document.write( "Question 1208421: Stuart and Dave each had some money. If Stuart spends 3 times the amount Dave spends each day, Stuart would have left $280 by the time Dave spends all his money. If Stuart spends 1/3 the amount that Dave spends each day, Stuart would have $568 by the time Dave spends all his money. How much does each of them have? \n" ); document.write( "

Algebra.Com's Answer #846824 by ikleyn(52800) $\"\"$ $\"About$

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\n" ); document.write( "Stuart and Dave each had some money. If Stuart spends 3 times the amount Dave spends each day,
\n" ); document.write( "Stuart would have left $280 by the time Dave spends all his money.
\n" ); document.write( "If Stuart spends 1/3 the amount that Dave spends each day, Stuart would have $568 by the time
\n" ); document.write( "Dave spends all his money. How much does each of them have?
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document.write( "Let S be initial amount of Stuart's money; D is the initial amount of Dave's money.\r\n" );
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document.write( "Let x (dollars per day) be the regular rate at which Dave spends his money each day.\r\n" );
document.write( "Then in the first scenario, the rate at which Stuart spends his money is 3x dollars per day.\r\n" );
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document.write( "For the first scenario, we have these equations \r\n" );
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document.write( "    S - 3x*d   = 280,     (1)  \r\n" );
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document.write( "    D - x*d =  0,         (2)\r\n" );
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document.write( "where d is the number of days in the first scenario.\r\n" );
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document.write( "We can exclude the unknown d from equations (1) and (2).  For it, multiply equation (2) by 3 (both sides).\r\n" );
document.write( "Keep equation (1) as is.  You will get\r\n" );
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document.write( "     S - 3x*d = 280,     (1')\r\n" );
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document.write( "    3D - 3x*d =   0.     (2')\r\n" );
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document.write( "Now from eq.(1') subtract equation (2').  You will get\r\n" );
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document.write( "    S - 3D = 280.    (3)\r\n" );
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document.write( "In the second scenario, Dave spends x dollars per day;  Stuart spends    dollars per day.\r\n" );
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document.write( "For the second scenario, the equations are\r\n" );
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document.write( "     = 568,    (4)\r\n" );
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document.write( "    D - x*d = 0.                (5)\r\n" );
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document.write( "Multiply equation (4) by 3; keep equation (5) as is.\r\n" );
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document.write( "    3S - x*d = 1704,            (4')\r\n" );
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document.write( "    D - x*d = 0.                (5')\r\n" );
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document.write( "Subtract eq.(5') from eq.(4').  You will get\r\n" );
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document.write( "    3S - D = 1704.              (6)\r\n" );
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document.write( "Thus we have two equations (3) and (6)\r\n" );
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document.write( "    S  - 3D = 280,              (3)\r\n" );
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document.write( "    3S -  D = 1704.             (6)\r\n" );
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document.write( "From (3), express  S = 3D + 280  and substitute it into (6)\r\n" );
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document.write( "    3(3D+280) - D = 1704,\r\n" );
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document.write( "    9D + 840 - D = 1704,\r\n" );
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document.write( "    9D - D = 1704 - 840,\r\n" );
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document.write( "       8D  =    864,\r\n" );
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document.write( "        D  =    864/8 = 108.\r\n" );
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document.write( "Now from (3)  S = 280 + 3*D = 280 + 3*108 = 604.\r\n" );
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document.write( "ANSWER.  Stuart has $604;  Dave has $108.\r\n" );
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