document.write( "Question 1120244: Alice and Bob started to walk towards each other’s home and then back to theirs, with steady speeds. Alice passed by a bus station at 25 m away from her home, while at the same time Bob was passing by an abandoned old car. Afterwards, they met at 55 m away from Bob’s home and then they met again at 85 m from Alice’s home. What is the distance between the bus station and the abandoned car? \n" ); document.write( "

Algebra.Com's Answer #735909 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Let  S  be the distance between their homes, in meters.\r\n" );
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document.write( "Let    be Alice's rate, and   be Bob's rate.\r\n" );
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document.write( "Let    be the time after their start to their first meeting.\r\n" );
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document.write( "Let    be the time after their start to their second meeting.\r\n" );
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document.write( "     Notice I follow your  (your ?)  designations in your  (your ?)  set up in\r\n" );
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document.write( "          https://math.stackexchange.com/questions/2859310/distances-and-speeds\r\n" );
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document.write( "          https://math.stackexchange.com/questions/2859310/distances-and-speeds\r\n" );
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document.write( "Then you have \r\n" );
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document.write( "     = S - 55;      = 55;           (1)\r\n" );
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document.write( "     = 2S - 85;      = S + 85.      (2)\r\n" );
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document.write( "From (1) you have   = ;               (3)\r\n" );
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document.write( "From (2) you have   = ,              (4)\r\n" );
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document.write( "which gives you an equation for S\r\n" );
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document.write( "     = .\r\n" );
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document.write( "Simplify and solve for S:\r\n" );
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document.write( "    (S-55)*(S+85) = (2S-85)*55\r\n" );
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document.write( "    S^2 -55*S + 85*S - 55*85 = 110*S - 55*85\r\n" );
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document.write( "    S^2 - 55*S + 85*S - 110*S = 0\r\n" );
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document.write( "    S^2 - 80*S = 0\r\n" );
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document.write( "    S*(S-80) = 0  ====>  The only positive root is   S = 80.\r\n" );
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document.write( "Thus we found the distance between the homes: it is 80 meters.\r\n" );
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document.write( "Having this known,  you van calculate the ratio of their rates from (3)\r\n" );
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document.write( "     =  =  =    \r\n" );
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document.write( "and hence calculate the distance \"x\" from the Bob's home to the abandoned car from the proportion\r\n" );
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document.write( "     =   ======>  x =  = 55 m.\r\n" );
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document.write( "Then the distance under the question is  80 - 25 - 55 meters = 0 meters.\r\n" );
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