document.write( "Question 1163541: john and nat were given some money. if john spends $50 and nat spends $100 each day, john would still have $2500 left while nat would have spent all her money. if john spends $100 and nat spends $50 each day, john would still have $1000 left while nat would have spent all her money. How much money john and nat were given each? solution \n" ); document.write( "

Algebra.Com's Answer #787677 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "john and nat were given some money. if john spends $50 and nat spends $100 each day, john would still have $2500 left
\n" ); document.write( "while nat would have spent all her money. if john spends $100 and nat spends $50 each day, john would still have $1000 left
\n" ); document.write( "while nat would have spent all her money. How much money john and nat were given each?
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document.write( "Let J be the amount John had initially and let x be the number of days in the 1-st scenario.\r\n" );
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document.write( "Let N be the amount Nat  had initially and let y be the number of days in the 2-nd scenario.\r\n" );
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document.write( "For the first scenario, we have these two equations\r\n" );
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document.write( "    J = 50x + 2500,     (1)\r\n" );
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document.write( "    N = 100x.           (2)\r\n" );
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document.write( "For the second scenario, we have these two equations\r\n" );
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document.write( "    J = 100y + 1000,     (3)\r\n" );
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document.write( "    N = 50y.             (4)\r\n" );
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document.write( "In all, we have 4 equations in 4 unknown  J, N, x and y, so we have a chance to solve it.\r\n" );
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document.write( "From eq(2), express x =   and substitute it into eq(1).  You will have\r\n" );
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document.write( "    J = 50*(N/100) + 2500,   or  J = (1/2)*N + 2500,   or   2J - N = 5000  (5)\r\n" );
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document.write( "From eq(4), express y =   and substitute it into eq(3).  You will have\r\n" );
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document.write( "    J = 100*(N/50) + 1000,    or  J = 2N + 1000,       or    J -2N = 1000  (6)\r\n" );
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document.write( "Thus, we reduced the problem to two equations in 2 unknowns\r\n" );
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document.write( "    2J -  N = 5000    (5)\r\n" );
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document.write( "     J - 2N = 1000    (6)\r\n" );
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document.write( "From eq(6), express  J = 2N + 1000  and substitute it into eq(5).  You will get\r\n" );
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document.write( "    2*(2N + 1000) - N = 5000,\r\n" );
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document.write( "which implies\r\n" );
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document.write( "    4N + 2000 - N = 5000\r\n" );
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document.write( "    3N            = 5000 - 2000 = 3000\r\n" );
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document.write( "     N            = 3000/3 = 1000.\r\n" );
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document.write( "Then from eq(5)\r\n" );
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document.write( "    2J = 5000 + 1000 = 6000\r\n" );
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document.write( "     J = 6000/2 = 3000.\r\n" );
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document.write( "ANSWER.  John had 3000;  Nat had 1000.\r\n" );
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