Ramesh travels 760 km to his home partly by train and partly by car.

Solution:

Let the speed of the train be x km/hour that of the car be y km/hr, we have the following cases

Case I: When Ramesh travels 760 Km by train and the rest by car

Time taken by Ramesh to travel $160 \mathrm{Km}$ by train $=\frac{160}{x} h r s$

Time taken by Ramesh to travel $(760-160)=600 \mathrm{Km}$ by car $=\frac{600}{y} h r s$

Total time taken by Ramesh to cover $760 \mathrm{Km}=\frac{160}{x}+\frac{600}{y}$

It is given that total time taken in 8 hours

$\frac{160}{x}+\frac{600}{y}=8$

$8\left(\frac{20}{x}+\frac{75}{y}\right)=8$

$\left(\frac{20}{x}+\frac{75}{y}\right)=\frac{8}{8}$

$\frac{20}{x}+\frac{75}{y}=1 \cdots(i)$

Case II: When Ramesh travels 240Km by train and the rest by car

Time taken by Ramesh to travel $240 \mathrm{Km}$ by train $=\frac{240}{x} \mathrm{hrs}$

Time taken by Ramesh to travel $(760-240)=520 \mathrm{Km}$ by car $=\frac{520}{y} h r s$

In this case total time of the journey is 8 hours 12 minutes

$\frac{240}{x}+\frac{520}{y}=8 h r s 12 \mathrm{~min}$ utes

$\frac{240}{x}+\frac{520}{y}=8 \frac{12}{60}$

$\frac{240}{x}+\frac{520}{y}=\frac{41}{5}$

$40\left(\frac{6}{x}+\frac{13}{y}\right)=\frac{41}{5}$

$\frac{6}{x}+\frac{13}{y}=\frac{41}{5} \times \frac{1}{40}$

$\frac{6}{x}+\frac{13}{y}=\frac{41}{200}$ $\ldots$ (ii)

Putting $\frac{1}{x}=u$ and, $\frac{1}{y}=u$, the equations $(i)$ and $(i i)$ reduces to

$20 u+75 v=1 \cdots($ iii $)$

$6 u+13 v=\frac{41}{200} \cdots(i v)$

Multiplying equation (iii) by 6 and (iv)by 20 the above system of equation becomes

$120 u+450 v=6 \cdots(v)$

$120 u+260 v=\frac{41}{10} \cdots(v i)$

Subtracting equation (vi) from (v) we get

$190 v=\frac{60-41}{10}$

$190 v=\frac{19}{10}$

$v=\frac{19}{10} \times \frac{1}{190}$

$v=\frac{1}{100}$

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

$120 u+\frac{45}{10}=6$

$120 u=6-\frac{45}{10}$

$120 u=\frac{60-45}{10}$

$120 u=\frac{15}{10}$

$u=\frac{15}{10} \times \frac{1}{120}$

$u=\frac{1}{80}$

Now

$u=\frac{1}{80}$

$\frac{1}{x}=\frac{1}{80}$

$x=80$

and

$v=\frac{1}{100}$

$\frac{1}{y}=\frac{1}{100}$

$y=100$

Hence, the speed of the train is $80 \mathrm{~km} / \mathrm{hr}$,

The speed of the car is $100 \mathrm{~km} / \mathrm{hr}$