Question

Mahesh travels 250 km to his home partly by train and partly by bus. He takes 6 hours if he travels 50 km by train and the remaining distance by bus. If he travels 100 km by train and the remaining distance by bus he takes 7 hours. Find the speed of the train and the bus separately.

Answer 1

Hint: We will let the speed of the train is $x$ km/h and let the speed of the bus be $y$ km/h. We will then find the time taken by bus and train using the formula, ${\text{time}} = \dfrac{{{\text{distance}}}}{{{\text{speed}}}}$. Form the equations according to the given conditions. Then, solve the equations to find the value of $x$ and $y$.

Complete step-by-step answer:
We have to find the speed of the train and the bus.
Let the speed of the train is $x$ km/h and let the speed of the bus is $y$ km/h
The total distance travelled by Mahesh is 250 km.
If the distance travelled by train is 50 km and the distance travelled by bus will be $250 - 50 = 200km$
We know that ${\text{time}} = \dfrac{{{\text{distance}}}}{{{\text{speed}}}}$
Then, time taken by train will be $\dfrac{{50}}{x}$ and time taken by bus $\dfrac{{200}}{y}$
We are given that a person takes 6 hours to travel 50 km by train and 200 km by bus.
That is, $6 = \dfrac{{50}}{x} + \dfrac{{200}}{y}$ eqn. (1)
Similarly, we are given that it takes 7 hours when 100 km distance is covered by train and the remaining distance, that is $250 - 100 = 150km$, is covered by bus.
$7 = \dfrac{{100}}{x} + \dfrac{{150}}{y}$ eqn. (2)
We will solve equation (1) and (2) to find the value of $x$ and $y$.
Multiply equation (1) by 2 and subtract equation (1) and (2)
$
  12 - 7 = \dfrac{{100}}{x} + \dfrac{{400}}{y} - \dfrac{{100}}{x} - \dfrac{{150}}{y} \\
   \Rightarrow 5 = \dfrac{{250}}{y} \\
   \Rightarrow y = \dfrac{{250}}{5} \\
   \Rightarrow y = 50 \\
$
Now, substitute the value of $y$ in equation (1) to find the value of $x$
$
  6 = \dfrac{{50}}{x} + \dfrac{{200}}{{50}} \\
   \Rightarrow 6 = \dfrac{{50}}{x} + 4 \\
   \Rightarrow 2 = \dfrac{{50}}{x} \\
   \Rightarrow x = 25 \\
$
Hence, the speed of the train is 25 km/h and the speed of the bus is 50 km/h.

Note: Equation should be formed correctly to avoid mistakes. We have used elimination to solve for the value of $x$ and then substitution for finding the value of $y$. We can also use substitution to find the value of $y$. Do not forget to mention the units of speed.