Question

Roohi travels

300 km

to her home partly by train and partly by bus. She takes

4

hours if she travels

60 km

by train and the remaining by bus. If she travels

100 km

by train and the remaining by bus, she takes

10

minutes longer. Find the speed of the train and the bus separately.

Accepted Answer

Let the speed of the train be $x km / hr$ that of the bus be $y km / hr,$ we have the following cases.
Case I: When Roohi travels $60 km$ by train and the rest by bus
Time taken by Roohi to travel $60 km$ by train $= \frac{60}{x} hrs$
Time taken by Roohi to travel $(300 - 60) = 240 km$ by bus $= \frac{240}{y} hrs$
Total time taken by Roohi to cover $300 km$ $= \frac{60}{x} + \frac{240}{y}$
It is given that total time taken in $4$ hours,
$\frac{60}{x} + \frac{240}{y} = 4$
$60 (\frac{1}{x} + \frac{4}{y}) = 4$
$(\frac{1}{x} + \frac{4}{y}) = \frac{4}{60}$
$\frac{1}{x} + \frac{4}{y} = \frac{1}{5} \dots (i)$

Case II: When Roohi travels $100 km$ by train and the rest by bus
Time taken by Roohi to travel $100 km$ by train $= \frac{100}{x} hrs$
Time taken by Roohi to travel $(300 - 100)$ = $200 km$ by bus $= \frac{200}{y} hrs$
In this case total time of the journey is $4$ hours $10$ minutes,
$\frac{100}{x} + \frac{200}{y} = 4 hrs 10 minutes$
$\frac{100}{x} + \frac{200}{y} = 4 \frac{10}{60}$
$\frac{100}{x} + \frac{200}{y} = \frac{25}{6}$
$100 (\frac{1}{x} + \frac{2}{y}) = \frac{25}{6}$
$\frac{1}{x} + \frac{2}{y} = \frac{25}{6} \times \frac{1}{100}$
$\frac{1}{x} + \frac{2}{y} = \frac{1}{24} \dots (i i)$
Putting $\frac{1}{x} = u$ and $\frac{1}{y} = v$ the equations $(i)$ and $(i i)$ reduces to,
$1 u + 4 v = \frac{1}{15} \dots (i i i)$
$1 u + 2 v = \frac{1}{24} \dots (i v)$
Subtracting equation $(i v)$ from $(i i i)$ we get,
$1 u + 4 v = \frac{1}{15}$
$- 1 u - 2 v = - \frac{1}{24}$
$2 v = \frac{1}{15} - \frac{1}{24}$
$2 v = \frac{24 - 15}{360}$
$2 v = \frac{9}{360}$
$v = \frac{1}{40} \times \frac{1}{2}$
$v = \frac{1}{80}$

Putting $v = \frac{1}{80}$ in equation $(i i i)$ , we get,
$1 u + 4 v = \frac{1}{15}$
$1 u + 4 \times \frac{1}{80} = \frac{1}{15}$
$1 u + \frac{4}{80} = \frac{1}{15}$
$1 u = \frac{1}{15} - \frac{4}{80}$
$1 u = \frac{1}{15} - \frac{1}{20}$
$1 u = \frac{20 - 15}{300}$
$1 u = \frac{5}{300}$
$u = \frac{1}{60}$
$\frac{1}{x} = \frac{1}{60}$
$x = 60$
and
$v = \frac{1}{80}$
$\frac{1}{y} = \frac{1}{80}$
$y = 80$
Hence, the speed of the train is $60 km / hr$ .
The speed of the bus is $80 km / hr$ .

R. D. Sharma Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 10: EXERCISE 3.10

R. D. Sharma Mathematics Solutions for Exercise - R. D. Sharma Solutions for Chapter: Pair of Linear Equations in Two Variables, Exercise 10: EXERCISE 3.10

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