document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171535>Question 1171535</A>: <I><SPAN STYLE=\"word-break: break-all;\"> April loaned her friend 750 to buy guitar he agreed to pay back the money with 5% annual interest at the end of 6 month how much did her friend give her at the end of 6 months </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #796496 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=796496\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if the interest rate was compounded semi-annually, then 5% / 2 = 2.5% and he would have paid her back 750 * 1.025 = 768.75<BR>\n" );
document.write( "the effective annual rate, in that case, would have been 1.025^2 = 1.050625 - 1 = .050625 * 100 = 5.0625%.\r<BR>\n" );
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document.write( "if the interest rate was not compounded semi-annually, then he would have paid her back 750 * 1.05^(1/2) = 768.5213074.\r<BR>\n" );
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document.write( "this is slightly less than 768.75.\r<BR>\n" );
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document.write( "the difference is in the compounding.\r<BR>\n" );
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document.write( "with annual compounding, the annual interest rate is divided by 1 and the number of years is multiplied by 1.<BR>\n" );
document.write( "with semi-annual compounding, the annual interest rate is divided by 2 and the number of years is multiplied by 2.<BR>\n" );
document.write( "with quarterly compounding, the annual interest rate is divided by 4 and the number of years is multiplied by 4.<BR>\n" );
document.write( "with monthly compounding, the annual interest rate is divided by 12 and the number of years is multiplied by 12.<BR>\n" );
document.write( "with daily compounding, the annual interest rate is divided by 365 and the number of years is multiplied by 365.\r<BR>\n" );
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document.write( "the higher the number of compounding periods per year, the higher the effective interest rate per year.\r<BR>\n" );
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document.write( "the highest number of compounding periods per year it can go is continuous compounding.<BR>\n" );
document.write( "that's a different formula.<BR>\n" );
document.write( "that formula is f = p * e ^ (r * t)<BR>\n" );
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( "t is the number of time periods.\r<BR>\n" );
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document.write( "with continuous compounding, the rate is usually left as the annual rate and the number of years is left as the number of year.<BR>\n" );
document.write( "the formula will give you the same effective interest rate regardless of the time periods you used.\r<BR>\n" );
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document.write( "with discrete compounding, the formula is:<BR>\n" );
document.write( "f = p * (1 + r) ^ n<BR>\n" );
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( "all of the above assumes compound interest.<BR>\n" );
document.write( "if you are taking about simple interest, then a different formula applies.<BR>\n" );
document.write( "that formula is:<BR>\n" );
document.write( "f = p + p * r * n which can also be shown as:<BR>\n" );
document.write( "f = p * (1 + r * n)\r<BR>\n" );
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document.write( "in your case, simple interest formula would give you:<BR>\n" );
document.write( "f = 750 * (1 + .05 * .5) = 768.75.\r<BR>\n" );
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document.write( "while you get the same answer as if the interest rate was compounded semi-annually, it is not the same.\r<BR>\n" );
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document.write( "to see the difference, assume the loan was for 20 years.<BR>\n" );
document.write( "at simple interest, he would have had to pay 750 * (1 + .05 * 20) = 1500.<BR>\n" );
document.write( "at 5% compounded semi-annually, he would have had to pay 750 * (1 + .025)^40 = 2013.7978879.\r<BR>\n" );
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document.write( "the difference gets larger, the longer the term of the loan.\r<BR>\n" );
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document.write( "the following table shows you the difference as the length of the loan increases.<BR>\n" );
document.write( "for your problem you would be looking at nyrs = .5.\r<BR>\n" );
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