A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.

Explanation

Let the speed of the train be x km/hr and that of the car be y km/hr.
Case I Distance covered by train = 250 km.
Distance covered by car = ` ( 370 - 250 ) km = 120 km `
Time taken to cover 250 m by train `= ( 250) /(x)` hours.
Time taken to cover 120 km by car = `(120) /( y) ` hours.
Total time takes = 4 hours.
` therefore ( 250 )/(x) + ( 120 )/( y) = 4 rArr ( 125 ) /(x) + ( 60 ) /( y )= 2 `
Case II Distance covered by train = 130 km.
Distance covered by car = ` ( 370 - 130) km = 240 km `
Time taken to cover 130 km by train = `( 130 ) /(x) ` hours.
Time taken to cover 240 km by car = ` ( 240 ) /( y)` hours.
Total time taken = ` 4 ( 18)/(60 ) ` hours = ` 4(3)/(10)`hours = ` ( 43)/(10)` hours.
` therefore ( 130 )/(x) + (240 ) /( y ) = ( 43) /(10) rArr ( 1300)/(x) + ( 2400)/( y) = 43" " `... (ii)
Putting `( 1)/(x) = u and (1)/( y) = v` , equations (i) and (ii) become
` 125u + 60 v = 2 " " `...(iii) and ` 1300 u + 2400 v = 43. " " ` ... (iv)
On multiplying (iii) by 40 and subtracting (iv) from the result, we get
` 5000 u - 1300 u = 80 - 43 rArr 3700 u = 37 `
` rArr u = ( 37) /( 3700) = (1)/(100) rArr (1)/(x) = (1)/(100) rArr x = 100 `
Putting ` u = (1)/(100) ` in (iv), we get
` ( 1300 xx (1)/(100)) + 2400 v = 43 rArr 2400 v = 43 - 13 = 30 `
` rArr v = (30)/( 2400) = (1)/(80) rArr (1)/(y) = ( 1)/(80) rArr y = 80 `
`therefore x = 100 and y = 80 `
Hence, the speed of the trains is 100 km/hr and that of the car is 80 km/hr.