document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=846572>Question 846572</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I want to accumulate $120,000 in 30 years. Plan A is a single deposit into an account that compounds annually and an apr of 5% plan B is a single deposit into an account with continuous compounding and an apr of 4.8%   <BR>\n" );
document.write( "How much do I need to deposit into each account in order to reach my goal  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #509848 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=509848\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you want $120,000 in 30 years.<BR>\n" );
document.write( "plan A compounds annually at an apr of 5%.<BR>\n" );
document.write( "plan B compounds continuously at an apr of 4.8%\r<BR>\n" );
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document.write( "the formula for annual compounding is:<BR>\n" );
document.write( "f = p * (1+r)^n\r<BR>\n" );
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document.write( "f is the future value which is equal to 120,000<BR>\n" );
document.write( "p is the present value.<BR>\n" );
document.write( "r is the interest rate per time period.<BR>\n" );
document.write( "n is the number of time periods.\r<BR>\n" );
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document.write( "since the compounding is annually, than the interest rate is the apr divided by 100 which is equal to .05.<BR>\n" );
document.write( "since the time periods are annual, then the number of time periods is the same as the number of years which is equal to 30.\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "120,000 = p * (1.05)^30.<BR>\n" );
document.write( "divide both sides of this formula by (1.05)^30 to get:<BR>\n" );
document.write( "120,000 / (1.05)^30 = p which makes p equal to $27,765.29.\r<BR>\n" );
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document.write( "that's what you'd have to invest today at 5% compounded annually in order to have $120,000 thirty years from now.\r<BR>\n" );
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document.write( "the formula for continuous compounding is:<BR>\n" );
document.write( "f = p * e^(r*n)\r<BR>\n" );
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document.write( "r is the interest rate per time period which is in years.<BR>\n" );
document.write( "n is the number of time periods which is in years.\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "120,000 = p * e^(.048 * 30).<BR>\n" );
document.write( "divide both sides of this equation by 3^(.048 * 30) to get:<BR>\n" );
document.write( "120,000 / (e^(.048*30)) = p which makes p equal to 28,431.33.\r<BR>\n" );
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document.write( "that's what you'd have to invest today at 5% compounded continuously in order to have $120,000 thirty years from now.\r<BR>\n" );
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