$\implies \mathsf{Time\;taken\;by\;person\;to\;reach\;Bus\;Stop\;with\;Speed\;60\;kmph : \dfrac{D}{60}\;hour}$

$\textsf{In order to solve the Problem, We need to realize that : The Difference}\\\textsf{in the Time taken by the person while traveling with Speeds 50 kmph and}\\\textsf{60 kmph should be equal to Difference in Time by which the person missed}\\\textsf{the bus while traveling with Speeds 50 kmph and 60 kmph}$

$\implies \mathsf{\bigg(\dfrac{D}{50} - \dfrac{D}{60}\bigg)\;hour = (40 - 20)\;minutes}$

$\implies \mathsf{\bigg(\dfrac{D}{50} - \dfrac{D}{60}\bigg)\;hour = (20)\;minutes}$

$\implies \mathsf{\bigg(\dfrac{D}{50} - \dfrac{D}{60}\bigg)\;hour = \bigg(\dfrac{1}{3} \bigg)\;hour}$

$\implies \mathsf{\bigg(\dfrac{60D - 50D}{3000} \bigg) = \bigg(\dfrac{1}{3} \bigg)}$

$\implies \mathsf{\bigg(\dfrac{60D - 50D}{1000} \bigg) = 1}$

$\implies \mathsf{(60D - 50D) = 1000}$

$\implies \mathsf{10D = 1000}$

$\implies \mathsf{D = 100\;km}$

$\implies \mathsf{Time\;taken\;by\;person\;to\;reach\;Bus\;Stop\;with\;Speed\;50\;kmph : \dfrac{100}{50}\;hour}$

$\implies \mathsf{Time\;taken\;by\;person\;to\;reach\;Bus\;Stop\;with\;Speed\;50\;kmph : 2\;hours}$

$\implies \mathsf{Time\;taken\;by\;person\;to\;reach\;Bus\;Stop\;with\;Speed\;50\;kmph : 120\;mins}$

$\textsf{Given : Person traveled at a Speed of 50 kmph and Missed the Bus by 40 Min}$

$\textsf{It means : If the Person would have reached the Bus Stop in (120 - 40)}\\\textsf{Minutes, He would be at the Bus Stop at Right time to catch the Bus.}$

$\textsf{It means : If the Person would have reached the Bus Stop in 80 Minutes,}\\\textsf{He would be at the Bus Stop at Right time to catch the Bus.}$

$\mathsf{We\;know\;that,\;80\;Minutes\;is : \bigg(1 + \dfrac{1}{3}\bigg)\;hour = \dfrac{4}{3}\;hour}$

$\implies \mathsf{Minimum\;Speed\;to\;catch\;the\;Bus = \dfrac{100}{\frac{4}{3}}\;kmph}$

$\implies \mathsf{Minimum\;Speed\;to\;catch\;the\;Bus = \bigg(100 \times \dfrac{3}{4}\bigg)\;kmph}$

$\implies \mathsf{Minimum\;Speed\;to\;catch\;the\;Bus = (25 \times 3)\;kmph}$

$\implies \mathsf{Minimum\;Speed\;to\;catch\;the\;Bus = 75\;kmph}$