Question

Ankita travels $14$ 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. She takes $9\min .$ more to travel if she travel $2\mathrm{km}$ by rickshaw.

Answer 1

Step 1: Determine the required equation to further find the speed of the rickshaw and partly by bus

Ankita covers total distance$14\mathrm{km}$ (home partly by rickshaw and partly by bus)

She takes $\frac{1}{2}\mathrm{h}$ when she travels $2\mathrm{km}$ by rickshaw and the remaining distance by bus.

$\begin{array}{rcl}\frac{1}{2}& =& \frac{2}{x}+\frac{14-2}{y}\\ & \Rightarrow & \frac{1}{2}=\frac{2}{x}+\frac{12}{y}...\left(i\right)\end{array}$

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

Since,$\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}}$

Thus, the above statement can be expressed as follow:

$\begin{array}{rcl}\mathrm{Total}\mathrm{Time}& =& \mathrm{Time}\mathrm{taken}\mathrm{by}\mathrm{rickshaw}+\mathrm{Time}\mathrm{taken}\mathrm{bus}\\ \frac{1}{2}+\frac{9}{60}& =& \frac{4}{\mathrm{x}}+\frac{\left(14-4\right)}{\mathrm{y}}\\ & & \Rightarrow \frac{39}{60}=\frac{4}{\mathrm{x}}+\frac{10}{\mathrm{y}}...\left(\mathrm{ii}\right)\end{array}$

Also, she takes $9\min .$ more to travel if she travel $4\mathrm{km}$ by rickshaw.

Step 2: Find the speed of the bus.

Multiply the equation $\left(\mathrm{i}\right)$ by $2$:

$\begin{array}{rcl}1& =& \frac{4}{\mathrm{x}}+\frac{24}{\mathrm{y}}...\left(iii\right)\end{array}$

Subtract the equation $\left(\mathrm{iii}\right)$ from equation $\left(\mathrm{ii}\right)$ as follow:

$$\begin{gathered} \begin{array}{rcl}& & \frac{39}{60}\end{array}-1\begin{array}{rcl}& =& \end{array}\frac{4}{x}+\frac{10}{y}-\frac{4}{x}-\frac{24}{y} \\ \Rightarrow \frac{14}{y}=\frac{21}{60} \\ \Rightarrow y=\frac{60\times 14}{21} \\ \therefore y=40\mathrm{km}/\mathrm{hr} \end{gathered}$$

Step 3: Find the speed of the rickshaw.

Substitute the value of $\mathrm{y}=40\mathrm{km}/\mathrm{hr}$ in $\mathrm{eq}.\left(i\right)$:

$\begin{array}{rcl}\frac{1}{2}& =& \frac{2}{\mathrm{x}}+\frac{12}{\mathrm{y}}\\ & \Rightarrow & \frac{1}{2}=\frac{2}{\mathrm{x}}+\frac{12}{40}\\ & \Rightarrow & \frac{1}{2}=\frac{2}{\mathrm{x}}+\frac{3}{10}\\ & \Rightarrow & \frac{1}{2}-\frac{3}{10}=\frac{2}{\mathrm{x}}\\ & \Rightarrow & \frac{2}{10}=\frac{2}{\mathrm{x}}\\ \therefore x& =& 10\mathrm{km}/\mathrm{hr}\end{array}$

Hence, the speed of the rickshaw is $10\mathrm{km}/\mathrm{hr}$ and the speed of the bus is $40\mathrm{km}/\mathrm{hr}$.