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 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.

#### Solution

Let the speed of train and bus be u km/h and v km/h respectively.

According to the given information,

`60/u + 240/v = 4 ... (i)`

`100/u + 200/v = 25/6 ... (ii)`

Putting `1/u = p ` in the equations, we get

60p + 240q = 4 ... (iii)

100p + 200q = 25/6

600p + 1200q = 25 ... (iv)

Multiplying equation (iii) by 10, we get

600p + 2400q = 40 .... (v)

Subtracting equation (iv) from (v), we get1200q = 15

`q = 15/200 = 1/80 ... (vi)`

Putting equation (iii), we get

60p + 3 = 4

60p = 1

p = 1/60

`p = 1/u = 1/60 `

u = 60 and v = 80

Hence, speed of train = 60 km/h and speed of bus = 80 km/h.