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# Abdul travelled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes. But it he travels 260 km by train and 240 km by taxi he takes 6 minutes longer. Find the speed of the train and that of the taxi.

Given:

Abdul travelled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes. But it he travels 260 km by train and 240 km by taxi he takes 6 minutes longer.

To do:

We have to find the speed of the train and that of the taxi.

Solution:

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

We know that,

Time $=$ Speed $\div$ Distance

In the first case, Abdul travelled 300 km by train and 200 km by taxi, it took him 5 hours 30 minutes.

Time taken $=\frac{300}{x}+\frac{200}{y}$

$\Rightarrow \frac{300}{x}+\frac{200}{y}=5+\frac{30}{60}$

$\Rightarrow \frac{300}{x}+\frac{200}{y}=5+\frac{1}{2}$

$\Rightarrow \frac{300}{x}+\frac{200}{y}=\frac{2\times5+1}{2}$

$\Rightarrow \frac{300}{x}+\frac{200}{y}=\frac{11}{2}$.....(i)

In the second case, he travels 260 km by train and 240 km by taxi he takes 6 minutes longer.

Time taken $=$ 5 hours 30 minutes $+$ 6 minutes

$=5\ hours\ 36\ minutes$

$=5+\frac{36}{60}$

$=5+\frac{3}{5}$

$=\frac{5\times5+3}{5}$

$=\frac{28}{5}$ hours

Time taken $=\frac{260}{x}+\frac{240}{y}$

$\Rightarrow \frac{260}{x}+\frac{240}{y}=\frac{28}{5}$......(ii)

Multiplying equation (i) by 6, we get,

$6(\frac{300}{x}+\frac{200}{y})=6(\frac{11}{2})$

$\frac{1800}{x}+\frac{1200}{y}=33$....(iii)

Multiplying equation (i) by 5, we get,

$5(\frac{260}{x}+\frac{240}{y})=5(\frac{28}{5})$

$\frac{1300}{x}+\frac{1200}{y}=28$....(iv)

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

$\frac{1800}{x}+\frac{1200}{y}-\frac{1300}{x}-\frac{1200}{y}=33-28$

$\frac{1800-1300}{x}=5$

$\frac{500}{x}=5$

$x=\frac{500}{5}$

$x=100$

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

$\frac{300}{100}+\frac{200}{y}=\frac{11}{2}$

$3+\frac{200}{y}=\frac{11}{2}$

$\frac{200}{y}=\frac{11}{2}-5$

$\frac{200}{y}=\frac{11-2\times5}{2}$

$\frac{200}{y}=\frac{1}{2}$

$y=2\times200$

$y=400$

Therefore, the speed of the train is $100\ km/hr$ and the speed of the taxi is $400\ km/hr$ respectively.