Question

Anuj, Varun, Vansh can walk at the rates 3, 4, and 5 km an hour respectively. They start from a place at 1, 2, 3 hours respectively. When Varun catches Anuj, Varun sends him back with a message to Vansh. At what time does Vansh get the message?
A) 5hrs 15 mins
B) 3 hrs 30 mins
C) 6 hrs 15 mins
D) 5 hrs 45 mins

Answer 1

Hint:
You should be able to calculate their positions at different intervals of time. First, calculate the distance travelled by Anuj in the first two hours. Then calculate the extra distance covered by Varun to catch Arun. Calculate distance travelled by Vansh. Then, calculate the distance between Vansh and Anuj. Then add the total time taken to convey the message.

Complete step by step solution:
After 2 hrs, distance travelled by Anuj \[ = 3 \times 1 = 3km\]
For Varun, time \[T\] when Varun catches Anuj \[4T - 3T = 3\] as the extra distance to be covered by Varun \[ = 3km\]
\[ \Rightarrow T = 3km\]
After Varun Catches Anuj, 3hrs have passed and during this time Vansh started walking and has been walking for 2 hrs
Thus, Distance travelled by Vansh \[ = 5 \times 2 = 10km\]
Now, Distance travelled by Anuj when he was caught by Varun \[ = 12km\]
Now, distance between Vansh and Anuj \[ = 12 - 10 = 2km\]
Time \[{T_1}\] taken by Anuj to meet Vansh
\[
   = 5{T_1} + 3{T_1} = 2km \\
   \Rightarrow {T_1} = \dfrac{2}{8} = \dfrac{1}{4} = 15\min \\
 \]
Total Time taken to Convey the message
\[ = 2hrs\] (Varun Starts after this Time) \[ + 3hrs\] (Varun Catches Anuj in this Time) \[ + 15\min \] (Anuj meet Vansh in this Time) \[ = 5hrs15\min \]

Therefore, the correct answer is (A).

Note:
Distance is a scalar quantity. Distance is the product of speed and time. Since distance is a scalar quantity it can be added just like time. When an environment with more than one people is given, try to visualise the distance covered by them using simple graphs. It will help a lot while solving distance-time problems.