Question

A man travels 600 km partly by a train and partly by a car. If he covers 400 km by train and the rest by car it takes him 6 hour 30 min. What 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

Let x = speed of train
Let y = speed of car

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) * 

. . . 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) . from * 

y = 400 * 100 / (7 * 100 - 200)

y = 80 km / h = car speed