Question 1027372
.
Distance between two stations X and Y is 220 km. Trains P and Q leave station X at 8 am and 9.51 am respectively 
at the speed of 25 km/hr and 20 km/hr respectively for journey towards Y. Train R leaves station Y at 11.30 am 
at a speed of 30 km/hr for journey towards X. When and where will P be at equal distance from Q and R ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



<pre>

 P: 8:00 am 25 km/h --->
 Q: 9:51 am 20 km/h --->                               <--- R: 11:30 am 30 km/h

    -|------------------------------------------------------|-

     X (0)                                                  Y  (220 km)


Since the trains start at different time, the whole problem for analyzing is non-linear.
We should analyze it step by step separately for different time intervals, as presented in my solution below.


(1)  At t1 = 9:51 am, the positions relative point A are

         P(t1) = 25 * 1{{{51/60}}} = 46.25 km;    (train P moved 1 hour and 51 minutes at the rate 25 km/h)

         Q(t1) = 0;

         R(t1) = 220 km.

         So, train P still did not get midpoint between Q and R, and we shall continue our analysis.



(2)  At t2 = 11:30 am, the positions relative point A are

         P(t2) = 25 * 3.5 = 87.5 km;         (train P moved 3.5 hours at the rate 25 km/h)

         Q(t2) = 20 * 1{{{39/60}}} = 33 km;  (train Q moved 1{{{39/60}}} hours at the rate 20 km/h)

         R(t2) = 220 km.

         So, train P still did not get midpoint between Q and R, and we shall continue our analysis.



(3)  After 11:30 am, the positions relative point A are  (here 't' is the time after 11:30 am)

         P(t) = 87.5 + 25*t kilometers;

         Q(t) = 33 + 20*t   kilometers;

         R(t) = 220 - 30*t kilometers.


     We want to have  

         P(t) - Q(t) = R(t) - P(t),  which is an equation for P(t) to be the midpoint between Q(t) and R(t).


     It gives us this equation

         (87.5 + 25*t) - (33 + 20*t) = (220 - 30*t) - (87.5 + 25*t).


     Simplify it step by step and find 't'

         2(87.5 + 25*t) = (220 - 30*t) + (33 + 20*t),

         175 + 50*t = 253 - 10*t,

         50t + 10t = 253 - 175,

             60t   = 78,

               t   = {{{78/60}}}  of an hour, or 1 hour and 18 minutes. 


Thus, train P will be at midpoint between trains Q and R in 1 hour and 18 minutes after 11:30 am, i.e. at 12:48 pm.


The location of train P will be  4 h 48 min * 25 km/h = {{{288*(25/60)}}} = 120 kilometers from point A.

(here 4 h 48 min is the travel time for train P from 8:00 am to 12:48 pm).


<U>ANSWER</U>.  Train P will be at midpoint between trains Q and R at 12:48 pm, i.e. 48 minutes after noon.

         The position of train P will be 120 km from point A at this time moment.
</pre>

Solved.