Question

A man travels 600km apart by train and partly by car. It takes 8 hours and 40 minutes if he travels 320 km by train and rest by car. It would take 30 minutes more if he travels 200 km by train and the rest by the car/. Find the speed of the train and by car

Answer 1

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

Given,
Total distance travelled by man = $600\: km$

$\sf\textsf{Here, There are two Cases:}\\ \sf\textsf{Depending on the Distance}\\ \sf\textsf{(1) Travelled by Train}\\ \sf\textsf{(2) Travelled by Car}$

$\underline{\Large{\textsf{Step-1:}}}$
Note the distances travelled by train and by car

$\begin{tabular}{|c|c|c|}\cline{1-3}& Distance travelled by train& Distance travelled by Car\\\cline{1-3}&&\\Case-(1)&320 km & 600-320 = 280 km\\&&\\Case-(2)&200 km& 600-200 = 400 km\\&&\\\cline{1-3}\end{tabular}$

$\underline{\Large{\textsf{Step-2:}}}$
Assume a variable for speed of the train and car

$\sf\textsf{Let speed of the train be \textbf{x\: kmph}}$

$\sf\textsf{Let speed of the car be \textbf{y\: kmph}}$

$\underline{\Large{\textsf{Step-3:}}}$
Note the time taken by train and by car

$\sf\textsf{We have,}$
$\bigstar\: \: \boxed{\displaystyle\Large{\mathbf{Speed = \dfrac{Distance\: Covered}{time\: taken}}}}$

$\implies \sf\textsf{time taken = \dfrac{Distance\: covered}{Speed} }$

Substituting Values

$\begin{tabular}{|c|c|c|}\cline{1-3} &time taken by train& time taken by Car\\\cline{1-3}&&\\Case-(1)& \frac{320}{x} & \frac{280}{y} \\&&\\Case-(2)& \frac{200}{x} & \frac{400}{y}\\&&\\\cline{1-3}\end{tabular}$

$\underline{\Large{\textsf{Step-4:}}}$
Note the total time taken for both cases

$\begin{tabular}{|c|c|}\cline{1-2}&total time taken\\\cline{1-2}&&case-(1)&8 Hrs 40 min = \frac{26}{3} hrs\\&&case-(2)&8 Hrs 40 min + 30 min = \frac{55}{6} Hrs\\&&\cline{1-2}\end{tabular}$

$\sf\textsf{Here, time taken in expressed in Hrs}$

$\bigstar\: \: \boxed{\displaystyle\Large{\textbf{1\: Hour = 60\: Minutes}}}$

$\implies \sf\large{\textsf{1 min = \frac{1}{60} Hr}}$

$40 min= \frac{40}{60}\: Hr= \frac{2}{3}\: Hr$

$30 min= \frac{30}{60}\: Hr = \frac{1}{2} Hr$

$\implies \sf\textsf{8 Hrs 40 min=(8+ \frac{2}{3} ) Hrs = \frac{26}{3} Hrs }$

$\implies \sf\textsf{8 Hrs 40 min + 30 min= \frac{26}{3} Hrs + \frac{1}{2} Hr= \frac{55}{6} Hrs }$

$\underline{\Large{\textsf{Step-5:}}}$
Add the time taken by the train and by car to equate with the total time taken for journey

$\sf\frac{320}{x} + \frac{280}{y}=\frac{26}{3}$

$\sf\frac{200}{x} + \frac{400}{y}=\frac{55}{6}$

$\sf\textit{Solving the above Equations gives Speed of the train and the car}$

$\underline{\Large{\textsf{Step-6:}}}$
Substitute $\sf\textsf{a= \frac{1}{x} and b= \frac{1}{y} }$

$\sf\textsf{a = \frac{1}{x} }$ ————[1]

$\sf\textsf{b = \frac{1}{y} }$ ————[2]

$\sf\textsf{320a+280b= \frac{26}{3} }$ ————[3]

$\sf\textsf{200a+400b= \frac{55}{6} }$ ————[4]

$\underline{\Large{\textsf{Step-7:}}}$
Solving [3] for a

$\implies 320a= \frac{26}{3} - 280\, b$

$\implies a= \dfrac{\frac{26}{3}}{320} -\dfrac{280}{320}\: b$

$\implies \sf\textsf{a= \dfrac{13}{480} - \dfrac{7}{8} \: b}$ ————[6]

$\underline{\Large{\textsf{Step-8:}}}$
Solving [4] for b

$\implies 400b= \frac{55}{6} -200a$

$\implies b= \dfrac{\frac{55}{6}}{400} -\dfrac{200}{400}\: a$

$\implies \sf\textsf{b= \dfrac{11}{480} - \dfrac{1}{2} \: a}$ ————[7]

$\underline{\Large{\textsf{Step-9:}}}$
$\sf\textsf{Substitute [6] in [7] to get Value of b}$

$\implies b= \dfrac{11}{480}-\dfrac{1}{2}\left(\dfrac{13}{480}-\dfrac{7}{8}\, b\right)$

$\implies b=\dfrac{11}{480}-\dfrac{13}{960}+\dfrac{7}{16}\, b$

$\implies b= \dfrac{3}{320}+\dfrac{7}{16}\, b$

$\implies b-\dfrac{7}{16}\, b= \dfrac{3}{320}$

$\implies \dfrac{9}{16}\, b = \dfrac{3}{320}$

$\implies b=\dfrac{4}{3}\times \dfrac{1}{80}$

$\implies \sf\textsf{b= \dfrac{1}{60} }$

$\underline{\Large{\textsf{Step-10:}}}$
$\sf\textsf{plug in \frac{1}{60} for b into eq.[6] to get value of a}$

$\implies \sf\textsf{a= \dfrac{13}{480} - \left(\dfrac{7}{8}\times\dfrac{1}{60}\right) }$

$\implies \sf\textsf{a= \dfrac{13}{480} - \dfrac{7}{480} }$

$\implies \sf\textsf{a= \dfrac{6}{480} }$

$\implies \sf\textsf{a= \dfrac{1}{80} }$

$\underline{\Large{\textsf{Step-11:}}}$
$\sf\textsf{plug in values of a and b into [1] and [2] to get values of x and y}$

$\sf\textsf{x = \dfrac{1}{\frac{1}{80}} }$

$\implies \sf\textsf{x = 80\: kmph}$

$\sf\textsf{y = \frac{1}{\frac{1}{60}} }$

$\implies \sf\textsf{y = 60\: kmph}$

$\therefore$
$\blacksquare\: \: \textsf{Speed of the train = \underline{\Large{\textbf{80\: kmph}}}}$

$\blacksquare\: \: \textsf{Speed of the Car = \underline{\Large{\textbf{60\: kmph}}}}$