Answer:

Let's call the speed of the train v_train and the speed of the bus v_bus.

The time it takes Roohi to travel 60 km by train is t_train = 60 km / v_train.

The time it takes Roohi to travel 100 km by train is t_train' = 100 km / v_train.

The time it takes Roohi to travel the remaining distance (240 km) by bus is t_bus = 240 km / v_bus.

The total time it takes Roohi to travel 300 km by train and bus is t = t_train + t_bus = 4 hours.

The total time it takes Roohi to travel 300 km by train and bus when she travels 100 km by train is t' = t_train' + t_bus = 4 hours + 10 minutes.

We can set up two equations to represent these two situations:

60/v_train + 240/v_bus = 4 hours

100/v_train + 240/v_bus = 4 hours + 10 minutes

We can solve this system of equations to find the values of v_train and v_bus.

First, let's convert the time units to hours. 4 hours is the same as 4 + (10/60) hours, where 10/60 represents the number of hours in 10 minutes.

This gives us:

60/v_train + 240/v_bus = 4 + (10/60) hours

100/v_train + 240/v_bus = 4 + (10/60) hours

Next, we can multiply both sides of the first equation by 60 and both sides of the second equation by 100 to eliminate the fractions:

60(60/v_train) + 60(240/v_bus) = 60(4 + (10/60))

100(100/v_train) + 100(240/v_bus) = 100(4 + (10/60))

This simplifies to:

3600 + 14400 = 240 + 600

10000 + 24000 = 400 + 1000

Then, we can solve the system of equations by subtracting the first equation from the second equation:

10000 + 24000 - 3600 - 14400 = 400 + 1000 - 240 - 600

6400 + 9600 = 160 + 400

16000 = 560

Dividing both sides by 560 gives:

v_train = 25 km/hr

Substituting this value back into the first equation gives:

60/25 + 240/v_bus = 4 + (10/60)

This simplifies to:

240/v_bus = 4 - (60/25)

Multiplying both sides by v_bus gives:

240 = v_bus(4 - (60/25))

Substituting 25 for v_train in the expression (60/25) gives:

240 = v_bus(4 - (60/25))

240 = v_bus(4 - (60/25))

240 = v_bus(4 - (60/25))

240 = v_bus(4 - 2.4)

240 = v_bus(1.6)

Dividing both sides by 1.6 gives:

v_bus = 150 km/hr

Therefore, the speed of the train is 25 km/hr and the speed of the bus is 150 km/hr.

Verified answer

[tex]{ \boxed{ \large \green \bigstar \: \pmb{ \rm {\red{Qu} \: \blue{es} \pink{ tion}}}}}[/tex]

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

[tex]{ \boxed{ \large \green \bigstar \: \pmb{ \rm {\red{An} \: \blue{sw} \pink{er}}}}}[/tex]

Step-1 : Assume the speed of train and bus to be equal to some variables

$$\textbf{& then apply relevant formula

among time, distance & speed}$$

Let the speed of the train be x

km/hr and the speed of the bus is y km/hr.

So, according to question and using Time =

[tex] \frac{distance}{speed} [/tex]

Total distance = 300 km

Roohi travels 60 km by train and 300-60

= 240 by bus in 4 minute,

[tex] \frac{100}{x} + \frac{200}{y} = 4 - \frac{10}{60} [/tex]

and 100 km by train, 300 - 100 = 200

by bus, and takes 10 minutes more,

[tex] \frac{100}{x} + \frac{200}{y} = 4 + \frac{10}{60} = \frac{100}{x} + \frac{200}{y} = \frac{25}{6} [/tex]

Now,

[tex]let \: \frac{1}{x} = u \: and \: \frac{1}{y} = v[/tex]

[tex]then \: 60u \: + 240v = 4.............eq1[/tex]

[tex]100u + 200v = \frac{25}{6} ............eq2[/tex]

Step - 2: Simplify unknown

multiply eq1 by 5 and eq2 by 6 we get,

300u+1200v=20..........eq3

600u+1200v=25...........eq4

Subtracting eq3 and eq4 we get,

−300u=−5

[tex]u = \frac{5}{300} = \frac{1}{6} [/tex]

Putting the value of u in eq1 we get,

[tex]60 \times \frac{1}{60} + 240v = 4 [/tex]

240v=3

[tex]v = \frac{3}{240} = \frac{1}{80} [/tex]

[tex]now \: \frac{1}{x} = u = \frac{1}{60} [/tex]

∴ x = 60

[tex]and \: \frac{1}{y} = v = \frac{1}{80} [/tex]

∴ y = 80

So, the speed of the train is 60 km/hr and the speed of the bus is 80 km/hr.