Question

A train covered a certain distance at a uniform speed. If the train would have been $10km/hr$ faster, it would have taken $2hr$ less than the scheduled time. And, if the train were slower by $10km/hr$ faster, it would have taken $3hr$ more than the scheduled time. Find the distance covered by the train.

Answer 1

Step 1: Obtain a system of linear equations with the given details:

Let us assume the uniform speed of the train to be $x$ km/hr.

Let the time taken to travel the given distance be $t$ hours.

Then, the distance can be calculated as follows:

$\begin{array}{rcl}\text{Distance}& =& \text{Speed}\times \mathrm{time}\\ & =& xt\end{array}$.

It is given that if the train would have been $10km/hr$ faster, it would have taken $2hr$ less than the scheduled time.

$$\begin{gathered} \Rightarrow \left(x+10\right)\left(t-2\right)=xt \\ \Rightarrow xt-2x+10t-20=xt \\ \Rightarrow -2x+10t-20=0.......\left(1\right) \end{gathered}$$

Also, if the train was slower by $10km/hr$, it would have taken $3hr$ more than the scheduled time.

$$\begin{gathered} \Rightarrow \left(x-10\right)\left(t+3\right)=xt \\ \Rightarrow xt+3x-10t-30=xt \\ \Rightarrow 3x-10t-30=0.......\left(2\right) \end{gathered}$$

Hence, we get the system of equations as follows:

$\begin{array}{rcl}-2x+10t-20& =& 0\\ 3x-10t-30& =& 0\end{array}$

Step 2: Solve the system of linear equations using the elimination method.

Now add equations (1) and (2) to get:

$\begin{array}{rcl}\left(-2x+10t\right)+\left(3x-10t\right)& =& 20+30\\ & \Rightarrow & x=50\end{array}$

Now substitute the value of $x$ in equation (1) to get:

$\begin{array}{rcl}& \Rightarrow & -2x+10t-20=0\\ & \Rightarrow & -2\left(50\right)+10t-20=0\\ & \Rightarrow & -100+10t-20=0\\ & \Rightarrow & -120+10t=0\\ & \Rightarrow & 10t=120\\ & \Rightarrow & t=12\end{array}$

Now, the distance will be:

$\begin{array}{rcl}\text{Distance}& =& \text{Speed}\times \mathrm{time}\\ & =& xt\\ & =& 50\times 12\\ & =& 600\text{km}\end{array}$

Hence, the distance covered by the train is $600\text{km}$.