document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=983869>Question 983869</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You decide to put $150 in a savings account to save for a $3,000 down payment on a new car. If the account has an interest rate of 2.5% per year and is compounded monthly, how long does it take you to earn $3,000 without depositing any additional funds? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #604656 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=604656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The best way to see questions like this is to break up time. \r<BR>\n" );
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document.write( "Let's first look at time 0.\r<BR>\n" );
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document.write( "At time 0: We put $150 in hopes to accumulate to $3,000 so that we can put a down payment on our car.\r<BR>\n" );
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document.write( "We earn 2.5% interest compounded monthly.\r<BR>\n" );
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document.write( "So effectively we earn 2.5/12% or .2083% per month.\r<BR>\n" );
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document.write( "So at time t (in months): we have accumulated to  150(1.002083)^t\r<BR>\n" );
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document.write( "And we want to figure out at what t value do we accumulate to 3000?\r<BR>\n" );
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document.write( "So if we set the two equal:\r<BR>\n" );
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document.write( "150(1.002083)^t = 3000\r<BR>\n" );
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document.write( "(1.002083)^t = 3000/150 = 20\r<BR>\n" );
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document.write( "(1.002083)^t = 20\r<BR>\n" );
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document.write( "Take the natural log of both sides.\r<BR>\n" );
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document.write( "ln(1.002083^t) = ln(20)\r<BR>\n" );
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document.write( "Use properties of logs to finish this:\r<BR>\n" );
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document.write( "t * ln(1.002083) = ln(20)\r<BR>\n" );
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document.write( "t = ln(20)/ln(1.002083) = 1439.68 months  or as an integer:  1440 months.\r<BR>\n" );
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document.write( "1440 months is kind of a nasty answer, so let's divide that by 12 to get 120 years.<BR>\n" );
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document.write( "NOTE:\r<BR>\n" );
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document.write( "I find 120 years to be an unreasonable amount of time to save for a down payment, so maybe there is a typo in your question. If so, let me attempt to address that.\r<BR>\n" );
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document.write( "Let's assume you meant to say $1500 in a savings account (that would make much more sense).\r<BR>\n" );
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document.write( "Then all the math would be the same except 3000/1500 = 2 not 20.\r<BR>\n" );
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document.write( "So our answer would be t = ln(2)/ln(1.002083) = 28 months\r<BR>\n" );
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document.write( "In terms of years, this is 2 1/3 years. Much more reasonable.\r<BR>\n" );
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document.write( "--------------------------------------------------------------<BR>\n" );
document.write( "If you have additional questions, please contact me at swincher4391@yahoo.com.\r<BR>\n" );
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document.write( "Thanks,\r<BR>\n" );
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document.write( "Devin\r<BR>\n" );
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document.write( " </SPAN>\n" );
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