document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=171285>Question 171285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If $1500 is deposited in a savings account paying 5% compounded quarterly, how long will take the account to increase to $2000? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #126444 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=126444\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let PV be present value of the investment.<BR>\n" );
document.write( "let FV be future value of the investment.<BR>\n" );
document.write( "let i be the interest rate<BR>\n" );
document.write( "let n be the number of time periods.<BR>\n" );
document.write( "standard equation to solve this is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=FV+=+PV+%2A+%281%2Bi%29%5En ALIGN=MIDDLE  ALT=\"FV+=+PV+%2A+%281%2Bi%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "PV = 1500<BR>\n" );
document.write( "FV = 2000<BR>\n" );
document.write( "annual interest rate = 5% / 100% = .05<BR>\n" );
document.write( "quarterly interest rate = annual interest rate / 4 = .05 / 4 = .0125<BR>\n" );
document.write( "1 + .0125 = 1.0125<BR>\n" );
document.write( "your number of time periods is n which will be in quarters of a year.<BR>\n" );
document.write( "you need to solve for number of time periods.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your equation to solve is:<BR>\n" );
document.write( "2000 = (1500)*(1.0125)^n<BR>\n" );
document.write( "divide both sides of equation by 1500:<BR>\n" );
document.write( "1.0125^n = 2000/1500<BR>\n" );
document.write( "you should be able to use logarithms to solve this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.0125%5En+=+%282000%2F1500%29 ALIGN=MIDDLE  ALT=\"1.0125%5En+=+%282000%2F1500%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%281.0125%2C%282000%2F1500%29%29+=+n ALIGN=MIDDLE  ALT=\"log%281.0125%2C%282000%2F1500%29%29+=+n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "in order to solve this using a calculator, you need to convert the base to either 10 or e, whichever the calculator handles.<BR>\n" );
document.write( "we'll use the base of 10 which is the log function on the calculator.<BR>\n" );
document.write( "the general form of the base conversion formula is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2Cx%29+=+log%2810%2Cx%29%2Flog%2810%2Ca%29 ALIGN=MIDDLE  ALT=\"log%28a%2Cx%29+=+log%2810%2Cx%29%2Flog%2810%2Ca%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "in your logarithmic equation,<BR>\n" );
document.write( "a = 1.0125<BR>\n" );
document.write( "x = (2000/1500)<BR>\n" );
document.write( "your logarithmic equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%281.0125%2C%282000%2F1500%29%29+=+n+=+%28log%2810%2C%282000%2F1500%29%29%29%2F%28log%2810%2C1.0125%29%29+=+23.15810902 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "to prove this is correct, use your calculator and take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.0125%5E23.15810902 ALIGN=MIDDLE  ALT=\"1.0125%5E23.15810902\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you'll see that the answer comes out to be 1.3333333333333 which is the same as 2000/1500.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "it will take 23.15810902 quarters to reach 2000.<BR>\n" );
document.write( "divide that by 4, and it will take 5.789527255 years.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the investment is made at the beginning of each time period.<BR>\n" );
document.write( "the time periods are quarters of a year (4 time periods per year).<BR>\n" );
document.write( "the division of the annual rate by 4 to get a quarterly rate is the industry standard way to do this.  if you were compounding monthly, you would take the annual interest rate and divide by 12.  the number of periods, in that case, would be 12 periods per year.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );