document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103994>Question 1103994</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A population increases from 21,000 to 27,000 over a 5-year period at a constant annual percent growth rate. <BR>\n" );
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document.write( "a) By what percent did the population increase in total? <BR>\n" );
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document.write( "5.155% (round to the nearest 0.001%) <BR>\n" );
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document.write( "b) At what constant percent rate of growth did the population increase each year? <BR>\n" );
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document.write( "5.155% (round to the nearest 0.001%) <BR>\n" );
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document.write( "c) At what continuous annual growth rate did this population grow? <BR>\n" );
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document.write( "0.050% (round to the nearest 0.001%)<BR>\n" );
document.write( "I included the answers I got, but I know that they are not all right.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #718712 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=718712\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A population increases from 21,000 to 27,000 over a 5-year period at a constant annual percent growth rate. \r<BR>\n" );
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document.write( "a) By what percent did the population increase in total? \r<BR>\n" );
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document.write( "27,000 / 21,000 = 1.285714286 - 1 = .285714286 * 100 = 28.5714286% rounded to 3 decimal digits = 28.571%.\r<BR>\n" );
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document.write( "they were not asking for the annual rate of increase.<BR>\n" );
document.write( "they just wanted the total rate of increase.\r<BR>\n" );
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document.write( "b) At what constant percent rate of growth did the population increase each year? \r<BR>\n" );
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document.write( "27,000 / 21,000 = 1.285714286 ^ (1/5) = 1.051547497 - 1 = .051547497 * 100 = 5.1547497% rounded to 3 decimal digits = 5.155%.\r<BR>\n" );
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document.write( "you got this one right.\r<BR>\n" );
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document.write( "you could also have solved it this way.\r<BR>\n" );
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document.write( "the discrete compounding formula is f = p * (1 + r) ^ n\r<BR>\n" );
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document.write( "r 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( "your time periods are in years, therefore no adjustment is made to r or n.\r<BR>\n" );
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document.write( "your formula becomes:\r<BR>\n" );
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document.write( "27,000 = 21,000 * (1 + r) ^ 5\r<BR>\n" );
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document.write( "divide both sides of this equation by 21,000 to get:\r<BR>\n" );
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document.write( "27,000 / 21,000 = (1 + r) ^ 5\r<BR>\n" );
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document.write( "take the fifth root of both sides of this equation to get:\r<BR>\n" );
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document.write( "(27,000 / 21,000) ^ (1/5) = 1 + r\r<BR>\n" );
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document.write( "subtract 1 from both sides of this equation to get:\r<BR>\n" );
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document.write( "(27,000 / 21,000) ^ (1/5) - 1 = r\r<BR>\n" );
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document.write( "solve for r to get:\r<BR>\n" );
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document.write( "r = .0515474968 * 100 = 5.15474968% rounded to 2 decimal places = 5.155%.\r<BR>\n" );
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document.write( "c) At what continuous annual growth rate did this population grow? \r<BR>\n" );
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document.write( "the continuous growth rate formula is f = p * e^(r*n)\r<BR>\n" );
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document.write( "p is the present value.<BR>\n" );
document.write( "f is the future value.<BR>\n" );
document.write( "e is the scientific constant of 2.718281828.......<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( "your present value is 21,000 and your future value is 27,000 and your number of years is 5.\r<BR>\n" );
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document.write( "the formula becomes 27,000 = 21,000 * e^(r * 5)\r<BR>\n" );
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document.write( "divide both sides of this equation by 21,000 to get:\r<BR>\n" );
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document.write( "27,000 / 21,000 = e^(r * 5)\r<BR>\n" );
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document.write( "take the natural log of both sides of this equation to get:\r<BR>\n" );
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document.write( "ln(27,000 / 21,000) = ln(e^(5r))\r<BR>\n" );
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document.write( "since ln(e^x) = x*ln(e), your equation becomes:\r<BR>\n" );
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document.write( "ln(27,000 / 21,000) = 5r * ln(e)\r<BR>\n" );
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document.write( "since ln(e) = 1, your equation becomes:\r<BR>\n" );
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document.write( "ln(27,000 / 21,000) = 5r\r<BR>\n" );
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document.write( "divide both sides of this equation by 5 to get:\r<BR>\n" );
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document.write( "ln(27,000 / 21,000) / 5 = r\r<BR>\n" );
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document.write( "solve for r to get:\r<BR>\n" );
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document.write( "r = .0502628857 * 100 = 5.026728857% rounded to 3 decimal places = 5.027%.\r<BR>\n" );
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document.write( "the more number of compounding periods per year you have, the higher the effective interest rate.\r<BR>\n" );
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document.write( "that is why you get the same future value with the interest rate of 5.155% compounded annually as you get with the interest rate of 5.027% compounded continuously.\r<BR>\n" );
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document.write( "compounding continuously is the most compounding periods per year you can get.\r<BR>\n" );
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document.write( "if you did the discrete compounding formula with monthly compounding, you would have gotten an interest rate between 5.155% and 5.027%.\r<BR>\n" );
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document.write( "with monthly compounding, you formula would have become:\r<BR>\n" );
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document.write( "27,000 = 21,000 * (1 + r/12) ^ (5*12)\r<BR>\n" );
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document.write( "that would have become:\r<BR>\n" );
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document.write( "27,000 / 21,000 = (1 + r/12) ^ 60)\r<BR>\n" );
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document.write( "you would have taken the 60th root of both sides of this equation to get:\r<BR>\n" );
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document.write( "(27,000 / 21,000) ^ (1/60) = 1 + r/12\r<BR>\n" );
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document.write( "subtract 1 from both sides of this equation to get:\r<BR>\n" );
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document.write( "(27,000 / 21,000) ^ (1/60) - 1 = r/12.\r<BR>\n" );
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document.write( "solve for r/12 to get:\r<BR>\n" );
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document.write( "r/12 = .0041973581\r<BR>\n" );
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document.write( "solve for r to get:\r<BR>\n" );
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document.write( "r = .0503682977 * 100 = 5.03682977% rounded to 3 decimal digits = 5.037%.\r<BR>\n" );
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document.write( "with annual compounding, your interest rate had to be 5.155%.<BR>\n" );
document.write( "with monthly compounding, your interest rate had to be 5.037%.<BR>\n" );
document.write( "with continuous compounding, your interest rate had to be 5.027%.\r<BR>\n" );
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document.write( "the formula for discrete compounding is:\r<BR>\n" );
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document.write( "f = p * (1 + r) ^ n\r<BR>\n" );
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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( "you divide the annual rate by the number of compounding periods per year to get the interest rate per time period.\r<BR>\n" );
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document.write( "you multiply the number of years by the number of compounding periods per year to get the number of time periods.\r<BR>\n" );
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document.write( "this formula requires the time period adjustments be made before using the formula.\r<BR>\n" );
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document.write( "the formula can also be shown as:\r<BR>\n" );
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document.write( "f = p * (1 + r/c) ^ (n*c)\r<BR>\n" );
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document.write( "in this version of the formula, r is the annual interest rate and n is the number of years and c is the number of compounding periods per year.\r<BR>\n" );
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document.write( "this version of the formula assumes annual interest rate and number of year and adjusts for the number of compounding periods per year as part of the formula.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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