Question

A man misses a train by 1 hour, if he travels at 8 km/h. If he increases his speed to 10 km/h he still misses his train by 24 minutes. At what minimum speed should he travel so that he catches the train?

Answer 1

$\underline{\underline{\Huge{\textbf{Solution:}}}}$

$\sf\textsf{Given,}\\ \underline{\textit{Statement-1:}}\\ \sf\textsf{ A man misses a train by 1 hour, if he} \\\sf\textsf{travels at 8 km/h}$

$\underline{\textit{Statement-2:}}\\ \sf\textsf{ If he increases his speed to 10 km/h,}\\\sf\textsf{he still misses his train by 24 minutes}$

$\sf\textsf{Minimum speed required to catch the}\\\sf\textsf{train on time = ? kmph }$

$\text{\textdagger} \: \sf\textsf{ It is clear from the given date that,} \\\sf\textbf{Minimum Speed Required > 10 kmph}$

$\textsf{We have,}$

$\boxed{\begin{minipage}{7 cm}\begin{aligned}\bigstar \: \bf Speed \: \: &= \: \: \tt \dfrac{Distance\: covered}{time\: taken} \\\\\bf \implies time\: taken\: \: &=\: \: \tt \dfrac{Distance\: covered}{Speed}\end{aligned}\end{minipage}}$

$\underline{\LARGE{\texttt{Step-1:}}}$

$\sf\textsf{Assume a variable for distance to the train}\\\sf\textsf{ and time taken to catch the train on Time}$

$\sf\textsf{Let distance to the train be \textbf{'x' km}}$

$\sf\textsf{Let the time taken by man to catch the train}\\\sf\textsf{on time be \textbf{'t' s}}$

$\underline{\LARGE{\texttt{Step-2:}}}$

$\sf\textsf{Rewrite Statement-1}$

$\sf\mathsf{t \: \: \: = \: \: \: \dfrac{x}{4}\underbrace{ - \: \: \: \: 1}_{missed \: by \: 1 \: Hr}}$

$\implies \mathbf{t+1 = \dfrac{x}{10}}$ ———[1]

$\underline{\LARGE{\texttt{Step-3:}}}$

$\sf\textsf{Express 24 minutes in Hours}$

$\boxed{\begin{minipage}{5 cm}\begin{aligned} \bigstar \: \; \bf 1 \: Hour \: &= \bf \: 60\: minutes\\\implies \: \sf 1 \: Minute \: &= \: \sf \dfrac{1}{60} Hr \end{aligned}\end{minipage}}$

$\implies \mathsf{24 minutes = 24 \times \dfrac{1}{60} Hour}$

$\therefore \mathbf{24 minutes = \dfrac{2}{5} Hour}$

$\underline{\LARGE{\texttt{Step-4:}}}$

$\sf\textsf{Rewrite Statement-2}$

$\sf\mathsf{t \: \: \: = \: \: \: \dfrac{x}{10}\underbrace{ - \: \: \: \: \frac{2}{5}}_{missed \: by \: \frac{2}{5} \: Hr(=\: 24 min)}}$

$\implies \bf \mathbf{t+\dfrac{2}{5} = \dfrac{x}{10}}$ ———[2]

$\underline{\LARGE{\texttt{Step-5:}}}$

$\sf\textsf{Do [2] - [1] to get value of \textbf{x}}$

$\begin{aligned}t+1 &= \dfrac{x}{8}\\ \\ t+\dfrac{2}{5}&= \dfrac{x}{10}\\ \underline{ - \quad - \quad } & \underline{\: \: \: \: - \qquad \quad } \\ 1 - \dfrac{2}{5}& =\dfrac{x}{8} - \dfrac{x}{10}\end{aligned}$

$\begin{aligned}\implies \mathsf{\dfrac{3}{5}} &= x\mathsf{\left(\dfrac{1}{8}-\dfrac{1}{10}\right)}\\ \\ \implies \mathsf{\dfrac{3}{5}} &= x\mathsf{\left(\dfrac{2}{80}\right)} \\ \\ \implies \mathsf{\dfrac{3}{5}}& = x\mathsf{\left(\dfrac{1}{40}\right)} \end{aligned}$

$\sf\textsf{Cross Multiply}$

$\implies \mathsf{5x = 120}$

$\implies \mathbf{x = 24 kmph}$

$\underline{\LARGE{\texttt{Step-6:}}}$

$\sf\textsf{Plug in 24 for \textbf{x} in [1] to get value of \textbf{t}}$

$\implies \mathsf{t+1 = \dfrac{24}{8}}$

$\implies \mathsf{t+1 = 3}$

$\therefore \mathbf{t = 2 Hrs}$

$\underline{\LARGE{\texttt{Step-7:}}}$

$\sf\textsf{Find the minimum speed with which}\\\sf\textsf{man has to travel to catch the train on time}$

$\boxed{\mathbf{\bigstar \: \; Required\: Speed = \dfrac{x}{t}\: kmph}}$

$\sf\textsf{Substitute Values}$

$\implies \textsf{Required Speed = \dfrac{24}{2} }$

$\implies \textsf{Required Speed = 12 kmph}$

$\blacksquare\: \; \sf\textsf{The minimum speed with which}\\\sf\textsf{man has to travel so that he can catch} \\ \sf \textsf{the train on time = \: }\Large{\underline{\Large{\textbf{12 kmph}}}}\quad\boxed{ \large{{ \cdot}_{ \smile}{\cdot}}}$

Answer 2

Answer:

Explanation:

Let the distance travelled be X km.

As per given conditions

X/4 -X/5 =3/5

On solving,

X= 12 km.

X/4 = 3 hrs

He has to reach station in 2 hrs .

He has to travel at the Speed of 6 kmph in order to reach in time exactly.