document.write( "Question 1187744: Sam Long anticipates he will need approximately $226,800 in 15 years to cover his 3-year-old daughter’s college bills for a 4-year degree.\r
\n" ); document.write( "\n" ); document.write( "How much would he have to invest today at an interest rate of 8% compounded semiannually? (Use the Table provided.) (Do not round intermediate calculations. Round your answer to the nearest cent.)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #818762 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
i don't have a table, but i do have a formula and a calculator that can handle this.
\n" ); document.write( "he will need 226,800 in 15 years.
\n" ); document.write( "he will invest today at 8% per year compounded semi-annually.
\n" ); document.write( "the formula to use is f = p * (1 + r) ^ n
\n" ); document.write( "f is the future value
\n" ); document.write( "p is the present value
\n" ); document.write( "r is the interest rate per time period.
\n" ); document.write( "n is the number of time periods.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the time periods are semi-annual.
\n" ); document.write( "you multiply the number of years by 2 to get the number of semi-annual periods.
\n" ); document.write( "you divide the interest rate per year by 2 to get the interest rate per semi-annual period.
\n" ); document.write( "the future value is 226,800.
\n" ); document.write( "you want to find the present value\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the formula becomes 226,800 = p * (1 + .08/2) ^ (15 * 2)
\n" ); document.write( "simplify to get:
\n" ); document.write( "226,800 = p * 3.24339751.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "solve for p to get:
\n" ); document.write( "p = 226,800 / 3.24339751 = 69,926.6739.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "confirm by replacing p in the original equation and solving for f to get:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "f = 69,926.6739 * 3.24339751 = 226,800.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "if you have a table, it is probably a factor.
\n" ); document.write( "for 15 years compounded semi-annually, you are talking about 30 semi-annual periods with a future value factor of 1.04^30 = 3.24339751 which you round to whatever the table is saying.
\n" ); document.write( "for the same 15 years compounded semi-annually, you are talking about 30 semi-annual periods with a present value factor of 1/1.04^30 = .308318668 which you round to whatever the table is saying.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "for example, a future value of 226,800 would be multiplied by .308318668 to get a present value of 69,926.6739 and a present value of 69,926.6739 would be multiplied by 3.24339751 to get a future value of 226,800.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "not seeing your table, i don't know what it's in there and can only make an assumption.
\n" ); document.write( "if you can send me a copy of the table, i would probably be able to figure out how you would use it.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note that, to find the effective annual interest rate, you would take 1.04 ^ 2 = 1.0816/
\n" ); document.write( "that's your effective annual growth rate.
\n" ); document.write( "1. 0816^15 = 3.24339751.
\n" ); document.write( "this is the same growth factor as 1.04^30.
\n" ); document.write( "this makes sense because 1.04^30 = (1.04^2)^15.
\n" ); document.write( " \n" ); document.write( "
\n" );