Algebra.Com's Answer #853949 by ikleyn(53742)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "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
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document.write( "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
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document.write( "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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document.write( " The solution in the post by @mananth is conceptually incorrect.\r
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document.write( " @mananth assumes from the very beginning that the time spent by trains to get the meeting point \r
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document.write( " is the same for all three trains.\r
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document.write( " But in reality it is not so, since the trains started at different time moments.\r
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document.write( " So, his assumption is inadequate to the problem.\r
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document.write( " See my correct solution below.\r
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document.write( " P: 8:00 am 25 km/h --->\r\n" );
document.write( " Q: 9:51 am 20 km/h ---> <--- R: 11:30 am 30 km/h\r\n" );
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document.write( " X (0) Y (220 km)\r\n" );
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document.write( "Since the trains start at different time, the whole problem for analyzing is non-linear.\r\n" );
document.write( "We should analyze it step by step separately for different time intervals, as presented in my solution below.\r\n" );
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document.write( "(1) At t1 = 9:51 am, the positions relative point A are\r\n" );
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document.write( " P(t1) = 25 * 1 = 46.25 km; (train P moved 1 hour and 51 minutes at the rate 25 km/h)\r\n" );
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document.write( " Q(t1) = 0;\r\n" );
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document.write( " R(t1) = 220 km.\r\n" );
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document.write( " So, train P still did not get midpoint between Q and R, and we shall continue our analysis.\r\n" );
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document.write( "(2) At t2 = 11:30 am, the positions relative point A are\r\n" );
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document.write( " P(t2) = 25 * 3.5 = 87.5 km; (train P moved 3.5 hours at the rate 25 km/h)\r\n" );
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document.write( " Q(t2) = 20 * 1 = 33 km; (train Q moved 1 hours at the rate 20 km/h)\r\n" );
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document.write( " R(t2) = 220 km.\r\n" );
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document.write( " So, train P still did not get midpoint between Q and R, and we shall continue our analysis.\r\n" );
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document.write( "(3) After 11:30 am, the positions relative point A are (here 't' is the time after 11:30 am)\r\n" );
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document.write( " P(t) = 87.5 + 25*t kilometers;\r\n" );
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document.write( " Q(t) = 33 + 20*t kilometers;\r\n" );
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document.write( " R(t) = 220 - 30*t kilometers.\r\n" );
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document.write( " We want to have \r\n" );
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document.write( " 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).\r\n" );
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document.write( " It gives us this equation\r\n" );
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document.write( " (87.5 + 25*t) - (33 + 20*t) = (220 - 30*t) - (87.5 + 25*t).\r\n" );
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document.write( " Simplify it step by step and find 't'\r\n" );
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document.write( " 2(87.5 + 25*t) = (220 - 30*t) + (33 + 20*t),\r\n" );
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document.write( " 175 + 50*t = 253 - 10*t,\r\n" );
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document.write( " 50t + 10t = 253 - 175,\r\n" );
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document.write( " 60t = 78,\r\n" );
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document.write( " t =  of an hour, or 1 hour and 18 minutes. \r\n" );
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document.write( "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.\r\n" );
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document.write( "The location of train P will be 4 h 48 min * 25 km/h =  = 120 kilometers from point A.\r\n" );
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document.write( "(here 4 h 48 min is the travel time for train P from 8:00 am to 12:48 pm).\r\n" );
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document.write( "ANSWER. Train P will be at midpoint between trains Q and R at 12:48 pm, i.e. 48 minutes after noon.\r\n" );
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document.write( " The position of train P will be 120 km from point A at this time moment.\r\n" );
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document.write( "Solved correctly.\r
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