document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175340>Question 1175340</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The formula for an investment worth with interest compounded annually is A = P(1 + i)^n, where P represents the initial investment, i is the interest rate, and A is the worth of the investment after n years.\r<BR>\n" );
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document.write( "a) Solve the formula for P. What was the initial investment of an investment worth $1000 that compounded 10% interest for 10 years?\r<BR>\n" );
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document.write( "b) Solve the formula for i. What is the interest rate of an investment whose worth went from $1000 to $1200 in 2 years?\r<BR>\n" );
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document.write( "c) Explain a method with which you could estimate how many years it would take for an investment to reach a certain worth at a certain interest rate.\r<BR>\n" );
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document.write( "d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800933 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=800933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula you show is:\r<BR>\n" );
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document.write( "A = P * (1 + i) ^ n\r<BR>\n" );
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document.write( "in this formula:\r<BR>\n" );
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document.write( "A is the future value<BR>\n" );
document.write( "P is the present value<BR>\n" );
document.write( "i 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( "your problems will be taken in turn below:\r<BR>\n" );
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document.write( "a) Solve the formula for P. What was the initial investment of an investment worth $1000 that compounded 10% interest for 10 years?\r<BR>\n" );
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document.write( "in this problem, A is equal to 1000 and i is equal to .10 per year and n is equal to 10 years.\r<BR>\n" );
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document.write( "the interest rate is not shown as percent.<BR>\n" );
document.write( "the interest rate percent divided by 100 is equal to the interest rate.\r<BR>\n" );
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document.write( "the formula becomes 1000 = P * (1 + .10) ^ 10\r<BR>\n" );
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document.write( "solve for P to get P = 1000 / (1 + .10) ^ 10 = 385.5432894.\r<BR>\n" );
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document.write( "confirm by solving for A using that value for P.\r<BR>\n" );
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document.write( "you get A = 385.5432894 * (1 + .10) ^ 10 = 1000.\r<BR>\n" );
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document.write( "the value of P is confirmed to be good.\r<BR>\n" );
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document.write( "b) Solve the formula for i. What is the interest rate of an investment whose worth went from $1000 to $1200 in 2 years?\r<BR>\n" );
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document.write( "the same basic form of the equation is used.\r<BR>\n" );
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document.write( "A = P * (1 + i) ^ n becomes:\r<BR>\n" );
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document.write( "1200 = 1000 * (1 + i) ^ 2\r<BR>\n" );
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document.write( "divide both sides of this equation by 1000 to get:\r<BR>\n" );
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document.write( "1.2 = (1 + i) ^ 2\r<BR>\n" );
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document.write( "take the square root of both sides of the equation to get:\r<BR>\n" );
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document.write( "1.2 ^ (1/2) = 1 + i\r<BR>\n" );
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document.write( "solve for 1 + i to get:\r<BR>\n" );
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document.write( "1.095445115 = 1 + i\r<BR>\n" );
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document.write( "subtract 1 from both sides of the equation to get:\r<BR>\n" );
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document.write( ".095445115 = i\r<BR>\n" );
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document.write( "that's your answer.\r<BR>\n" );
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document.write( "confirm by replacing i in the original equation and solve for A to get:\r<BR>\n" );
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document.write( "A = 1000 * (1 + .095445115) ^ 2 = 1200.\r<BR>\n" );
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document.write( "this confirms the rate is good.\r<BR>\n" );
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document.write( "multiply the rate by 100 and you get 9.5445115%.\r<BR>\n" );
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document.write( "c) Explain a method with which you could estimate how many years it would take for an investment to reach a certain worth at a certain interest rate.\r<BR>\n" );
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document.write( "the formula is, once again, A = P * (1 + i) ^ n\r<BR>\n" );
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document.write( "you want to solve for  n.\r<BR>\n" );
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document.write( "divide both sides of the equation by P to get:\r<BR>\n" );
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document.write( "A/P = (1 + i) ^ n\r<BR>\n" );
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document.write( "take the log of both sides of the equation to get:\r<BR>\n" );
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document.write( "log(A/P) = log((1+i)^n)\r<BR>\n" );
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document.write( "by the law of logs that says log(x^a) = a * log(x), your equation becomes:\r<BR>\n" );
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document.write( "log(A/P) = n * log(1 + i)\r<BR>\n" );
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document.write( "divide both sides of the equation by log(1 + i) to get:\r<BR>\n" );
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document.write( "log(A/P) / log(1 + i) = n\r<BR>\n" );
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document.write( "that's the procedure you would take to find n.\r<BR>\n" );
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document.write( "d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225?\r<BR>\n" );
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document.write( "you would use the method described in (c) to find the value of n.\r<BR>\n" );
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document.write( "the basic formula is, once again, A = P * (1 + i) ^ n\r<BR>\n" );
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document.write( "the formula becomes 5225 = 2100 * (1 + .2) ^ n\r<BR>\n" );
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document.write( "divide both sides of the equation by 2100 to get:\r<BR>\n" );
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document.write( "5225/2100 = (1 + .2) ^ n\r<BR>\n" );
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document.write( "take the log of both sides of the equation to get:\r<BR>\n" );
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document.write( "log(5225/2100) = log((1 + .2)^n)\r<BR>\n" );
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document.write( "by the law of logs that says log(x^a) = a * log(x), your equation becomes:\r<BR>\n" );
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document.write( "log(5225/2100) = n * log(1.2)\r<BR>\n" );
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document.write( "divide both sides of the equation by log(1.2) to get:\r<BR>\n" );
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document.write( "log(5225/2100) / log(1.2) = n\r<BR>\n" );
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document.write( "solve for n to get:\r<BR>\n" );
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document.write( "n = 4.999504552.\r<BR>\n" );
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document.write( "confirm by replacing n with 4.999504552 in the equation and solve for A to get:\r<BR>\n" );
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document.write( "A = 2100 * (1 + .2) ^ 4.999504552 = 5225.\r<BR>\n" );
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document.write( "this confirms the value of n is good.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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