Question

A man travels 600km by train and car. If he covers 400km by train and rest by car it takes 6hrs &30mins. If he travels 200km by train and rest by car. It takes half an hour longer. Find the speed of train and car...

Answer 1

Let x=train speedLet y = car speed 
speed = distance / time time = distance / speed 
6 hours 30 minutes = 6.5 hours <== 400 km by train ; 200 km by car 1/2 hour longer = 7 hours <== 200 km by train ; 400 km by car 
Train time + car time = total time 
400 / x + 200 / y = 6.5 <== two equations and two unknowns 200 / x + 400 / y = 7 . . . . . . solve for x and y 
400y + 200x = 6.5 xy 200y + 400x = 7 xy 
400y - 6.5 xy = - 200x 200y - 7xy = - 400x 
y ( 400 - 6.5x) = -200x y ( 200 - 7x) = -400x 
y (6.5x - 400) = 200x y (7x - 200) = 400x 
y = 200x / (6.5x - 400) y = 400x / (7x - 200) . . . note this equation ... for later use 
. . . since both equal y, the difference is zero 
200x / (6.5x - 400) - 400x / (7x - 200) = 0 
200x ( 7x - 200) - 400x (6.5x - 400) = 0 
1400x^2 - 40000x - 2600x^2 + 160000x = 0 
120000 x - 1200 x^2 = 0 
100 - x = 0 
x = 100 km / h <===== train speed 
y = 400x / (7x - 200) . . . see note above 
y = 400 * 100 / (7 * 100 - 200) 
y = 80 km / h <===== car speed
hope I helped u
thank u

Answer 2

Given 
Total distance = 600 km
Consider speed of Train = x km/h
Consider speed of Car = y km/h
According to the question:
We know Time = distance/speed

Situation 1 
He takes 8 hours if he travels 120 km by train and rest by car

120/x + 480/y = 8-------(1)

Situation (2)
when he travels 200 km by train and rest by car he takes 8 hr 20 min
8 hr 20 min = 25/3 hr

200/x + 400/y = 25/3 ---------(2)

solve equation 1 and 2

you'll get x = 60
so the speed of the train = 60 km/h

There's an alternate method too:

Le the speed of the train be x km/hr and speed of car be y km/hr
Vijay travels 120 km by train, the distance covered by car 600-120 = 480 km
Now, time taken to cover 120 km by train = 120/xhr
And time taken to cover 480 km by car = 480/y 
Total time taken to travel 600 km = 120/x + 480/y = 8 hr (i)
Similarly 
200/x + 400/y = 8hr. 20minute = 8(20/60) (ii)

solve (i) and (ii) 
we get 
x = 60 km/hr and y = 80 km/hr

∴ Speed of train 60 km/hr

Hope This Helps :)