Algebra.Com's Answer #845274 by ikleyn(52834)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Keys Company has a target of establishing a fund of $50,000. If $10,000 is deposited
\n" );
document.write( "at the end of every six months, and the fund earns interest at 4% compounded quarterly,
\n" );
document.write( "how long will it take to reach the target?
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "It is a non-traditional accumulative saving plan with $10,000 deposited semi-annually and compounded \r\n" );
document.write( "quarterly at 4% per annum. \r\n" );
document.write( "\r\n" );
document.write( "It means that the quarterly effective rate is 0.04/4 = 0.01 and the equivalent\r\n" );
document.write( "semi-annual effective rate is  =  = 1.0201.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " +-------------------------------------------------------------------+\r\n" );
document.write( " | So, this non-traditional accumulative saving plan is equivalent |\r\n" );
document.write( " | to the ordinary annuity with semi-annual deposits of $10,000 |\r\n" );
document.write( " | and the semi-annual effective rate of compounding r = 1.0201. |\r\n" );
document.write( " +-------------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now use the general formula for a classic Ordinary Annuity saving plan\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " FV =  , (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "where FV is the future value of the annuity; P is the semi-annual deposit; r is the semi-annual \r\n" );
document.write( "effective percentage yield presented as a decimal; n is the number of deposits.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Under the given conditions, P = 10000; r = 0.0201. So, according to (1), the formula for\r\n" );
document.write( "the future value is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " FV =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we should find n, the number of deposits (or the number of semi-annual periods) from this equation\r\n" );
document.write( "\r\n" );
document.write( " 50000 =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Simplify it by dividing both sides by 10000\r\n" );
document.write( "\r\n" );
document.write( "  =  ,\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( " 5 = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Simplify it further, step by step\r\n" );
document.write( "\r\n" );
document.write( " 5*0.0201 =  ,\r\n" );
document.write( "\r\n" );
document.write( " 0.1005 = ,\r\n" );
document.write( "\r\n" );
document.write( " 0.1005 + 1 =  ,\r\n" );
document.write( "\r\n" );
document.write( " 1.1005 = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Take logarithm base 10 of both sides\r\n" );
document.write( "\r\n" );
document.write( " log(1.1005) = n*log(1.0201)\r\n" );
document.write( "\r\n" );
document.write( "and find n\r\n" );
document.write( "\r\n" );
document.write( " n =  = 4.81 (approximately).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Finally, round the decimal value of 4.81 to the closest GREATER integer value of 5 \r\n" );
document.write( "in order for the bank be in position to complete the last semi-annual compounding.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, the solution is complete.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The ANSWER is: 5 semi-annual periods are needed.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "-----------------\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "On Ordinary Annuity saving plans, see the lessons\r
\n" );
document.write( "\n" );
document.write( " - Ordinary Annuity saving plans and geometric progressions\r
\n" );
document.write( "\n" );
document.write( " - Solved problems on Ordinary Annuity saving plans\r
\n" );
document.write( "\n" );
document.write( "in this site.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The lessons contain EVERYTHING you need to know about this subject, in clear and compact form.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "When you learn from these lessons, you will be able to do similar calculations in semi-automatic mode.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |