document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1008342>Question 1008342</A>: <I><SPAN STYLE=\"word-break: break-all;\">  If we have $4000  to put in the bank earning 4.5 % interest compounded 2  times per year; how long will it take before we double our money? \r<BR>\n" );
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document.write( "Your money would double in        years.\r<BR>\n" );
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document.write( "If we have $P o   to put in the bank earning 4.5 % interest compounded 2  times per year; how long will it take before we double our money? \r<BR>\n" );
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document.write( "Your money would double in       years<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854449 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(53878)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "If we have $4000 to put in the bank earning 4.5 % interest compounded 2 times per year; <BR>\n" );
document.write( "how long will it take before we double our money?<BR>\n" );
document.write( "Your money would double in how many years?<BR>\n" );
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document.write( "Formal calculations in the post by @lwsshak3 are correct and lead to value <BR>\n" );
document.write( "time = 15.6 years approximately.  It is about 15 years and 7 months.\r<BR>\n" );
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document.write( "But his interpretation of this value is incorrect and contradicts to the meaning of the problem.\r<BR>\n" );
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document.write( "We will not get the doubled amount in 15 years and 7 months from the bank, <BR>\n" );
document.write( "because the bank will make the last relevant compounding only after 16 years.\r<BR>\n" );
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document.write( "16 years is the closest time when the amount first time will get and will overcome the doubled value.\r<BR>\n" );
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document.write( "It is because the amount at the account IS NOT a continuous exponential function of time.\r<BR>\n" );
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document.write( "The continuous exponential function is only approximation to the real function, which is piecewise constant<BR>\n" );
document.write( "and changes the values only at the ends of compounding periods, i.e. semi-annually.\r<BR>\n" );
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document.write( "So, the correct answer in this problem is not 15.6 years: the correct answer is 16 years.\r<BR>\n" );
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document.write( "I saw similar incorrect interpretation many times (uncounted number of times) <BR>\n" );
document.write( "at this forum from different tutors.\r<BR>\n" );
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document.write( "They all as one person make and repeat the same error.\r<BR>\n" );
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document.write( "Such uniformity in the repetition of the very same error simply shocks me.<BR>\n" );
document.write( "It turns out that so-called tutors — people who are, in many other aspects, perfectly normal individuals<BR>\n" );
document.write( "— are, on this particular issue, completely incapable of independent thought.\r<BR>\n" );
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document.write( "There is a certain mystery here for me.\r<BR>\n" );
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document.write( "Today, April 28, 2026, I submitted this problem to two artificial intelligence web-sites:<BR>\n" );
document.write( "one to Google Overview and other to math-gpt.org - for checking purposes.\r<BR>\n" );
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document.write( "Both web-sites returned incorrect answers, identical to the incorrect solution provided by @lwsshak3.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It &nbsp;CONFIRMS &nbsp;the fact that the whole community of tutors \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;INCORRECTLY &nbsp;treats all similar problems\r<BR>\n" );
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