document.write( "Question 1166208: A contract valued at $81500.00 requires payment of $2430.00 at the beginning of every 3 months. If interest is 6.5% compounded quarterly, calculate the term of this contract (round to years and months). \n" ); document.write( "

Algebra.Com's Answer #851392 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "A contract valued at $81500.00 requires payment of $2430.00 at the beginning of every 3 months.
\n" ); document.write( "If interest is 6.5% compounded quarterly, calculate the term of this contract (round to years and months).
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "This problem is about a sinking fund compounded quarterly.\r\n" );
document.write( "\r\n" );
document.write( "The initial/starting value of the fund is $81,500. The fund makes outpayments\r\n" );
document.write( "of $2430 quarterly and is compounded quarterly at the annual rate of 6.5%.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "They want to know the term of this contract, i.e. the number of quarters.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The feature of this fund is that it makes outpayments at the beginning of each quarter,\r\n" );
document.write( "while compounding are made at the end of each quarter.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For such a fund, the formula connecting the starting value, the periodical regular \r\n" );
document.write( "outpayment value, the number of payments and the rate of compounding is \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    A = ,    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "where A is the starting amount, W is the regular quarterly outpayment amount, \r\n" );
document.write( "r is the annual compounding rate, m is the number of outpayments per year (m= 4 in this problem), \r\n" );
document.write( "n is the total number of outpayments/compounding (the number of quarters, in this problem),\r\n" );
document.write( "  is the effective rate of compounding per the quarter (3 months) period.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "With the given data, formula (1) takes the form\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    81500 = .    (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The unknown in this equation is 'n'.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To find 'n', simplify equation (2) step by step\r\n" );
document.write( "\r\n" );
document.write( "      = \r\n" );
document.write( "\r\n" );
document.write( "    0.536295486 = \r\n" );
document.write( "\r\n" );
document.write( "     = 1 - 0.536295486\r\n" );
document.write( "\r\n" );
document.write( "     = 0.463704514\r\n" );
document.write( "\r\n" );
document.write( "     = 1/0.463704514\r\n" );
document.write( "\r\n" );
document.write( "     = 2.156545754\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, take logarithm of both sides\r\n" );
document.write( "\r\n" );
document.write( "    n*log(1+0.065/4) = log(2.156545754)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Express n and then calculate its value\r\n" );
document.write( "\r\n" );
document.write( "    n =  = 47.6760064\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The number of quarters is about 47.676, and the number of years is close to 12.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "After completing calculations, I think that the problem is posed in a bad way.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Had the problem be posed correctly, the final number of quarters would be VERY close to an integer number.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "But in any case, the logic of solution and the steps should be similar to that shown in this my post.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );