document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106912>Question 1106912</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that you want to purchase a used car for $12000. You currently have $8000. If you invest your $8000 at an annual interest rate of 3%, compounded quarterly, how long will it take (in years) before you can purchase the car? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #721915 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=721915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the price of the car is 12,000.<BR>\n" );
document.write( "you have 8,000.<BR>\n" );
document.write( "the annual interest rate is 3% compounded quarterly.<BR>\n" );
document.write( "how many years before you can purchase the car?\r<BR>\n" );
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document.write( "the formula to use is f = p * (1 + r) ^ n\r<BR>\n" );
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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" );
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document.write( "since you are compounding the interest rate quarterly, your interest rate per quarter is equal to 3% / 4 which is equal to .75% expressed as a percent and .0075 expressed as a rate.\r<BR>\n" );
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document.write( "in the formula, you use the rate, not the percent.\r<BR>\n" );
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document.write( "n is also expressed in quarters of a year, so if you want to know the number of years, you would have to divide n by 4 as well, once you find it.\r<BR>\n" );
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document.write( "your formula becomes 12,000 = 8,000 * (1 + .0075) ^ n.\r<BR>\n" );
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document.write( "divide both sides of this equation by 8,000 to get 12,000 / 8,000 = (1 + .0075) ^ n.\r<BR>\n" );
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document.write( "take the log of both sies of this equation to get log(12,000/8,000) = log(1.0075^n).\r<BR>\n" );
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document.write( "since log(1.0075^n) is equal to n * log(1.0075), your formula becomes log(12,000/8,000) = n * log(1.0075).\r<BR>\n" );
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document.write( "divide both sides of this equation by log(1.0075) and you get log(12,000/8,000) / log(1.0075) = n.\r<BR>\n" );
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document.write( "solve for n to get n  = 54.2644945.\r<BR>\n" );
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document.write( "replace n in the original equation with this to get 12,000 = 8,000 * (1.0075)^54.2644945.\r<BR>\n" );
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document.write( "the result is 12,000 = 12,000.\r<BR>\n" );
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document.write( "recall that n is in quarters, so divide 54.2644945 by 4 to get 13.56612362 years.\r<BR>\n" );
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document.write( "an alternate way of analyzing this is to use the formula f = p * (1 + r/c) ^ (n*c).\r<BR>\n" );
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document.write( "this formula assumes annual interest rate and number of years and incorporates the number of compounding periods per year into it.\r<BR>\n" );
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document.write( "using this formula, you would get 12,000 = 8,000 * (1 + .03/4) ^ (n*4).\r<BR>\n" );
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document.write( "you would ue the same procedure to get log(12,000 / 8,000) / log(1 + .03/4) = n*4.\r<BR>\n" );
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document.write( "you would solve for n * 4 to get n * 4 = 54.2644945.\r<BR>\n" );
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document.write( "you would then solve for n to get n = 13.56612362.\r<BR>\n" );
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document.write( "in the formula of f = p * (1 + r/c) ^ (n*c), .....\r<BR>\n" );
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document.write( "f = future value as before<BR>\n" );
document.write( "p = present value as before<BR>\n" );
document.write( "r = annual interest rate (not the annual interest rate percent).<BR>\n" );
document.write( "n = number of years<BR>\n" );
document.write( "c = number of compounding periods per year.\r<BR>\n" );
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document.write( "in the first formula of f = p * (1 + r) ^ n,<BR>\n" );
document.write( "r is assumed to be per time period and n is assumed to be number of time periods.<BR>\n" );
document.write( "the adjustment from years to time periods is done prior to using the formula.\r<BR>\n" );
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document.write( "in the second formula of f = p * (1 + r/c) ^ (n*c),<BR>\n" );
document.write( "r is assumed to be per year and n is assumed to be number of years and c is the number of compounding periods per year.<BR>\n" );
document.write( "the adjustment from years to time periods is incorporated as part of the formula.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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