document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1140028>Question 1140028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You deposit $4000 into an account that earns 5% compounded annually. A friend deposits $3750 into an account that earns 4.95% annual interest, compounded continuously. Will your friend's balance ever equal yours? If so, when?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #760515 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=760515\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f = p * (1 + r) ^ n\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f is the future value.<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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the formula for discrete compounding.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula for continuous compounding is f = p * e ^ (r * n)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f is the future value.<BR>\n" );
document.write( "p is the present value.<BR>\n" );
document.write( "e is the scientific constant equal to 2.718281828..... <BR>\n" );
document.write( "it is shown as the e ^ x key on your calculator.<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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "their balance will be the same when the future value of their respective accounts are equal to each other.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this occurs when p * (1 + r) ^ n is equal to p * e ^ (r * n)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "given your inputs, the formulas become:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4000 * (1 + .05) ^ n = 3750 * e ^ (.0495 * n)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "divide both sides of this equation by 3750 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4000 / 3750 * (1 + .05) ^ n = e ^ (.0495 * n)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "divide both sides of this equation by (1 + .05) ^ n to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4000 / 3750 = (e ^ (.0495 * n) / ((1 + .05) ^ n)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "take the natural log of both sides of this equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ln(4000 / 3750) = ln((e ^ (.0495 * n) / ((1 + .05) ^ n))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since ln(a/b) is equal to ln(a) - ln(b), your equation becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ln(4000 / 3750) = ln(e ^ (.0495 * n)) - ln((1 + .05) ^ n)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since ln(a^b) = b * ln(a), your equation becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ln(4000 / 3750) = .0495 * n * ln(e) - n * ln(1 + .05)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since ln(e) is equal to 1, your equation becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ln(4000 / 3750) = .0495 * n - n * ln(1 + .05)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "factor out the n to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ln(4000 / 3750) = n * (.0495 - ln(1 + .05))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for n to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n = ln(4000 / 3750) / (.0495 - ln(1 + 05) = 90.92034856.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the balance in both accounts will be equal in 90.92034856 years.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4000 * (1 + .05) ^ 90.92034856 = 337,752.4038\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3750 * e ^ (.0495 * 90.92034856) = 3347,752.4038\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your solution is that the account balances will be equal in 90.92034856 years.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the steps involved might be easier to see in my hand drawn worksheet shown below.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/042801.jpg\" alt=\"$$$\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the transition from step 3 to step 4 takes advantages of the fact that ln(a/b) = ln(a) - ln(b).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the transition from step 4 to step 5 takes advantage of the fact that ln(a^b) = b * ln(a) and also takes advantage of the fact that ln(e) = 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that allows ln(e^(.0495*n)) to become equal to .0495 * n * ln(e) which then becomes equal to .0495 * n.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that also allows ln(1.05 ^ n) to become equal to n * ln(1.05).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "step 7 factors out the n and then divides both sides of the equation by (.0495 - ln(1.05).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "step 8 shows the result.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "note that (1 + .05) is the same as 1.05.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "showing 1.05 as (1 + .05) is done to reinforce the concept that it comes from the general expression of (1 + r).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );