Question

An express train takes $1\mathrm{hour}$ less than a passenger train to travel $132\mathrm{km}$ between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is $11\mathrm{km}/\mathrm{h}$ more than that of the passenger train, find the average speed of the two trains.

Answer 1

Find the average speed of the two trains.

Let the average speed of the passenger train be $x\mathrm{km}/\mathrm{h}$.

Therefore, according to the question, the average speed of the express train will be $(x+11)\mathrm{km}/\mathrm{h}$.

As per the given data, the time taken by the express train to cover $132\mathrm{km}$ is $1\mathrm{hour}$ less than the passenger train i.e.,

$\left(\mathrm{Time}\mathrm{taken}\mathrm{by}\mathrm{the}\mathrm{passenger}\mathrm{train}\mathrm{to}\mathrm{cover}132\mathrm{km}\right)-\left(\mathrm{Time}\mathrm{taken}\mathrm{by}\mathrm{the}\mathrm{express}\mathrm{train}\mathrm{to}\mathrm{cover}132\mathrm{km}\right)=1\mathrm{hour}$

Therefore,

$\begin{array}{rcl}& & \begin{array}{lcl}\frac{132}{x}-\frac{132}{\left(x+11\right)}& =& 1\left[\because \mathrm{Time}\mathrm{taken}=\frac{\mathrm{Total}\mathrm{distance}}{\mathrm{Average}\mathrm{speed}}\right]\end{array}\\ & & \begin{array}{c}\frac{132\left(x+11\right)-132x}{x\left(x+11\right)}=1\\ \frac{132x+\left(132\times 11\right)-132x}{x\left(x+11\right)}=1\\ \frac{1452}{x\left(x+11\right)}=1\\ 1452=x\left(x+11\right)\\ x^2+11x-1452=0\\ x^2+44x-33x-1452=0\\ x\left(x+44\right)-33\left(x+44\right)=0\\ \left(x+44\right)\left(x-33\right)=0\\ x=-44,33\end{array}\end{array}$

Since speed cannot take negative values, therefore consider only the positive value of $x$ i.e. $x=33\mathrm{km}/\mathrm{h}$.

Hence, the average speed of passenger train is $33\mathrm{km}/\mathrm{h}$.

The average speed of the express train is

$\begin{array}{rcl}\left(x+11\right)\mathrm{km}/\mathrm{h}& =& \left(33+11\right)\mathrm{km}/\mathrm{h}\\ & =& 44\mathrm{km}/\mathrm{h}\end{array}$

Hence, the average speed of the passenger train is $33\mathrm{km}/\mathrm{h}$ and the average speed of the express train is $44\mathrm{km}/\mathrm{h}$.