Question

Rahul travels 600 km to his home, partly by train and partly by car. He takes 8 hours if he travels 120 km by train and the rest by car. He takes 20 minutes longer if he travels 200 km by train and the rest by car. Find the speed of the train and the car.
A. Speed of train = $ 20\ km/hr $ & Speed of car = $ 40\ km/hr $ .
B. Speed of train = $ 60\ km/hr $ & Speed of car = $ 80\ km/hr $ .
C. Speed of train = $ 45\ km/hr $ & Speed of car = $ 52\ km/hr $ .
D. Speed of train = $ 66\ km/hr $ & Speed of car = $ 88\ km/hr $ .

Answer 1

Hint: Distance and time are directly proportional to each other:
  $ Speed=\dfrac{Distance}{Time} $
Use two variables and form two equations from the given two cases.
Solve the pair of simultaneous equations by either elimination or Cramer's rule etc.

Complete step-by-step answer:
Let's say that the speeds of the train and car are $ x\ km/hr $ and $ y\ km/hr $ , respectively.
The values of time, distance and speed in the two cases are shown below:

	CASE 1	CASE 2
	Train	Car	Train	Car
Speed	x	y	x	y
Distance	120	480	200	400
Time	$ \dfrac{120}{x} $	$ \dfrac{480}{y} $	$ \dfrac{200}{x} $	$ \dfrac{400}{y} $

According to the question:
  $ \dfrac{120}{x}+\dfrac{480}{y}=8 $ ... (1)
  $ \dfrac{200}{x}+\dfrac{400}{y}=8+\dfrac{1}{3} $ ... (2)
Multiplying equation (1) by 5 and equation (2) by 3, we get:
  $ \dfrac{600}{x}+\dfrac{2400}{y}=40 $ ... (3)
  $ \dfrac{600}{x}+\dfrac{1200}{y}=24+1 $ ... (4)
Subtracting equation (4) from equation (3), we get:
  $ \dfrac{1200}{y}=15 $
⇒ $ y=80 $
Substituting this in equation (1) [or equation (2)], gives us:
  $ \dfrac{120}{x}+\dfrac{480}{80}=8 $
⇒ $ \dfrac{120}{x}+6=8 $
Subtracting 6 from both the sides, we get:
⇒ $ \dfrac{120}{x}=2 $
⇒ $ x=60 $
The correct answer is B. Speed of train = $ 60\ km/hr $ & Speed of car = $ 80\ km/hr $ .
So, the correct answer is “Option B”.

Note: The units of time and distance should be the same in all the cases.
There are 60 minutes in 1 hour and 60 seconds in 1 minute.
There are 1000 meters in 1 kilometer.
The overall average speed is: $ Average\ speed=\dfrac{Total\ distance}{Total\ time} $ .