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.
A. Speed of the train is: 40km/hr and speed of bus: 90 km/hr
B. Speed of the train is: 60km/hr and speed of bus: 80 km/hr
C. Speed of the train is: 90km/hr and speed of bus: 120 km/hr
D. Speed of the train is: 1000km/hr and speed of bus: 70 km/hr

Answer 1

Hint: In order to find the speed of bus and train, first we will assume the speed of train and bus as variables, then we will make two equations with two variables with the help of a given statement and by simplifying it we will proceed further to get the perfect answer.

Complete Step-by-Step solution:
Let the speed of the train be x km/hr and the speed of the bus is y km/hr.
As we know the formula for time.
\[{\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}\]
Total distance = 300 km
Roohi travels 60 km by train and 300 – 60 = 240 by bus in 4 minute,
$ \Rightarrow \dfrac{{60}}{x} + \dfrac{{240}}{y} = 4$
And 100 km by train, 300 – 100 = 200 by bus and takes 10 minutes more,
$
   \Rightarrow \dfrac{{100}}{x} + \dfrac{{200}}{y} = 4 + \dfrac{{10}}{{60}} \\
   \Rightarrow \dfrac{{100}}{x} + \dfrac{{200}}{y} = \dfrac{{240 + 10}}{{60}} \\
   \Rightarrow \dfrac{{100}}{x} + \dfrac{{200}}{y} = \dfrac{{250}}{{60}} = \dfrac{{25}}{6} \\
 $
Now, let $\dfrac{1}{x} = u{\text{ and }}\dfrac{1}{y} = v$
Then,
$
  60u + 240v = 4............(1) \\
  100u + 200v = \dfrac{{25}}{6}.........(2) \\
 $
In order to find the value of u and v let us solve the equation.
Multiplying equation (1) by 5 and equation (2) by 6 we get:
$
  300u + 1200v = 20............(3) \\
  600u + 1200v = 25.........(4) \\
 $
Subtract equation (3) from equation (4) we get:
$
  300u = 5 \\
   \Rightarrow u = \dfrac{5}{{300}} = \dfrac{1}{{60}} \\
 $
Putting the value of u in equation (1) we get:
$
   \Rightarrow 60 \times \dfrac{1}{{60}} + 240v = 4 \\
   \Rightarrow 1 + 240v = 4 \\
   \Rightarrow 240v = 4 - 1 = 3 \\
   \Rightarrow 240v = 3 \\
   \Rightarrow v = \dfrac{3}{{240}} = \dfrac{1}{{80}} \\
 $
Now let us find the value of x and y from the value of u and v
\[
  \dfrac{1}{x} = u = \dfrac{1}{{60}} \\
  \therefore x = 60 \\
 \]
And
\[
  \dfrac{1}{y} = v = \dfrac{1}{{80}} \\
  \therefore y = 80 \\
 \]
Hence the speed of the train is 60 km/hr and the speed of the bus is 80 km/hr.
So, option B is the correct option.

Note: In order to solve such types of problems it is easier to solve the problem by the means of algebraic methods. Students must remember the formula for speed of and should make some modification in the formula where required. Also some substitutions can be made in order to get the answer.