Ankita travels $ 14 \mathrm{~km} $ to her home partly by rickshaw and partly by bus. She takes half an hour if she travels $ 2 \mathrm{~km} $ by rickshaw, and the remaining distance by bus. On the other hand, if she travels $ 4 \mathrm{~km} $ by rickshaw and the remaining distance by bus, she takes 9 minutes longer. Find the speed of the rickshaw and of the bus.

Given:

Ankita travels \( 14 \mathrm{~km} \) to her home partly by rickshaw and partly by bus. She takes half an hour if she travels \( 2 \mathrm{~km} \) by rickshaw, and the remaining distance by bus. On the other hand, if she travels \( 4 \mathrm{~km} \) by rickshaw and the remaining distance by bus, she takes 9 minutes longer.

To do:

We have to find the speed of the rickshaw and the bus.

Solution:

Total distance to the home $=14\ km$.

Let the speed of the rickshaw  be $x$ km/hr and the speed of the bus be $y$ km/hr.

We know that,

Time $=$ Distance $\div$ Speed

In the first case, she takes $\frac{1}{2}$ hour if she travels 2 km by rickshaw and the remaining by bus.

Time taken $=\frac{2}{x}+\frac{14-2}{y}$

$\Rightarrow \frac{2}{x}+\frac{12}{y}=\frac{1}{2}$.....(i)

In the second case, she takes 9 minutes more if she travels 4 km by rickshaw and the remaining by bus. 

Time taken $=\frac{4}{x}+\frac{14-4}{y}$

$\Rightarrow \frac{4}{x}+\frac{10}{y}=\frac{1}{2}+\frac{9}{60}$

$\Rightarrow \frac{4}{x}+\frac{10}{y}=\frac{30+9}{60}$

$\Rightarrow \frac{4}{x}+\frac{10}{y}=\frac{39}{60}$

$\Rightarrow \frac{4}{x}+\frac{10}{y}=\frac{13}{20}$......(ii)

Multiplying equation (i) by 2 and subtracting it from (ii), we get,

$2(\frac{2}{x}+\frac{12}{y}-\frac{1}{2})-(\frac{4}{x}+\frac{10}{y}-\frac{13}{20})=0$

$\frac{4}{x}-\frac{4}{x}+\frac{24}{y}-\frac{10}{y}=1-\frac{13}{20}$

$\frac{24-10}{y}=\frac{20-13}{20}$

$\frac{14}{y}=\frac{7}{20}$

$y=\frac{14\times20}{7}$

$y=2times20$

$y=40$

Substituting $y=40$ in equation (i), we get,

$\frac{2}{x}+\frac{12}{40}=\frac{1}{2}$

$\frac{2}{x}+\frac{3}{10}=\frac{1}{2}$

$\frac{2}{x}=\frac{1}{2}-\frac{3}{10}$

$\frac{2}{x}=\frac{5-3}{10}$

$\frac{2}{x}=\frac{2}{10}$

$x=\frac{10\times}{2}$

$x=10$

Therefore, the speed of the rickshaw is $10\ km/hr$ and the speed of the bus is $40\ km/hr$ respectively.