Formulate the following problems as a pair of equations,and hence find their solutions:
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.

(N/A) Let the speed of the train be $u \ km/h$ and the speed of the bus be $v \ km/h$.
According to the given information:
Case $1$: Time taken = $4 \ hours$ for $60 \ km$ by train and $240 \ km$ by bus.
$\frac{60}{u} + \frac{240}{v} = 4 \quad ...(1)$
Case $2$: Time taken = $4 \ hours + 10 \ minutes = 4 + \frac{10}{60} = 4 + \frac{1}{6} = \frac{25}{6} \ hours$ for $100 \ km$ by train and $200 \ km$ by bus.
$\frac{100}{u} + \frac{200}{v} = \frac{25}{6} \quad ...(2)$
Let $\frac{1}{u} = p$ and $\frac{1}{v} = q$. The equations become:
$60p + 240q = 4 \quad ...(3)$
$100p + 200q = \frac{25}{6} \implies 600p + 1200q = 25 \quad ...(4)$
Multiply equation $(3)$ by $10$:
$600p + 2400q = 40 \quad ...(5)$
Subtract equation $(4)$ from $(5)$:
$(600p + 2400q) - (600p + 1200q) = 40 - 25$
$1200q = 15 \implies q = \frac{15}{1200} = \frac{1}{80}$
Substitute $q = \frac{1}{80}$ into equation $(3)$:
$60p + 240(\frac{1}{80}) = 4$
$60p + 3 = 4 \implies 60p = 1 \implies p = \frac{1}{60}$
Since $p = \frac{1}{u} = \frac{1}{60}$ and $q = \frac{1}{v} = \frac{1}{80}$,we get:
$u = 60 \ km/h$ and $v = 80 \ km/h$.
Thus,the speed of the train is $60 \ km/h$ and the speed of the bus is $80 \ km/h$.