document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=874076>Question 874076</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Isabella invested $500 at 6% annual interest, compounded quarterly. The value, A, of an investment can be calculated using the equation A=P(1+r/n)^nt, where P is the initial investment, r is the interest rate, n is the number of times the interest is compounded each year, and t is time in years. Exactly how long will it take for her investment to be worth four times as much (quadruple)in value? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #527291 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=527291\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A = P * (1 + (r/n))^(n*t)\r<BR>\n" );
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document.write( "replace what you know in the formula and solve for what you don't know.\r<BR>\n" );
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document.write( "what you know:\r<BR>\n" );
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document.write( "r = 6% divided by 100% = .06 because r is always the decimal equivalent of the % which is always the percent divided by 100%.\r<BR>\n" );
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document.write( "n = 4 because there are 4 compounding periods per year (quarterly compounding).\r<BR>\n" );
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document.write( "r/n = .06 / 4 = .015\r<BR>\n" );
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document.write( "P = $500.\r<BR>\n" );
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document.write( "the formula is, once again:\r<BR>\n" );
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document.write( "A = P * (1+(r/n)^(n*t\r<BR>\n" );
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document.write( "you are given that A is going to be a quadruple of P which means that A will be equal to 4 * $500, so A is equal to $2000.\r<BR>\n" );
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document.write( "to summarize what you know:\r<BR>\n" );
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document.write( "n = 4<BR>\n" );
document.write( "r = .06<BR>\n" );
document.write( "r/n = .015<BR>\n" );
document.write( "P = $500<BR>\n" );
document.write( "A = $2000<BR>\n" );
document.write( "t = unknown since that what you need to solve for.\r<BR>\n" );
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document.write( "your formula of:\r<BR>\n" );
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document.write( "A = P * (1 + (r/n))^(n*t) becomes:\r<BR>\n" );
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document.write( "$2000 = $500 * (1 + .015)^4t which becomes:\r<BR>\n" );
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document.write( "$2000 = $500 * (1.015)^4t\r<BR>\n" );
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document.write( "divide both sides of this formula by $500 to get:\r<BR>\n" );
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document.write( "$2000 / $500 = (1.015)^4t\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "4 = (1.015)^4t\r<BR>\n" );
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document.write( "since the unknown is in the exponent, this calls for using logarithms to solve for it.\r<BR>\n" );
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document.write( "take the log of both sides of this equation to get:\r<BR>\n" );
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document.write( "log(4) = log((1.015)^4t)\r<BR>\n" );
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document.write( "use law of logarithms formula number 1 (shown below the final answer) to convert this equation to:\r<BR>\n" );
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document.write( "log(4) = 4t * log(1.015)\r<BR>\n" );
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document.write( "divide both sides of this formula by log(1.015) to get:\r<BR>\n" );
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document.write( "log(4) / log(1.015) = 4t\r<BR>\n" );
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document.write( "use your calculator to solve for 4t to get:\r<BR>\n" );
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document.write( "4t = 93.11105126\r<BR>\n" );
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document.write( "divide both sides of this equation by 4 to get:\r<BR>\n" );
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document.write( "t = 23.27776282\r<BR>\n" );
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document.write( "that's your final answer.\r<BR>\n" );
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document.write( "logarithm law number 1 is equal to:\r<BR>\n" );
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document.write( "log(x^a) = a * log(x)\r<BR>\n" );
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document.write( "in your problem, a was equal to 4t and x was equal to 1.015.\r<BR>\n" );
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document.write( "that's how you got log(1.015^4t) = 4t*log(1.015)\r<BR>\n" );
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document.write( "to confirm your answer is correct, substitute for t in the original equation to see if that equation becomes true.\r<BR>\n" );
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document.write( "the original equation is:\r<BR>\n" );
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document.write( "$2000 = $500 * (1.015)^4t\r<BR>\n" );
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document.write( "replace t with 23.27776282 to get:\r<BR>\n" );
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document.write( "$2000 = $500 * (1.015)^(4 * 23.27776282)\r<BR>\n" );
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document.write( "simplify the right side of this equation to get:\r<BR>\n" );
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document.write( "$2000 = $2000.\r<BR>\n" );
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document.write( "This confirms the solution is good.\r<BR>\n" );
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document.write( "the solution is:\r<BR>\n" );
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document.write( "t = 23.27776282 which you can round off to 23.28 years depending on how accurate you need the solution to be.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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