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 ?
Found 3 solutions by mananth, ikleyn, n2:
Answer by mananth(16949)   (Show Source):
You can put this solution on YOUR website! 
The time taken by P, Q, R to reach some point enroute is the same.
They start at different times and speeds
Equate time of P & Q
Equate time of P & r
and solve the equations
P &Q
(x+d)/25 = x/20 + 1 51/60
(x+d)/25 -x/20 = 37/20
20x+20d -25x = 37*20*25/20
-5x +20d = 925
/-5
x-4d= -185.......................(1)
P& R
(x+d)/25 = (220-x-2d)/30 +7/2
(x+d)/25 -(220-x-2d)/30 =7/2
30x+30d -25*220+25x+50d = 30*25*7/2
55x+80d =8125......................(2)
V1.00 x -4.00 d = -185.00 .............1
Total value
55.00 x + 80.00 d = 8125.00 .............2
Eliminate y
multiply (1)by 20.00
Multiply (2) by 1.00
20.00 x -80.00 d = -3700.00
55.00 x + 80.00 d = 8125.00
Add the two equations
75.00 x = 4425.00
/ 75.00
x = 59.00
plug value of x in (1)
1.00 x -4.00 d = -185.00
59.00 -4.00 d = -185.00
-4.00 d = -185.00 -59.00
-4.00 d = -244.00
d = 61.00
you can calculate real time easily
m.ananth@hotmail.ca
Answer by ikleyn(53742)  (Show Source):
You can put this solution on YOUR website! .
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 ?
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The solution in the post by @mananth is conceptually incorrect.
@mananth assumes from the very beginning that the time spent by trains to get the meeting point
is the same for all three trains.
But in reality it is not so, since the trains started at different time moments.
So, his assumption is inadequate to the problem.
See my correct solution below.
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 = 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 = 33 km; (train Q moved 1 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 =  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 =  = 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).
ANSWER. 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.
Solved correctly.
Answer by n2(78)  (Show Source):
You can put this solution on YOUR website! .
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 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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 = 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 = 33 km; (train Q moved 1 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 = 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 = = 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).
ANSWER. 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.
Solved.
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