Question

The distance between two places is $900km$. An ordinary express train takes $5hours$ more than a superfast train to cover this distance. If the speed of the superfast express train is $15km/hr$ more than that of the ordinary express train, find the speeds of the train.

Answer 1

Step 1: Given data

Let the speed of the ordinary train be $xkm/hr$.

Distance between two places is $900km$.

Time taken to cover the distance by ordinary train will be $t=\frac{dis\tan ce}{speed}$.

$\Rightarrow t=\frac{900}{x}$

Speed of the superfast express train is $15km/hr$ more than that of the ordinary express train.

Time taken to cover the same distance by superfast express train will be,

$\Rightarrow T=\frac{900}{x+15}$

Step 2: Determining the quadratic equation

According to the question,

An ordinary express train takes $5hours$ more than a superfast train to cover the same distance.

$$\begin{gathered} \Rightarrow t-T=5 \\ \Rightarrow \frac{900}{x}-\frac{900}{x+15}=5 \\ \Rightarrow \frac{900\left(x+15\right)-900x}{x\left(x+15\right)}=5 \\ \Rightarrow \frac{900x+13500-900x}{x\left(x+15\right)}=5 \end{gathered}$$

After simplification,

$$\begin{gathered} \Rightarrow 13500=5x\left(x+15\right) \\ \Rightarrow 5x^2+75x-13500=0 \\ \Rightarrow 5\left(x^2+15x-2700\right)=0 \\ \Rightarrow x^2+15x-2700=0 \end{gathered}$$

Step 3: Determining the speeds of the train

To solve the obtained equation by split the middle term method, we choose two numbers such as their sum is equal to $b$ and product is equal to $a\times c$.

$$\begin{gathered} \Rightarrow x^2+\left(60-45\right)x-2700=0 \\ \Rightarrow x^2+60x-45x-2700=0 \\ \Rightarrow x\left(x+60\right)-45\left(x+60\right)=0 \\ \Rightarrow \left(x+60\right)\left(x-45\right)=0 \end{gathered}$$

After simplification,

$x=-60,45$

Here, $x=-60$ is not considerable as speed cannot be negative.

Therefore, speed of an ordinary express train is $45km/hr$.

Speed of superfast express train is $15km/hr$ more than that of the ordinary express train.

Therefore, speed of an superfast express train is $60km/hr$.