# Time, Speed and Distance. Moderate Level Question.

Ravi, who lives in the countryside, caught a train for home earlier than usual day. His wife normally drives to the station to meet him. But that day he set out on foot from the station(as he had reached the station earlier than usual) to meet his wife on the way. He reached home 12 minutes earlier than he would have reached, had he waited at the station for his wife. The car travels at a uniform speed, which is 5 times Ravi's speed on foot. Ravi reached home at exactly 6'O clock. At what time would he have reached home if his wife forewarned of his plan, had met him at the station?

My Attempt:

Let the distance between the station and the home be $D$.

Let the speed of the man be $x$.

Let the speed of the car be $5x$.

On normal days(usually):

Time taken by the car to reach the station and come back home = $t$

$$t=\frac{2D}{5x} \tag 1$$

On the particular day mentioned in the question:

The man starts walking and meets the car somewhere between the station and the home.

Let the time at which the man and the car meet be $t_1$.

Therefore,

$t_1=\frac{D}{x+5x}=\frac{D}{6x}$

Time required for the car to go back from the meeting point to home =$t_2(say)=t_1=\frac{D}{6x}$

Therefore, time required for the car to come to the meeting point and go back home=$$t'=t_1+t_2=\frac{D}{6x}+\frac{D}{6x}=\frac{D}{3x}$$

ATQ

$$t-t'=12$$

$$\frac{2D}{5x}-\frac{D}{3x}=12$$

$$\frac{D}{x}=180$$

Therefore,

$$t=\frac{2}{5}*180=72 min$$

$$t'=t-12=72-12=60 min$$

I have found out the time taken by the car to reach the station and come back home (on usual days) and the time required for the car to come to the meeting point and go back home(on the specific day mentioned in the question). However, I an unable to proceed after that.

Consider this in the formulation of your equations. The car journey has saved $6$ minutess in each direction due to meeting Ravi on the way. Because Ravi walks $5$ times slower than the car, he has been walking for $30$ minutes. If the car had left $30$ minutes earlier than normal it would have met Ravi at the station. So the time of arrival will be 30 minutes sooner than normal, where normal is the same as if Ravi had waited at the station. $6:12$pm minus $30$ minutes is $5:42$ pm.