Question

Ramesh travels 760 kms to his home partly by train and partly by car. He takes 8 hours if he travels 160 kms by train and the rest by car. He takes 12 minutes more if he travels 240 kms by train and the rest by car. Find the speed of the train and the car.
(A) Speed of train = 80 km/hr and speed of car = 100 km/hr
(B) Speed of train = 40 km/hr and speed of car = 50 km/hr
(C) Speed of train = 60 km/hr and speed of car = 120 km/hr
(D) Speed of train = 70 km/hr and speed of car = 130 km/hr

Answer 1

Hint: Assume both the speed of the train and that of the car to be some variables \[x\] and \[y\]. Frame two different equations in \[x\] and \[y\] from the given two conditions using formula ${\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$. Solve these equations simultaneously to get the required speeds.

Complete step-by-step solution:
According to the question, we have been given two conditions based on the journey of Ramesh of 760 kms partly by train and partly by car. We need to find their individual speeds.
Let the speed of the train be \[x{\text{ km/hr}}\] and the speed of the car be \[y{\text{ km/hr}}\].
Then from the first condition, it is given that he traveled 160 kms by train. So the remaining 600 kms must be traveled by car and it is also given that the journey took 8 hours. We know that the formula for time taken in a journey is given as:
$ \Rightarrow {\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$
So the time taken in covering 160 kms by train is $\dfrac{{160}}{x}$ and in covering the remaining 760 kms by car is $\dfrac{{600}}{y}$. Total time taken is 8 hours. So we have:
$ \Rightarrow \dfrac{{160}}{x} + \dfrac{{600}}{y} = 8$
This equation can further be simplified as:
$ \Rightarrow \dfrac{{20}}{x} + \dfrac{{75}}{y} = 1{\text{ }}.....{\text{(1)}}$
In the next condition, he traveled 240 kms by train and the remaining 520 kms by car. And in this case he took 12 more minutes to complete the journey. If we convert 12 minutes into hour, this is $\dfrac{{12}}{{60}}{\text{ hours}}$. So if we frame equation as we did above, we’ll get:
$ \Rightarrow \dfrac{{240}}{x} + \dfrac{{520}}{y} = 8 + \dfrac{{12}}{{60}}$
If we simplify this, we have:
$
   \Rightarrow \dfrac{{240}}{x} + \dfrac{{520}}{y} = 8 + \dfrac{1}{5} \\
   \Rightarrow \dfrac{{240}}{x} + \dfrac{{520}}{y} = \dfrac{{41}}{5}{\text{ }}.....{\text{(2)}}
 $
Now, multiplying equation (1) by 12 and subtracting equation (2) from it, we’ll get:
$ \Rightarrow 12\left( {\dfrac{{20}}{x} + \dfrac{{75}}{y}} \right) - \dfrac{{240}}{x} + \dfrac{{520}}{y} = 12 \times 1 - \dfrac{{41}}{5}$
Simplifying it further, we’ll get:
$
   \Rightarrow \dfrac{{240}}{x} + \dfrac{{900}}{y} - \dfrac{{240}}{x} - \dfrac{{520}}{y} = \dfrac{{60 - 41}}{5} \\
   \Rightarrow \dfrac{{380}}{y} = \dfrac{{19}}{5} \\
   \Rightarrow \dfrac{1}{y} = \dfrac{{19}}{{5 \times 380}} = \dfrac{1}{{100}} \\
   \Rightarrow y = 100
 $
Putting this value in equation (1), we’ll get:
$
   \Rightarrow \dfrac{{20}}{x} + \dfrac{{75}}{{100}} = 1 \\
   \Rightarrow \dfrac{{20}}{x} + \dfrac{3}{4} = 1 \\
   \Rightarrow \dfrac{{20}}{x} = \dfrac{1}{4} \\
   \Rightarrow x = 80
 $
Thus the speed of the train is 80 km/hr and the speed of the car is 100 km/hr.

Option A is the correct answer.

Note: If a linear equation is having only one variable, it can be solved directly to get the value of the variable. If it is a two variable equation then to determine the values of two different variables, we need two different equations in those variables. Similarly if there are $n$ different variables then we require $n$ different equations to find their values.