Question

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


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