Question

A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours and 30 minutes. But, if he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and that of the car.

Answer 1

Given:

Total distance = 600 km.

Case A: 400 km by train + 200 km by car → total time = 6 h 30 min = 6.5 h.

Case B: 200 km by train + 400 km by car → total time = 7 h.

Let train speed = x (km/h) and car speed = y (km/h).

Step-wise calculation:

1. Write the time equations:

`400/x + 200/y = 6.5`   ...(1) 

`200/x + 400/y = 7`   ...(2)

2. Multiply both equations by 2 to simplify:

`800/x + 400/y = 13`   ...(1a) 

`400/x + 800/y = 14`   ...(2a)

3. Let `u = 1/x` and `v = 1/y`. 

Then 800u + 400v = 13   ...(3) 

400u + 800v = 14   ...(4)

4. Multiply (4) by 2 and subtract (3):

(800u + 1600v) – (800u + 400v) = 28 – 13

1200v = 15

⇒ `v = 15/1200`

 ⇒ `v = 1/80`

⇒ y = 80 km/h

5. Substitute `v = 1/80` into (3): 

`800u + 400(1/80) = 13`

⇒ 800u + 5 = 13

⇒ 800u = 8

⇒ `u = 1/100` 

⇒ x = 100 km/h.

Verify: `400/100 + 200/80 = 4 + 2.5` = 6.5 h

`200/100 + 400/80 = 2 + 5` = 7 h

Speed of the train = 100 km/h.

Speed of the car = 80 km/h.