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

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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Total distance=600km

CASE 1

Distance travelled by train=400km

Distance travelled by car =600-400=200km

Time taken by train = D/S=400/x

Time taken by car = D/S=200/y

CASE 2

Distance travelled by train=200km

Distance travelled by car =600-200=400km

Time taken by train = D/S=200/x

Time taken by car = D/S=400/y

A.T.Q

__400__

_{+ }

__200__

_{ =}6+

__30__

x y 60

Let 1/x and 1/y be a and b

400a+200b=13/2

200a+100b=13/4..........(i)

AND

__200__

_{ + }

__400__

_{ = }7

x y

Let 1/x and 1/y be a and b

200a+400b=7......(ii)

In (i) & (ii)using elimination method

200a+100b=13/4

+200a+400b=+7

__- - -__

-300b=-15/4

b=-15/4*-300

b=1/80

and Putting in (ii) we get

200a=7-400(1/80)

a=2/200

a=1/100

NOW a=1/x and b=1/y

so x=>Speed of train=>100 km/hr

and y=>Speed of car=>80km/hr

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we need to find the speed so speed = dist/time ( d=s/t)

let the speed of train be x kmph and bus be y kmph

first case

train 400 km

bus 200km

time 13/2 hours

400/x + 200/y = 13/2

second case

train = 200 km

car 400 km

time = 7 hours

200/x + 400/y = 7

let 1/x be u and 1/y be v

400u+200v=13/2 or 800u+400v=13

200u+400v=7

solve them

x = train = 100kmph

y= car=80kmph

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- 4

Let 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 it helped u

- 4

Let 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 it helped u

- 3

CASE 1; he travels 400 km by train and 200 km by car

CASE 2; he travels 200 km by train and 400 km by car

let the speed of train = x kmph

let the speed of car = y kmph

400/x + 200/y=13/2

200/x + 400/y= 7

let 1/x = a , 1/y= b

400a + 200b =13/2 800a + 400b = 13 ----------(1)

200a + 400b = 7 ------------(2)

subtracting (1) and (2)

600a=6

a=1/100

substituiting the value of a in (2)

b=1/80

so,

the speed of train is 100 kmph

the speed of car is 80 kmph

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- 3

Let 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:)

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