Question

The speed of an ordinary train is x km per hr and that of an express train is \[\left( x+25 \right)\]km per hr. Find the time taken by each train to cover 300 km. If the ordinary train takes 2 hrs more than the express train. Calculate speed of express train.

Answer 1

Hint: In this question, we first need to calculate the time taken by each of the train to travel 300 km using the formula \[\text{speed}=\dfrac{\text{distance}}{\text{time}}\]. Now, we get the time taken by each of the trains and then add 2 hrs to the time taken by express train and equate it to the time taken by ordinary train. Now, on simplifying it further we get the value of x.

Complete step by step solution:
Now, as we already know that the relation between speed, time and distance is given by the formula
\[\text{speed}=\dfrac{\text{distance}}{\text{time}}\]
Now, let us assume that the time taken by ordinary train as t and the time taken by express train as T to travel 300 km
Now, form the given conditions we have speed of ordinary train as x and distance as 300 km
Now, from the above formula we have
\[\text{speed}=\dfrac{\text{distance}}{\text{time}}\]
Now, on substituting the respective values we get,
\[\Rightarrow x=\dfrac{300}{t}\]
Now, on rearranging the terms we get,
\[\Rightarrow t=\dfrac{300}{x}\]
Now, let us find the time taken by express train to travel 300 km given its speed as \[\left( x+25 \right)\]
Now, on substituting the respective values in the formula we get,
\[\Rightarrow x+25=\dfrac{300}{T}\]
Now, on rearranging the terms we get,
\[\Rightarrow T=\dfrac{300}{x+25}\]
Now, given that time taken by ordinary train is 2 hrs more than the time taken by express train
Now, this can be written as adding 2 hrs to time taken by express train is equal to time taken by ordinary train
\[\Rightarrow t=2+T\]
Now, on substituting the respective values in the above equation we get,
\[\Rightarrow \dfrac{300}{x}=2+\dfrac{300}{x+25}\]
Now, on rearranging the terms we get,
\[\Rightarrow \dfrac{300}{x}-\dfrac{300}{x+25}=2\]
Now, this can be further written in the simplified form as
\[\Rightarrow 300\left( \dfrac{x+25-x}{x\left( x+25 \right)} \right)=2\]
Now, on again rearranging the terms we get,
\[\Rightarrow 300\times 25=2\times x\times \left( x+25 \right)\]
Let us now divide with 2 on both sides and simplify further
\[\Rightarrow 150\times 25=x\left( x+25 \right)\]
Now, this can also be written as
\[\Rightarrow 75\times 50=x\left( x+25 \right)\]
Now, on comparing both the sides we get,
\[\therefore x=50\]
As we already know that speed of express train is given by
\[\Rightarrow \left( x+25 \right)\]
Now, on substituting the respective value of x we get,
\[\Rightarrow 50+25\]
Now, on further simplification we get,
\[\Rightarrow 75\]
Hence, the speed of the express train is 75 km per hr.

Note:
Instead of finding the time taken by each of the trains and then substituting in the given condition we can also solve it by writing the time in the condition in terms of speed and distance and then simplify further to get the speed directly. This method has fewer steps.
Here, while finding the speed of the express train instead of finding the value of x and then calculating \[x+25\]we can directly get the speed from the equation on comparing so that it decreases the number of steps.
It is important to note that the time taken by the ordinary train is more than the express train. Because while writing the equation if we consider the other way then we get the incorrect result.