Question

Roohi travels 300km to her home partly by train and partly by bus. She takes 4 hours if she travels 60km by train and the remaining by bus. If she travels 100km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

Answer 1

Total distance to be covered d = 300 km.
Let,
Speed of train = x km/h
Speed of bus = y km/h

Condition 1 :
:- 60 km by train(d₁) in t₁ hours + Remaining by bus(d₂) i.e 300-60 = 240 km in t₂ hours.
Time taken t = 4 hours.
time = distance/speed
t = t₁ + t₂ = 60/x + 240/y
60/x + 240/y = 4
240x + 60y = 4xy
=> 60x +15y = xy    -----------------------------(i)

Condition 2 :
:- 100 km by train(d₁) in t₁ hours + Remaining by bus(d₂) i.e 300-100 = 200 km in t₂ hours.
Time taken t = 4 hours + 10 minutes = 4+1/6 = 25/6 hours
time = distance/speed
t = t₁ + t₂ = 100/x + 200/y
=> 100/x + 200/y = 25/6
=> 4/x + 8/y = 1/6
=> 48x + 24y = xy   -----------------------------(ii)
From eqns (i) & (ii),
48x + 24y = 60x + 15y
=> 12x  = 9y
=> 4x = 3y
y =4x/x ----------------------------------------------(iii)

put the value of y in eqn (ii)
48x + 24*4x/3 = x*4x/3
=> 48x + 32x = 4x²/3
=> x² - 60x = 0
=> x(x-60) = 0
 x = 0                              | x is value of speed. it can't be 0 in this scenario.
x - 60 = 0 => x = 60        |  value of x = 60 is a valid value.
put the value of x in the eqn (iii)
 y = 4*60/3 = 80

hence x = 60 and y = 80.
Therefore speed of the train = 60 km/h and speed of the bus = 80 km/h

Answer 2

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$\bf{https://brainly.in/question/37581179}$

Let the speed of train and bus be u km/h and v km/h respectively.

According to the question,

$\frac{60}{u} + \frac{240}{v} = 4$....(i)

$\frac{100}{u} + \frac{200}{v} = 4 + \frac{10}{60} = \frac{25}{6}$....(ii)

Let $\frac{1}{u} = p\: and \frac{1}{v} = q$

The given equations reduce to:

60p + 240q = 4 ....(iii)

100p + 200q = $\frac{25}{6}$

600p + 1200q = 25....(iv)

Multiplying equation (iii) by 10, we obtain:

600p + 2400q = 40....(v)

Subtracting equation (iv) from equation (v), we obtain:

1200q = 15

q = $\frac{15}{1200} = \frac{1}{80}$

Substituting the value of q in equation (iii), we obtain:

60p + 3 = 4

60p = 1

p = $\frac{1}{60}$

:. p = $\frac{1}{u} = \frac{1}{60}$, q = $\frac{1}{v} = \frac{1}{80}$

u = 60 km/h , v = 80 km/h

Thus, the speed of train and the speed of bus are 60 km/h and 80 km/h respectively.

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