Question

A train is moving at an average speed of 40 kmph, reaches its destination on time. When its average speed reduces to 5 kmph, then it reaches its destination 15 minutes late. The distance traveled by train is
(A) 30 km
(B) 40 km
(C) 70 km
(D) 80 km

Answer 1

Hint: Assume that the distance traveled by train to reach its destination be x km. Here, we have two cases. In the first case, the average speed of the train is 40 km/hr. Now, calculate the time using the formula \[\text{Time=}\dfrac{\text{Distance}}{\text{Average}\,\text{Speed}}\] . In the second case, the average speed is decreased by 5 kmph. So, the average speed, in this case, is 35 km/hr. The distance to be traveled by train to reach its destination is x km. Now, calculate the time, in this case, using the formula, \[\text{Time=}\dfrac{\text{Distance}}{\text{Average}\,\text{Speed}}\] . It is also given that the train reaches its destination 15 minutes late. It means that the time taken by the train in the second case is 15 minutes more than the time taken by the train in the first case. So, \[\dfrac{x}{40}hr+15\min =\dfrac{x}{35}hr\] . Now, convert the term which is in minutes into hours using the relation \[1\min =\dfrac{1}{60}hr\] and solve it further to get the value of x.

Complete step by step solution:
First of all, let us assume that the distance traveled by train to reach its destination be x km.
Now, we have two cases.
In the \[{{1}^{st}}\] case, it is given that the train reaches its destination on time and the average speed of the train is 40 km/hr.
The distance to be covered by the train to reach its destination = x km ……………………………(1)
The speed of the train = 40 km/hr ……………………………..(2)
We know the formula, \[\text{Time=}\dfrac{\text{Distance}}{\text{Average}\,\text{Speed}}\] …………………………(3)
Using the formula shown in equation (3) to get the time required by the train to reach its destination.
The time taken by the train to reach its destination = \[\dfrac{x}{40}\] hrs …………………………(4)
In the \[{{2}^{nd}}\] case, it is given that the train reaches its destination on time and the average speed of the train reduces to 5 kmph.
The distance to be covered by the train to reach its destination = x km ……………………………(5)
The speed of the train = \[\left( 40-5 \right)\] km/hr = 35 km/hr ……………………………..(6)
Using the formula shown in equation (3) to get the time required by the train to reach its destination.
The time taken by the train to reach its destination = \[\dfrac{x}{35}\] hrs …………………………(7)
It is also given that the train reaches its destination 15 minutes late. It means that the time taken by the train in the second case is 15 minutes more than the time taken by the train in the first case.
From equation (4) and equation (7), we have the time taken by the train to reach its destination in the first case and the second case respectively.
So, \[\dfrac{x}{40}hr+15\min =\dfrac{x}{35}hr\] ………………………………..(8)
We know that \[60\min =1hr\] .
\[\Rightarrow 60\min =1hr\]
\[\Rightarrow 1\min =\dfrac{1}{60}hr\] ………………………………………(9)
From equation (8) and equation (9), we get
 \[\begin{align}
  & \Rightarrow \dfrac{x}{40}hr+\dfrac{15}{60}hr=\dfrac{x}{35}hr \\
 & \Rightarrow \dfrac{1}{4}=\dfrac{x}{35}-\dfrac{x}{40} \\
 & \Rightarrow \dfrac{1}{4}=\dfrac{40x-35x}{40\times 35} \\
 & \Rightarrow \dfrac{1}{4}=\dfrac{5x}{40\times 35} \\
 & \Rightarrow \dfrac{40\times 35}{4\times 5}=x \\
 & \Rightarrow 10\times 7=x \\
 & \Rightarrow 70=x \\
\end{align}\]
Therefore, the distance to be traveled by train to reach its destination is 70 km.
Hence, the correct option is (C).

Note: In this question, one might do a silly mistake while solving the equation, \[\dfrac{x}{40}hr+15\min =\dfrac{x}{35}hr\] . Here, one might miss the point that a term in the LHS of the equation is in minutes. So, we have to convert that term into hours using the relation \[1\min =\dfrac{1}{60}hr\] .