Question

A train left point A at noon sharp. Two hours later another train started from point A in the same direction. It overtook the first train at 8 p.m. Find the average speeds of the trains, if the sum of their average speeds is 70 km/h.

Answer 1

Hint: To do this question, we will first assume the speeds of the trains to be x and y. Then we will keep their sum equal to 70 km/h as it is given in the question. Then, we will see how much time the two trains traveled till 8:00 pm. Then using that, we will keep the distance they travelled in that time equal. Using that we will have another equation in x and y. Thus, solving both those equations we will find the values of x and y and hence we will get the required answer.

Complete step by step answer:
Here, we have been given two trains and the sum of their average speeds to be 70 km/h. we have been asked to find their individual speeds.
For that, let us first consider the speed of the first train to be ‘x’ km/h and the speed of the second train to be ‘y’ km/h. Now, we have been given their sum to be 70 km/h. Thus, we can say that:
 $ x+y=70 $ …..(i)
Now, we have been given that the first train leaves at noon sharp. We know that the time at noon sharp is 12:00 pm. We have also been given that the second train leaves after two hours of the departure of the first train. Thus, we can conclude that the second train leaves at 2:00 pm.
Now, we have been given that the second train overtakes the first train at 8:00 pm.
Hence, we can say that the distance traveled by the first train till 8:00 pm is equal to the distance traveled by the second train till the same time.
Now, as given to us in the question, we know that the first train departed at 12:00 pm. Hence, till 8:00 pm, it traveled for 8 hours. Similarly, we know that the second train departed at 2:00 pm. Hence, till 8:00 pm, it traveled for 6 hours.
Now, we know that the relation between speed, time and distance is given as:
 $ \text{speed}=\dfrac{\text{distance}}{\text{time}} $
This gives the distance in terms of speed and time as:
 $ \text{distance=speed}\times \text{time} $
Thus, from this formula, the distance travelled by the first train is given as follows:
Time for which the first train travelled = 8 hours
Speed of the first train = x km/h
Hence, distance travelled by the first train is given as:
 $ \begin{align}
  & 8\times x \\
 & \Rightarrow 8x \\
\end{align} $
Similarly, distance travelled by the second train is given as:
 $ \begin{align}
  & 6\times y \\
 & \Rightarrow 6y \\
\end{align} $
Now, as mentioned above, these two distances are equal. Thus, we can say that:
 $ 8x=6y $
Now, writing ‘y’ in terms of ‘x’ we get:
 $ \begin{align}
  & 8x=6y \\
 & \Rightarrow y=\dfrac{8x}{6} \\
 & \Rightarrow y=\dfrac{4x}{3} \\
\end{align} $
Now, putting the value of y in equation (i) we get:
 $ \begin{align}
  & x+y=70 \\
 & \Rightarrow x+\dfrac{4x}{3}=70 \\
 & \Rightarrow \dfrac{3x+4x}{3}=70 \\
 & \Rightarrow \dfrac{7x}{3}=70 \\
 & \therefore x=30 \\
\end{align} $
Thus, the value of x, i.e. the speed of the first train is 30 km/h.
Now, putting the value of x back in equation (i), we get the value of y as:
 $ \begin{align}
  & x+y=70 \\
 & \Rightarrow 30+y=70 \\
 & \therefore y=40 \\
\end{align} $
Thus, the value of y, i.e. the speed of the second train is 40 km/h.
Hence, the speeds of the given trains are 30 km/h and 40 km/h for the first and the second train respectively.

Note:
Here, we have used the substitution method to solve the equations but we could have used any other method according to our convenience. Also, whatever method we use, we should be careful as any mistake in solving the equations will result in the wrong values of x and y and hence the wrong answer.