Question

Bombay Express left Delhi for Bombay at 14:30 hours travelling at a speed of 60 km/hr and Rajdhani Express left Delhi for Bombay on the same day at a speed of 80 km/hr. How far from Delhi will the two trains meet?
(a) 120km
(b) 360km
(c) 480km
(d) 500km

Answer 1

Hint: Assume the distance from Delhi to the point at which they meet as ‘x’. Now, assume the time taken by the trains to cover this distance are \[{{t}_{1}}\] and \[{{t}_{2}}\]. Apply the formula: - distance = speed \[\times \] time to form a relation between \[{{t}_{1}}\] and \[{{t}_{2}}\]. Now, subtract \[{{t}_{2}}\] from \[{{t}_{1}}\] and equate it with 2 to form another relation between \[{{t}_{1}}\] and \[{{t}_{2}}\]. Solve for the value of \[{{t}_{1}}\] and equate the obtained values in the expression of distance x to get the answer.

Complete step-by-step answer:
Here, let us assume the distance of the point from Delhi at which the two trains meet is x. We are assuming that this distance covered by Bombay Express takes time \[{{t}_{1}}\] and Rajdhani Express takes time \[{{t}_{2}}\]. Assuming that there is no stoppage in between, we have,

In the above representation point D is Delhi, point B is Bombay and point C is where the two trains meet. So, we have,
1. Considering Bombay Express: -
Speed = 60 km/hr
Time = \[{{t}_{1}}\]
Distance = x
Applying the formula: - distance = speed \[\times \] time, we get,
\[\Rightarrow x=60{{t}_{1}}\] - (i)
2. Considering Rajdhani Express: -
Speed = 80 km/hr
Time = \[{{t}_{2}}\]
Distance = x
Applying the formula: - distance = speed \[\times \] time, we get,
\[\Rightarrow x=80{{t}_{2}}\] - (ii)
From equation (i) and (ii), we get,
\[\begin{align}
  & \Rightarrow 60{{t}_{1}}=80{{t}_{2}} \\
 & \Rightarrow 3{{t}_{1}}=4{{t}_{2}} \\
\end{align}\]
\[\Rightarrow {{t}_{2}}=\dfrac{3}{4}{{t}_{1}}\] - (iii)
Now, it is given that Bombay Express started the journey at 14:30 hours and Rajdhani Express started the same journey at 16:30 hours. That means Rajdhani Express took 2 hours less to travel the same distance x. So, we have,
\[\Rightarrow {{t}_{2}}={{t}_{1}}-2\] - (iv)
Now, substituting the value of \[{{t}_{2}}\] from equation (iii) in equation (iv), we get,
\[\begin{align}
  & \Rightarrow \dfrac{3}{4}{{t}_{1}}={{t}_{1}}-2 \\
 & \Rightarrow {{t}_{1}}-\dfrac{3}{4}{{t}_{1}}=2 \\
 & \Rightarrow \dfrac{1}{4}{{t}_{1}}=2 \\
\end{align}\]
\[\Rightarrow {{t}_{1}}=8\]hours
Therefore, substituting the value of \[{{t}_{1}}\] in equation (i), we get,
\[\Rightarrow x=8\times 60\]
\[\Rightarrow x=480\]km
Therefore, the two trains meet at a distance of 480km from Delhi.

So, the correct answer is “Option (c)”.

Note: One may note that we have found the value of \[{{t}_{1}}\] and substituted it in equation (i) to get the value of x. You may also obtain the value of \[{{t}_{2}}\] and substitute it in equation (ii) to get the value of x. The answer will be the same. Remember the speed – time formula to solve the question. It will be better if we will mark the points which are important to solve the question just like we did in the above solution.