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# Roohi travels 300km to her home partly by train and partly by bus. She takes 4 hours if she travels 60km by train and the remaining by bus. If she travels 100km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

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Hint- This question can be solved by converting the given statements into linear equations.

Given that, Roohi has to travel a total distance = 300km
And she has to travel partly by bus and partly by train.
Let the speed of the train and bus be $u{\text{ }}km{\text{ per }}h$ and $v{\text{ }}km{\text{ per }}h$ respectively.
According to the question,
She takes $4$ hours if she travels $60km$ by train and the remaining by bus.
Now distance travelled by train$= 60km$
Distance travelled by bus$= 240km$
Or $\dfrac{{60}}{u} + \dfrac{{240}}{v} = 4 - - - - - - - - \left( i \right)$
Also it is given that,
If she travels $100km$ by train and the remaining by bus, she takes $10$ minutes longer.
Now distance travelled by train$= 100km$
Distance travelled by bus$= 200km$
Or $\dfrac{{100}}{u} + \dfrac{{200}}{v} = 4 + \dfrac{{10}}{{60}} \\ \dfrac{{100}}{u} + \dfrac{{200}}{v} = \dfrac{{25}}{6} - - - - - - - - \left( {ii} \right) \\$
Let $\dfrac{1}{u} = p$ and $\dfrac{1}{v} = q$
Thus the given equation reduces to
$60p + 240q = 4 - - - - - - \left( {iii} \right) \\ 100p + 200q = \dfrac{{25}}{6} \\ 600p + 1200q = 25 - - - - - - \left( {iv} \right) \\$
Multiplying $\left( {iii} \right)$ by $10$ we get,
$600p + 2400q = 40 - - - - - - \left( v \right)$
Subtracting $\left( {v} \right)$ from $\left( iv \right)$ we get,
$600p + 2400q-600p - 1200q=40-25 \\ 1200q = 15 \\ q = \dfrac{{15}}{{1200}} \\ q = \dfrac{1}{{80}} \\$
Substituting the value of $q$ in $\left( {iii} \right)$ we get,
$60p + 3 = 4 \\ 60p = 1 \\ p = \dfrac{1}{{60}} \\ \therefore p = \dfrac{1}{u} = \dfrac{1}{{60}},q = \dfrac{1}{v} = \dfrac{1}{{80}} \\$
Or $u = 60km{\text{ per }}h{\text{ }},{\text{ }}v = 80km{\text{ per }}h$
Thus the speed of the train and the speed of the bus are $60{\text{ }}km{\text{ per }}h$ and $80{\text{ }}km{\text{ per }}h$ respectively.

Note- Whenever we face such types of questions the key concept is that we should mention what is given to us and assume some variables which we need to find and then convert the statements into linear equations like we did and solve the equations to get our desired answer.