document.write( "Question 1174471: To save for​ retirement, Karla Harby put ​$500 each month into an ordinary annuity for 10 years. Interest was compounded monthly. At the end of the 10 ​years, the annuity was worth ​$84,608. What annual interest rate did she​ receive?
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Algebra.Com's Answer #850674 by ikleyn(52786)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "To save for​ retirement, Karla Harby put ​$500 each month into an ordinary annuity for 10 years.
\n" ); document.write( "Interest was compounded monthly. At the end of the 10 ​years, the annuity was worth ​$84,608.
\n" ); document.write( "What annual interest rate did she​ receive?
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\n" ); document.write( "\n" ); document.write( "        The solution by @CPhill is  FAR  from to be accurate, is far from to be correct, and is far from to be acceptable.\r
\n" ); document.write( "\n" ); document.write( "        Instead,  see my solution below.\r
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document.write( "Use the formula for the future value of the ordinary annuity\r\n" );
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document.write( "    FV = \"PMT%2A%28%28%281%2Br%29%5En-1%29%2Fr%29\",\r\n" );
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document.write( "where r is the effective monthly rate of compounding, n the number of deposits ( = months, in this problem).\r\n" );
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document.write( "Substitute the numbers\r\n" );
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document.write( "    84608 = \"500%2A%28%28%281%2Br%29%5E120-1%29%2Fr%29\".\r\n" );
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document.write( "Simplify\r\n" );
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document.write( "    \"84608%2F500\" = \"%28%281%2Br%29%5E120-1%29%2Fr\",\r\n" );
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document.write( "or\r\n" );
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document.write( "    \"%28%281%2Br%29%5E120-1%29%2Fr\" = 169.216.\r\n" );
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document.write( "It is the equation, which you want to solve to find r.\r\n" );
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document.write( "Use any online solver for transcendent equations, for example, at this site www.desmos.com/calculator\r\n" );
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document.write( "The solution is r = 0.00549  as the effective monthly rate.\r\n" );
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document.write( "So, the annual nominal rate is  0.00549*12 = 0.06588,  or  6.588%.    <<<---===  \"highlight%28highlight%28ANSWER%29%29\".\r\n" );
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document.write( "CHECK.  Substitute this value of annual nominal rate into the formula for the future value.\r\n" );
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document.write( "         You will get  FV = \"500%2A%28%28%281%2B0.06588%2F12%29%5E120-1%29%2F%28%280.06588%2F12%29%29%29\" = $84,607.76.\r\n" );
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document.write( "         Now check the value of 7.91% = 0.0791 from the post by @CPhill.  Substitute it into the same formula\r\n" );
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document.write( "                       FV = \"500%2A%28%28%281%2B0.0791%2F12%29%5E120-1%29%2F%28%280.0791%2F12%29%29%29\" = $91,015.18\r\n" );
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document.write( "                            which is FAR from to be an acceptable precision.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Many times at this forum I pointed to @CPhill that his technology of working with Finance problems/applications
\n" ); document.write( "is   \"highlight%28highlight%28DEFECTIVE%29%29\"   and must be corrected/fixed/repaired/improved/replaced, since it provides wrong numbers.\r
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\n" ); document.write( "\n" ); document.write( "But the effect is the same as to talk with a blind-deaf-mute person,
\n" ); document.write( "or the same as to beat with a stick on a tree.\r
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\n" ); document.write( "\n" ); document.write( "                Regarding the post by @CPhill . . . \r
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\n" ); document.write( "\n" ); document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r
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\n" ); document.write( "\n" ); document.write( "The artificial intelligence is like a baby now. It is in the experimental stage
\n" ); document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r
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\n" ); document.write( "\n" ); document.write( "                It has no feeling of shame - it is shameless.\r
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\n" ); document.write( "\n" ); document.write( "This time, again,  it made an error.\r
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\n" ); document.write( "\n" ); document.write( "Although the @CPhill' solution are copy-paste  Google  AI solutions,  there is one essential difference.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  Google  AI  makes a note at the end of its solutions that  Google  AI  is experimental
\n" ); document.write( "and can make errors/mistakes.\r
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\n" ); document.write( "\n" ); document.write( "All @CPhill' solutions are copy-paste of  Google  AI  solutions, with one difference:
\n" ); document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.
\n" ); document.write( "So, he NEVER SAYS TRUTH.\r
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\n" ); document.write( "\n" ); document.write( "Every time,  @CPhill embarrassed to tell the truth.\r
\n" ); document.write( "\n" ); document.write( "But I am not embarrassing to tell the truth,  as it is my duty at this forum.\r
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\n" ); document.write( "\n" ); document.write( "And the last my comment.\r
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\n" ); document.write( "\n" ); document.write( "When you obtain such posts from @CPhill,  remember,  that  NOBODY  is responsible for their correctness,
\n" ); document.write( "until the specialists and experts will check and confirm their correctness.\r
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\n" ); document.write( "\n" ); document.write( "Without it,  their reliability is  ZERO and their creadability is  ZERO,  too.\r
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