document.write( "Question 1174374: Scott and Laura have both invested some money. Scott invested $2,500 more than Laura and at a 2% higher interest rate. If Scott received $900 annual interest and Laura received $400, how much did Scott invest? (There are two possibilities for how much Scott invested. \n" ); document.write( "

Algebra.Com's Answer #799831 by ikleyn(52925) $\"\"$ $\"About$

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\n" ); document.write( "Scott and Laura have both invested some money. Scott invested $2,500 more than Laura and at a 2% higher interest rate.
\n" ); document.write( "If Scott received $900 annual interest and Laura received $400, how much did Scott invest?
\n" ); document.write( "(There are two possibilities for how much Scott invested.
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\n" ); document.write( "\n" ); document.write( "            This problem has very interesting, elegant and unexpected solution.\r
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\n" ); document.write( "\n" ); document.write( "            In addition, it has beautiful core, that you rarely may find in such problems. \r
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\n" ); document.write( "\n" ); document.write( "            It is WORTH to learn.     It is WORTH to teach.     Watch my steps attentively.\r
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document.write( "Let x be the the annual interest of the Scott's investment (as decimal).\r\n" );
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document.write( "Then Laura's annual interest is (x-0.02).\r\n" );
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document.write( "    |  This variable x is the only unknown I will use in my solution.        |\r\n" );
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document.write( "Then the  Scott's principal investment is  {{900/x}}}  dollars,\r\n" );
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document.write( "while the Laura's principal investment is    dollars.\r\n" );
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document.write( "According to the condition, the difference of their principals is 2500;\r\n" );
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document.write( "it gives you the core equation\r\n" );
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document.write( "     -  = 2500.      (1)\r\n" );
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document.write( "At this point, the setup is just completed, and now I start solving this equation.\r\n" );
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document.write( "First, divide by 100 both sides\r\n" );
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document.write( "     -  = 25.\r\n" );
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document.write( "Next, multiply both sides by x*(x-0.02) and simplify, step by step\r\n" );
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document.write( "    9*(x-0.02) - 4x = 25x*(x-0.02)\r\n" );
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document.write( "    9x - 0.18  - 4x = 25x^2 - 0.5x\r\n" );
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document.write( "       5x -0.18     = 25x^2 - 0.5x\r\n" );
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document.write( "       25x^2 - 5.5x + 0.18 = 0\r\n" );
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document.write( "     =  =  = .\r\n" );
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document.write( "So, there are two roots:   =  =  = 0.18,\r\n" );
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document.write( "                  and      =  =  = 0.04.\r\n" );
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document.write( "Both the roots satisfy the basic equation (1):\r\n" );
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document.write( "    x= 0.18:    -  = 5000 - 2500 = 2500 dollars.      ! correct !\r\n" );
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document.write( "and\r\n" );
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document.write( "    x= 0.14:    -  = 22500 - 20000 = 2500 dollars.    ! correct !\r\n" );
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document.write( "Thus for Scott's investment, we have two possible answers: a)   = 5000 dollars  and/or  b)  = 22500 dollars.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Honestly, it is for the first time in my life I see so beatiful solution to this class of problems.\r
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