Question

A train \[X\] starts from New Delhi at \[4\] pm and reaches Ghaziabad at \[5\] pm. While another train \[Y\] starts from Ghaziabad at \[4\] pm and reaches New Delhi at 5:30 pm. At what time will the two trains cross each other?
A. \[4:30\]
B. \[4:24\]
C. \[4:36\]
D. \[4:41\]

Answer 1

Hint: Distance travelled by both the trains \[X\] and \[Y\] is the same but the time taken to travel is different. To get the time when the two trains cross each other, the amount of distances travelled by both the trains must be added to give a total distance between New Delhi and Ghaziabad. So, we assume total distance as \[D\] . Find time in hours for both trains. Apply speed, distance and time formula to get the distance covered by both trains.

Complete step-by-step answer:
Let us consider the distance between New Delhi and Ghaziabad to be \[D\] km.
We have been given that the train \[X\] started at 4 pm and reached its destination at 5 pm. So, we can infer that the train travelled \[D\] km in 1 hour (4 pm to 5 pm).
We know that the speed, distance and time formula is given as,
\[\text{distance}=\text{speed}\times \text{time}\]
Now let us find the speed of the train \[X\]. So, for the train \[X\], we know that the distance is \[D\] km and the time is 1 hour. So, from the formula, we get, $ \text{speed=}\dfrac{\text{distance}}{\text{time}} $ ,
So the speed of the train \[X\] will be \[\dfrac{D}{1}=D\] km/hr.
Similarly, we have been given that train \[Y\] started at 4 pm from Ghaziabad and reached New Delhi at 5 : 30 pm.
So, we can say that the distance travelled by train \[Y\] will be\[D\]km and the time taken will be 1 hour and 30 minutes. And we know that 30 minutes is equal to $ \dfrac{1}{2}hr $ , so we can write the time taken by train \[Y\] as $ \left( 1+\dfrac{1}{2} \right)hr=\left( \dfrac{2+1}{2} \right)hr=\dfrac{3}{2}hr $ . Therefore, the train \[Y\] travelled \[D\]km in \[\dfrac{3}{2}\]hours.
So, we can write the speed of the train \[Y\] as \[\text{speed=}\dfrac{\text{distance}}{\text{time}}=\dfrac{D}{\dfrac{3}{2}}=\dfrac{2D}{3}km/hr\]
Now, let the two trains \[X\] and \[Y\] cross each other at the time interval of \[x\] hours from \[4\]pm.
We can see the situation in the following figure.

Now, using the formula, \[\text{distance}=\text{speed}\times \text{time}\], we can write,
The distance travelled by train \[X\] in \[x\] hours will be equal to \[=x\times D\] km
The distance travelled by train \[Y\] in \[x\] hours will be equal to \[=x\times \dfrac{2D}{3}\] km
As it is understood, when the two trains cross each other the sum of distances travelled by both the trains will be equal to the distance between New Delhi and Ghaziabad. So, we can write,
\[\left( x\times D \right)+\left( x\times \dfrac{2D}{3} \right)=D\]
Dividing by \[D\] on both sides we have,
\[x+\dfrac{2x}{3}=1\]
On taking the LCM on the LHS, we get,
\[\dfrac{5x}{3}=1\]
On transposing $ \dfrac{5}{3} $ from LHS to RHS, we get,
\[x=\dfrac{3}{5}\]hours
We can express the time in minutes as,
\[x=\dfrac{3}{5}\times 60=36\] minutes
We have calculated that the two trains cross each other after an interval of \[36\] minutes from \[4\] pm. So, the time at which they will cross each other is \[4:36\] pm.
Hence, the correct option is C.

Note: In the speed, distance and time problem remember and apply the formula correctly and in this particular problem, it is better to draw a figure representing the given conditions so that the question will be easier to solve. Also, remember that all the values must be in a single unit only else it would lead to errors in the answer. In this question, the distance was the same, but in some questions, the time of journey would be the same and we would be asked to find the distance when they would meet. In that case, we can assume total time as T and then equate the sum of time taken for both trains.