Question

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

Accepted Answer

Let the speed of the rickshaw and the bus are $x$ and $y$ km/h, respectively.

Now, she has takes time to travel $2$ km by rickshaw, $t_{1} = \frac{2}{x} h$ . $[∵ s p e e d = \frac{d i s \tan c e}{t i m e}]$

and she has taken time to travel remaining distance i.e., $(14 - 2) = 12 k m$ by bus $= t_{2} = \frac{12}{y} h$ .

By first condition, $t_{1} + t_{2} = \frac{1}{2} \Rightarrow \frac{2}{x} + \frac{12}{y} = \frac{1}{2}$ $. . . (i)$

Now, she has taken time to travel $4$ km by rickshaw, $t_{3} = \frac{4}{x} h$

and she has taken time to travel remaining distance i.e., $(14 - 4) = 10 k m$ by bus $= t_{4} = \frac{10}{y} h$

By second condition, $t_{3} + t_{4} = \frac{1}{2} + \frac{9}{60} = \frac{1}{2} + \frac{3}{20}$

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

Let $\frac{1}{x} = u$ and $\frac{1}{y} = v$ , then Eqs. $(i)$ and $(i i)$ becomes

$2 u + 12 v = \frac{1}{2}$ $. . . (i i i)$

and $4 u + 10 v = \frac{13}{20}$ $. . . (i v)$

On multiplying in Eq. $(i i i)$ by $2$ and then subtracting, we get

$\begin{array}{l} \underline{\begin{array}{l} 4 u + 24 v = 1 \\ \underline{4 u} + \underline{10 v} = \underline{\frac{13}{20}} \end{array}} \\ 14 v = 1 - \frac{13}{20} = \frac{7}{20} \end{array}$

$\Rightarrow 2 v = \frac{1}{20} \Rightarrow v = \frac{1}{40}$

Now, put the value of $v$ in Eq. $(i i i)$ , we get

$2 u + 12 (\frac{1}{40}) = \frac{1}{2}$

$\Rightarrow 2 u = \frac{1}{2} - \frac{3}{10} = \frac{5 - 3}{10}$

$\Rightarrow 2 u = \frac{2}{10} \Rightarrow u = \frac{1}{10}$

$∴ \frac{1}{x} = u$

$\Rightarrow \frac{1}{x} = \frac{1}{10} \Rightarrow x = 10 k m / h$

and $\frac{1}{y} = v \Rightarrow \frac{1}{y} = \frac{1}{40}$

$\Rightarrow y = 40 k m / h$

Hence, the speed of rickshaw and the bus are $10$ km/h and $40$ km/h, respectively.

Neha Tyagi and Amit Rastogi Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 4: Exercise

Neha Tyagi Mathematics Solutions for Exercise - Neha Tyagi and Amit Rastogi Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 4: Exercise

Questions from Neha Tyagi and Amit Rastogi Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 4: Exercise with Hints & Solutions

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