Question

A man travels $370km$ partly by train and partly by car. If he covers $250km$ by train and the rest by car, it takes him $4$ hours. But, if he travels $130km$ by train and the rest by car, he takes $18$ minutes longer. Find the speed of the train and that of the car.

Answer 1

Let the speed of the train be x km/hr and that of the car be y km/hr. We have following cases:

Case I When he travels $250$ km by train and the rest by car.
In this case, we have
Time taken by the man to travel $250$ km by train $=\frac{250}{x}$hrs
Time taken by the man to travel $(370-250)=120$km by car $=\frac{120}{y}$hrs

$\therefore$ Total time taken by the man to cover $370$km $=\frac{250}{x}+\frac{120}{y}$

It is given that the total time taken is $4$ hours

$\therefore \frac{250}{x}+\frac{120}{y}=4$

$\Rightarrow \frac{125}{x}+\frac{60}{y}=2$ (i)

Case II When he travels $130$ km by train and the rest by car:
In this case, we have
Time taken by the man to travel $130$km by train $=\frac{130}{x}$ hrs
Time taken by the man to travel $(370-130)=240$km by car $=\frac{240}{y}$hrs.

In this case, total time of the journey is $4$ hrs $18$ minutes.

$\therefore \frac{130}{x}+\frac{240}{y}=4$hrs $18$ minutes

$\Rightarrow \frac{130}{x}+\frac{240}{y}=4\frac{18}{60}$

$\Rightarrow \frac{130}{x}+\frac{240}{y}=\frac{43}{10}$ .(ii)

Thus, we obtain the following system of equations:

$\frac{125}{x}+\frac{60}{y}=2$

$\frac{130}{x}+\frac{240}{y}=\frac{43}{10}$

Putting $\frac{1}{x}=u$ and $\frac{1}{y}=v$, the given system reduces to

$125u+60v=2$ (iii)

$130u+240v=\frac{43}{10}$ (iv)

Multiplying equation (iii) by $4$ the given system of equations becomes

$500u+240v=8$ ..(v)

$130u+240v=\frac{43}{10}$ ..(vi)

Subtracting equation (vi) from equation (v), we get

$370u=8-\frac{43}{10}\Rightarrow 370u=\frac{37}{10}\Rightarrow u=\frac{1}{100}$

Putting $u=\frac{1}{100}$ in equation (v), we get

$5+240v=8\Rightarrow 240v=3\Rightarrow v=\frac{1}{80}$

Now, $u=\frac{1}{100}$ and $v=\frac{1}{80}$

$\Rightarrow \frac{1}{x}=\frac{1}{100}$ and $\frac{1}{y}=\frac{1}{80}$

$\Rightarrow x=100$ and $y=80$

Hence, Speed of the train $=100$ km/hr
Speed of the car $=80$ km/hr.