Question

An express train takes 1 hour less than a passenger train to travel
$132\;{\rm{km}}$between Mysore and Bangalore. If the average speed of the express train is
$11\;{\rm{km}}/{\rm{hr}}$ more than that of the passenger train from the quadratic equation to find
the average speed of the express train.

Answer 1

Hint: In this problem, first we have to find the time taken by each train. Since the total distance
travel and the time and speed difference between both the train are given, so we have to find the
expression of time taken by both trains. After that we have to subtract the times for the given
condition. Form that we will get the speed of the passenger train. Finally, we have to calculate
the speed of the express train.
Let the speed of the passenger train is $x\;{\rm{km}}/{\rm{hr}}$.
Since the speed of the express train is $11\;{\rm{km}}/{\rm{hr}}$ more than the passenger train.
Thus the speed of express train will be $x + 11\;{\rm{km}}/{\rm{hr}}$
It is given that the distance from Mysore to Bangalore is $132\;{\rm{km}}$.
Therefore, the time taken by each train to travel from Mysore to Bangalore is,
Time taken by the passenger train $ = \dfrac{{132}}{x}\;{\rm{hr}}$
Time taken by the express train $ = \dfrac{{132}}{{x + 11}}\;{\rm{hr}}$
According to question, express train takes 1 hour less than a passenger train to travel between
Mysore and Bangalore.
$\therefore \dfrac{{132}}{x} - \dfrac{{132}}{{x + 11}} = 1$
Simplifying the above expression and solving for $x$.
$\begin{array}{l} \Rightarrow \dfrac{{132\left( {x + 11 - x} \right)}}{{x\left( {x + 11} \right)}} = 1\\
\Rightarrow \dfrac{{1452}}{{{x^2} + 11x}} = 1\\ \Rightarrow 1452 = {x^2} + 11x\\ \Rightarrow {x^2} + 11x
- 1452 = 0\\ \Rightarrow {x^2} + 44x - 33x - 1452 = 0\\ \Rightarrow x\left( {x + 44} \right) - 33\left( {x +
44} \right) = 0\\ \Rightarrow \left( {x - 33} \right)\left( {x + 44} \right) = 0\\\therefore x = 33\;{\rm{or}}\;x
= - 44\end{array}$

Since speed cannot be negative, so $ - 44$ is not possible.
Therefore, the speed of the passenger train is 33 km/hour.
Hence, the speed of the express train $ = 33 + 11\;{\rm{km}}/{\rm{hr}} = 44\;{\rm{km}}/{\rm{hr}}$
Note: Here we have to determine the speed of the express train for the given data. The distance
travelled and the speed of the passenger train is given. So, from that data we can easily calculate
the speed of the express train by using the relation of speed, distance and time.