Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hours if he travels 160 km. by train and the rest by car. He takes 12 minutes more if the travels 240 km by train and the rest by car. Find the speed of the train and car respectively.

Given:

Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hours if he travels 160 km. by train and the rest by car. He takes 12 minutes more if the travels 240 km by train and the rest by car. To do: We have to find the speed of the train and car respectively.

Solution:

Total distance to the home $=760\ km$.

Let the speed of the train be $x$ km/hr and the speed of the car be $y$ km/hr.

We know that,

Time $=$ Speed $\div$ Distance

In the first case, he takes 8 hours if he travels 160 km. by train and the rest by car.

Time taken $=\frac{160}{x}+\frac{760-160}{y}$

$\Rightarrow \frac{160}{x}+\frac{600}{y}=8$.....(i)

In the second case, he takes 12 minutes more if the travels 240 km by train and the rest by car.

Time taken $=\frac{240}{x}+\frac{760-240}{y}$

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

$\Rightarrow \frac{240}{x}+\frac{520}{y}=8+\frac{1}{5}$

$\Rightarrow \frac{240}{x}+\frac{520}{y}=\frac{8\times5+1}{5}$

$\Rightarrow \frac{240}{x}+\frac{520}{y}=\frac{41}{5}$......(ii)

Multiplying equation (i) by 3 and equation (ii) by 2, we get,

$3(\frac{160}{x}+\frac{600}{y})=3(8)$

$\frac{480}{x}+\frac{1800}{y}=24$....(iii)

$2(\frac{240}{x}+\frac{520}{y})=2(\frac{41}{5})$

$\frac{480}{x}+\frac{1040}{y}=\frac{82}{5}$.....(iv)

Subtracting equation (iv) from (iii), we get,

$\frac{480}{x}+\frac{1800}{y}-\frac{480}{x}-\frac{1040}{y}=24-\frac{82}{5}$

$\frac{1800-1040}{y}=\frac{5(24)-82}{5}$

$\frac{760}{y}=\frac{38}{5}$

$y=\frac{760\times5}{38}$

$y=20\times5=100$

Substituting $y=100$ in equation (i), we get,

$\frac{160}{x}+\frac{600}{100}=8$

$\frac{160}{x}+6=8$

$\frac{160}{x}=8-6=2$

$x=\frac{160}{2}$

$x=80$

Therefore, the speed of the train is $80\ km/hr$ and the speed of the car is $100\ km/hr$ respectively.

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