Question

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.​

Answer 1

• $\huge\pink{\mid{\fbox{\tt(answer)}\mid}}$
• 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=
• Speed
• Distance
• Total distance =300 km
• Roohi travels 60 km by train and 300−60=240 by bus in 4 minute,
• x
• 60
• +
• y
• 240
• =4
• and 100 km by train, 300−100=200 by bus, and takes 10 minutes more,
• x
• 100
• +
• y
• 200
• =4+
• 60
• 10
• ⇒
• x
• 100
• +
• y
• 200
• =
• 6
• 25
• Now, let
• x
• 1
• =u and
• y
• 1
• =v
• then 60u+240v=4.............eq1
• 100u+200v=
• 6
• 25
• ..............eq2
• multiply eq1 by 5 and eq2 by 6 we get
• 300u+1200v=20..........eq3
• 600u+1200v=25...........eq4
• Subtracting eq3 qnd eq4 we get
• −300u=−5
• u=
• 300
• 5
• =
• 60
• 1
• Putting the value of u in eq1 we get
• 60×
• 60
• 1
• +240v=4
• 240v=3
• v=
• 240
• 3
• =
• 80
• 1
• Now
• x
• 1
• =u=
• 60
• 1
• ∴x=60
• and
• y
• 1
• =v=
• 80
• 1
• ∴y=80
• Hence the speed of the train is 60 km/hr and the speed of the bus is 80 km/hr.

Answer 2

Let the speed of the train be x

and the speed of the bus be y

Case-1

Distance travelled by train =60 km

$\rm time = \frac{distance}{speed} = \frac{60}{x} \\ \implies \rm \boxed{ t_{1} = \frac{60}{ \rm \: x} }$

Distance travelled by bus=300-60=240

$\rm time = \frac{distance}{speed} = \frac{240}{y} \\ \rm \implies \boxed {t_{2} = \frac{240}{ \rm \: y} }$

According to first condition

$\rm t_{1} + t_{2} = \frac{60}{x} + \frac{240}{y} \\ \boxed{ \frac{60}{x} + \frac{240}{y} = 4}....1$

Case-2

Distance travelled by train=100km

$\rm time = \frac{distance}{speed} = \frac{100}{x} \\ \implies \rm \boxed{ t_{3} = \frac{100}{x}}$

Distance travelled by bus=200km

$\rm \: time = \frac{distance}{speed} = \frac{200}{y} \\ \rm \: \implies \boxed{ t_{4} = \frac{200}{y} }$

According to 2nd condition

$\rm t_{3} + t_{4} = \frac{ 100}{x} + \frac{200}{y} \\ \implies \boxed{ \frac{25}{6} = \frac{100}{x} + \frac{200}{y} }..2$

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

So the equation is

$\rm 60u + 240v = 4 \\ 100u + 200v = \frac{25}{6}$

Using elimination method

$\rm 600u + 2400v = 40 \\ \rm \: 600u + 1200v = 25 \\ - - - - - - - - - \\ \rm \: 1200v = 15 \\ \rm\implies \boxed{v = \frac{1}{80} and \: u = \frac{1}{60} }$

Therefore

$\boxed{ \rm x =80 }and \boxed{ \rm \: y = 60}$

Therefore speed of train is 80km/h and speed of bus is 60km/h