Question

Maheep travels 600 km partly by train and partly by car. He takes 8 hours to cover the whole distance if he travels 120 km by train and rest by car. But he takes 20 minutes longer if he travels 200 km by train and the rest by car. Find the speed of train and car.

Answer 1

Here, let the Speed of the train = x km/h,

And Speed of car = y km/h,

According to the given condition,

(i) Distance by train = 120 km, Time = `120/x`,

Distance by car = 600 − 120 = 480 km, Time = `480/y`,

Total time = 8 hours,

`120/x + 480/y = 8`     ...(1)

(ii) Distance by train = 200 km, Time = `200/x`,

Distance by car = 600 − 200 = 400 km, Time = `400/y`,

Total time = 8 hours 20 minutes = `25/3` hours

`200/x + 400/y = 25/3`     ...(2)

Solving equations by multiplying them to eliminate fractions,

Multiply equation (1) by 3xy:

`3xy(120/x + 480/y) = 3xy(8)`

3y(120) + 3x(480) = 24xy

360y + 1440x = 24xy     ...(3)

Multiply equation (2) by 3xy:

`3xy(200/x + 400/y) = 3xy(25/3)`

3y(200) + 3x(400) = 25xy

600y + 1200x = 25xy     ...(4)

Subtracting equation (3) from equation (4):

(600y + 1200x) − (360y + 1440x) = 25xy − 24xy

600y + 1200x − 360y − 1440x = xy

240y − 240x = xy

xy = 240(y − x)     ...(5)

Multiplying equation (1) by xy on both sides:

`xy(120/x + 480/y) = xy(8)`

120y + 480x = 8xy     ...(6)

Substitute xy = 240(y − x) in equation (6):

120y + 480x = 8 × 240(y − x)

120y + 480x = 1920y − 1920x

120y − 1920y + 480x + 1920x = 0

−1800y + 2400x = 0

2400x = 1800y

Divide both sides by 600,

4x = 3y

`x=(3y)/4`     ...(7)

Substituting equation (7) into equation (1):

`120/((3y)/4) + 480/y = 8`

`(120 xx 4)/(3y) + 480/y = 8`

`160/y + 480/y = 8`

`(160 + 480)/y = 8`

`640/y = 8`

`y = 640/8`

∴ y = 80

Now, substituting y = 80 in equation (7),

`x = (3 xx 80)/4`

x = 3 × 20

∴ x = 60

Hence, the Speed of the train is 60 km/h, and the Speed of the car is 80 km/h.