document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207197>Question 1207197</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi\r<BR>\n" );
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document.write( "Four friends Aisha Ben Cathy and Daniel shared lunch equally. Aisha paid 3/5 of the total amount Ben and Cathy paid. Cathy paid $10 more than Ben. Daniel repaid $24.00 to Aisha and some money to Ben and Cathy.<BR>\n" );
document.write( "How much did Daniel repay to Ben and Cathy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844934 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52778)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844934\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Four friends, Aisha, Ben, Cathy, and Daniel shared lunch equally. <BR>\n" );
document.write( "Aisha paid 3/5 of the total amount Ben and Cathy paid. <BR>\n" );
document.write( "Cathy paid $10 more than Ben. <BR>\n" );
document.write( "Daniel repaid $24.00 to Aisha and some money to Ben and Cathy.<BR>\n" );
document.write( "How much did Daniel repay to Ben and Cathy<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "It is clear that the major task in solution to this problem is to make setup.\r\n" );
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document.write( "I will make setup in several steps. \r\n" );
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document.write( "       At step 1, I introduce variables in general form.\r\n" );
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document.write( "We have a network (like a circuit). You can imagine it as a graph with vertices (nodes) A, B, C and D and edges, \r\n" );
document.write( "connecting the nodes. The nodes represent the people; the edges represent cash flows between people.\r\n" );
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document.write( "Aisha (node A) pays A dollars to the restaurant; Ben    (node B) pays B dollars to the restaurant;\r\n" );
document.write( "Cathy (node C) pays C dollars to the restaurant; Daniel (node D) pays D dollars to the restaurant.\r\n" );
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document.write( "Edge Ab represents cash flow from A to B; edge Ac represents cash flow from A to C; \r\n" );
document.write( ". . . and so on . . .                     edge Dc represents cash flow from D to C.\r\n" );
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document.write( "There is no cash flow from A to A; no cash flow from B to B; . . . no cash flow from D to D.\r\n" );
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document.write( "So, there are 3*4 = 12 edges Ab, Ac, . . . , Dc. \r\n" );
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document.write( "       At step 2, I will give the values to some variables, based on given data.\r\n" );
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document.write( "A,   Ab=0,  Ac=0, Ad=0                               \r\n" );
document.write( "B,   Ba=0,  Bc=0, Bd=0                               \r\n" );
document.write( "C,   Ca=0,  Cb=0, Cd=0                               \r\n" );
document.write( "D=0, Da=24, Db,   Dc                                 \r\n" );
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document.write( "As you see from this table, many variables are 0 (zero). Non-zero values are A, B, C, Da=24, Db and Dc.\r\n" );
document.write( "My next task is to write equations for non-zero unknowns.\r\n" );
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document.write( "       At step 3, I will write equations for non-zero variables\r\n" );
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document.write( "    A = 0.6B + 0.6C  (1)     A - 24 = x             (4)   part for A\r\n" );
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document.write( "                             B - Db = x             (5)   part for B\r\n" );
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document.write( "    C = B + 10       (2)     C - Dc = x             (6)   part for C\r\n" );
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document.write( "    Da = 24          (3)     24 + Db + Dc = x       (7)   part for D\r\n" );
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document.write( "Equations follow to given information LITERALLY, so I will not explain their origin and meaning.\r\n" );
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document.write( "Here x is the amount representing the PART of everyone.\r\n" );
document.write( "According to the problem, this part is the same for all 4 participants.\r\n" );
document.write( "As you see, there are 7 unknown and 7 independent equations - so, there is a hope that the problem is solvable to the end.\r\n" );
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document.write( "These equations, written in one column, are presented below one more time. \r\n" );
document.write( "Equation (3) is excluded, since it is just used in (7).\r\n" );
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document.write( "    A = 0.6B + 0.6C          (8)\r\n" );
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document.write( "    A - 24 = x               (9)\r\n" );
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document.write( "    B - Db = x              (10)\r\n" );
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document.write( "    C = B + 10              (11)\r\n" );
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document.write( "    C - Dc = x              (12)\r\n" );
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document.write( "    24 + Db + Dc = x        (13)\r\n" );
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document.write( "So, there are 6 equations for 6 unknowns.\r\n" );
document.write( "The unknown A can be easily excluded together with equation (8), by replacing A = 0.6B + 0.6C in equation (9).\r\n" );
document.write( "Thus we arrive to 5 equations in 5 unknowns\r\n" );
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document.write( "0.6B + 0.6C - 24 = x        (14)\r\n" );
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document.write( "B - Db = x                  (15)\r\n" );
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document.write( "C = B + 10                  (16)\r\n" );
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document.write( "C - Dc = x                  (17)       \r\n" );
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document.write( "24 + Db + Dc = x            (18)               \r\n" );
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document.write( "Now I rewrite these equations in canonical form of a system of linear equations\r\n" );
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document.write( "    0.6B + 0.6C             - x =  24    (19)\r\n" );
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document.write( "    B            - dB       - x =   0    (20)\r\n" );
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document.write( "    B    -    C                 = -10    (21)\r\n" );
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document.write( "              C -        dC - x =   0    (22)\r\n" );
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document.write( "                   dB  + dC - x = -24    (23)\r\n" );
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document.write( "At this point, my setup is complete.\r\n" );
document.write( "Probably, this system can be solved manually, step by step.\r\n" );
document.write( "But I decided do not spend my time for technical exercises and used an online solver\r\n" );
document.write( "<A  rel=nofollow HREF=http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi>http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi</A>\r\n" );
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document.write( "Below are the numbers from this online solver\r\n" );
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document.write( "B	=	55\r\n" );
document.write( "C	=	65\r\n" );
document.write( "Db	=	 7\r\n" );
document.write( "Dc	=	17\r\n" );
document.write( "x	=	48\r\n" );
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document.write( "The <U>ANSWER</U>  to the problem's question is: Daniel repaid $7 to Ben (Db=7) and $17 to Cathy (Dc=17).\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "You may check, that the obtained numbers satisfy the problem's conditions and equations (19) - (23).<BR>\n" );
document.write( "This check is easy straightforward task, so I leave it to you.\r<BR>\n" );
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document.write( "This problem and this my solution seems very educative to me, <BR>\n" );
document.write( "since they represent new type of thinking, which I never saw before in school Math problems.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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