document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197961>Question 1197961</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How long would it take to double $2000 in the bank deposit with 4% interest compounded monthly </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831439 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "We'll be using the compound interest formula<BR>\n" );
document.write( "A = P*(1+r/n)^(n*t)\r<BR>\n" );
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document.write( "The variables are<ul><li>A = final amount after t years</li><li>P = deposit amount</li><li>r = annual interest rate in decimal form</li><li>n = number of times the money is compounded per year</li><li>t = number of years</li></ul>In this case we know the following values:<ul><li>A = 4000 (we want the deposit of 2000 to double to 4000)</li><li>P = 2000</li><li>r = 0.04</li><li>n = 12  (compounding 12 times a year; aka compounding monthly)</li><li>t = unknown, which we'll be solving for</li></ul>If the variable is in the trees (aka exponent), then log it down.<BR>\n" );
document.write( "In other words, we use logarithms to isolate the exponent.\r<BR>\n" );
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document.write( "A = P*(1+r/n)^(n*t)<BR>\n" );
document.write( "4000 = 2000*(1+0.04/12)^(12*t)<BR>\n" );
document.write( "4000/2000 = (1.003333)^(12*t)<BR>\n" );
document.write( "2 = (1.003333)^(12*t)<BR>\n" );
document.write( "log( 2 ) = log(  (1.003333)^(12*t) ) .... apply logs to both sides<BR>\n" );
document.write( "log( 2 ) = 12*t*Log( 1.003333 ) .... use the rule log(A^B) = B*log(A)<BR>\n" );
document.write( "12*t*Log( 1.003333 ) = log( 2 )<BR>\n" );
document.write( "t = log(2)/(12*log(1.003333))<BR>\n" );
document.write( "t = 17.359278<BR>\n" );
document.write( "This value is approximate.\r<BR>\n" );
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document.write( "A shortcut through use of estimation:<BR>\n" );
document.write( "We can use the <a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Rule_of_72\">Rule of 72</a> to determine an approximate timeframe when the amount of money will double.<BR>\n" );
document.write( "This is where we divide 72 over the whole number form of the interest rate. We treat \"4%\" as simply \"4\", so we have 72/4 = 18<BR>\n" );
document.write( "The rule of 72 says we need about 18 years for the money to double. This is fairly close to the 17.359278 figure calculated earlier.\r<BR>\n" );
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document.write( "<font size=4>Answer: <font color=red>Approximately 17.359278 years</font></font>\r<BR>\n" );
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document.write( "Edit: <BR>\n" );
document.write( "Be careful. The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.04%2F12 ALIGN=MIDDLE  ALT=\"1.04%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > portion that the tutor @josgarithmetic wrote is not correct. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Br%2Fn+=+1%2B0.04%2F12+=+1.0033333 ALIGN=MIDDLE  ALT=\"1%2Br%2Fn+=+1%2B0.04%2F12+=+1.0033333\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately, which you'll find is not equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.04%2F12=0.086667 ALIGN=MIDDLE  ALT=\"1.04%2F12=0.086667\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately.\r<BR>\n" );
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document.write( "It's not valid to say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Br%2Fn ALIGN=MIDDLE  ALT=\"1%2Br%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Br%29%2Fn ALIGN=MIDDLE  ALT=\"%281%2Br%29%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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