Question

A train M leaves Meerut at 5 am and reaches Delhi at 9 am. Another train N leaves Delhi at 7 am and reaches Meerut at 10:30 am. At what times does the two trains cross each other?
 $ (a){\text{ }} $ 7:36 am
 $ (b){\text{ }} $ 7:56 am
 $ (c){\text{ }} $ 8:00 am
 $ (d){\text{ }} $ 8:26 am

Answer 1

Hint:In the above given question, calculate the number of hours taken to cover the specific distance and then by using the conditions provided in the question, formulate the equation and substitute the values to obtain the solution.

Complete step-by-step answer:
Let us assume the distance between Meerut and Delhi be $ x $ and let the trains meet $ y $ hours after 7 am.
Since, train M leaves Meerut at 5 am and reaches Delhi at 9 am, that is train M takes 4 hours to cover $ x $ km.
Also, train N leaves Delhi at 7 am and reaches Meerut at 10:30 am, that is train N takes 3.5 hours to cover $ x $ km.
Now, speed of train M $ = \dfrac{x}{4}km\backslash r $ .
Speed of train N $ = \dfrac{{2x}}{7}km\backslash hr $ .
According to the conditions given in the question,
Distance $ x $ = Distance covered by M in $ (y + 2){\text{ }}hrs $ + Distance covered in $ y{\text{ }}hrs $ …(1)
The distance covered by train M in $ (y + 2){\text{ }}hrs $ $ = \dfrac{x}{4} \times (y + 2) $ …(2)
The distance covered by train N in $ y{\text{ }}hrs $ $ = \dfrac{{2x}}{7} \times y $ …(3)
After putting the value from equation (2) and (3) in the equation (1), we get
 $ \Rightarrow x = \dfrac{x}{4}(y + 2) + \dfrac{{2x}}{7}(y) $
 $ \Rightarrow x = x\left( {\dfrac{1}{4}(y + 2) + \dfrac{1}{7}(y)} \right) $
 $ \Rightarrow \dfrac{1}{4}(y + 2) + \dfrac{1}{7}(y) = 1 $
 $ \Rightarrow \dfrac{{y + 2}}{4} + \dfrac{y}{7} = 1 $
\[ \Rightarrow \dfrac{{7y + 14 + 8y}}{{28}} = 1\]
\[ \Rightarrow \dfrac{{15y + 14}}{{28}} = 1\]
\[ \Rightarrow 15y + 14 = 28\]
\[ \Rightarrow 15y = 28 - 14\]
\[ \Rightarrow y = \dfrac{{14}}{{15}}\]hours
Now, we know that 1 hour = 60 minutes.
Therefore, we get,
\[y = \dfrac{{14}}{{15}} \times 60\min \]
 $ \therefore y = 56\min $
Hence, the train M and train N meet at 7:00 + 0.56 = 7:56 am.
So, the correct answer is the option $ (b) $ .

Note: Try to use the values in the fractional form as it simplifies the calculations. Also, make sure that you make the respective conversions when required, from hours to minutes and vice versa. That is, all the quantities belong to the same unit system.