Algebra.Com's Answer #740454 by ikleyn(52790)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A person invested $8200 for 1 year, part at 8%, part at 9%, and the remainder at 15%.
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document.write( "The total annual income from these investments was $987. The amount of money invested at 15% was $600 more
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document.write( "than the amounts invested at 8% and 9% combined. Find the amount invested at each rate.
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document.write( " At the first glance, this problem is for 3 unknowns and 3 equations.\r
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document.write( " But it is really big deal to solve such a system.\r
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document.write( " Fortunately, there is a way to reduce the solution to two unknowns and two equations, which is much easier to solve.\r
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document.write( " Below I will show you how to do it.\r
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document.write( "Notice this phrase of the condition\r\n" );
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document.write( " \"The amount of money invested at 15% was $600 more than the amounts invested at 8% and 9% combined\".\r\n" );
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document.write( "It means that if you subtract $600 from the total sum of $8200, and then divide this difference by 2, \r\n" );
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document.write( "you will get how much was invested at 8% and 9% combined:  = 3800 dollars.\r\n" );
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document.write( "Thus you just know that the combined amount invested at 8% and 9% is $3800.\r\n" );
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document.write( "Hence, the rest $8200 - $3800 = $4400 was invested at 15%.\r\n" );
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document.write( "Also, now you can easily calculate interest from $4400 invested at 15%: it is 0.15*4400 = 660 dollars.\r\n" );
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document.write( " At this point you just, actually reduced the original problem to 2 unknowns and 2 equations.\r\n" );
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document.write( "Let x = the amount invested at 8% and y = the amount invested at 9%.\r\n" );
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document.write( "Then \r\n" );
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document.write( " x + y = 3800 dollars (1)\r\n" );
document.write( "0.08x + 0.09y = 987 - 660 dollars (2)\r\n" );
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document.write( "Do you understand how I obtained the equation (2) and what does it mean ?\r\n" );
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document.write( "It simply means that combined interest from the 8% account and 9% account is equal to the total interest \r\n" );
document.write( "of the three accounts minus the interest of the 15% account, which we calculated above.\r\n" );
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document.write( "OK, very good. Any way, we just reduced the original problem to the standard equations for two investments:\r\n" );
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document.write( " x + y = 3800 (1)\r\n" );
document.write( "0.08x + 0.09y = 327 (2)\r\n" );
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document.write( "To solve it you can use the Substitution method. From eq(1) express x = 3800-y\r\n" );
document.write( "and substitute it into eq(2). You will get\r\n" );
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document.write( "0.08*(3800-y) + 0.09y = 327\r\n" );
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document.write( "0.08*3800 - 0.08y + 0.09y = 327\r\n" );
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document.write( "0.01y = 327 - 0.08*3800\r\n" );
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document.write( "y =  = (at this point I use Excel spreadsheet in my computer and get the answer in one click) = 2300 dollars.\r\n" );
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document.write( "Thus you got that the amount at 9% is 2300 dollars.\r\n" );
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document.write( "Then the amount at 8% is the rest 3800 - 2300 = 1500 dollars.\r\n" );
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document.write( "Answer. $1500 at 8%; $2300 at 9% and $4400 at 15%.\r\n" );
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document.write( "Check. a) the combined amount invested at 8% and 9% is 1500 + 2300 = 3800 dollars;\r\n" );
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document.write( " It is $600 dollars less than $4400 invested at 15%. ! Correct !\r\n" );
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document.write( " b) Total investment is 1500 + 2300 + 4400 = 8200 dollars. ! Correct !\r\n" );
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document.write( "Solved.\r
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document.write( "The lesson to learn from my solution is THIS :\r
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document.write( " At the first glance, this problem is for 3 unknowns and 3 equations.\r\n" );
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document.write( " But you learned, and I showed it to you, how to reduce the problem to 2 equation in 2 unknowns.\r\n" );
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document.write( "To see many other similar problems solved by the same method, look into the lesson\r
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document.write( " - HOW TO algebreze and solve this problem on 2 equations in 2 unknowns \r
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document.write( "in this site.\r
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