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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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
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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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