document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138716>Question 1138716</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Ram Company borrowed $20 000 at 10% compounded semi-annually and made payments toward the loan of $8000 after two years and $10 000 after three-and-a-half years. How much is required to pay off the loan one year after the second payment?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756535 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=756535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> they borrowed 20,000 at 10% compounded semi-annually.\r<BR>\n" );
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document.write( "the interest rate per half year time period was 10% / 2 = 5% / 100 = .05.\r<BR>\n" );
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document.write( "the number of half year periods was equal to 2 times the number of years.\r<BR>\n" );
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document.write( "they made payment of $8,000 at the end of 2 years and $10,000 at the end of 3.5 years.\r<BR>\n" );
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document.write( "the remaining balance on the loan started at 10,000.\r<BR>\n" );
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document.write( "at the end of 2 years after the loan was made, the remaining balance of the loan after interest was added but before payment was made was equal to $10,000 * 1.05 ^ 4 = $24,310.13.\r<BR>\n" );
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document.write( "they made their payment of $8,000 and the remaining balance on the loan after interest added and payment was subtracted was equal to $24,310.13 minus $8,000 = $16,310.13.\r<BR>\n" );
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document.write( "at the end of 3.5 years, which was 1.5 years after the end of 2 years, the remaining balance on the loan was $16,310.13 * 1.05 ^ 3 = $18,881.01 after interest was added but before payment was made.\r<BR>\n" );
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document.write( "they made their payment of $10,000 and the remaining balance on the loan after interest was added and payment was made became equal to $8,881.01.\r<BR>\n" );
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document.write( "one year after that, the remaining balance on the loan was 8,881.01 * 1.05 ^ 2  = 9,791.31.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "that's how much was required to pay off the loan one year after the last payment was made.\r<BR>\n" );
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document.write( "this can be seen in more detail in the following excel spreadsheet printout.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/041002.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the formula used 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( "the time period is in half years.<BR>\n" );
document.write( "the rate per half year is 10% / 100 = .10 / 2 = .05 per half year.<BR>\n" );
document.write( "the number of half years is 2 times the number of years.\r<BR>\n" );
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document.write( "1 + r becomes (1 + .05) which becomes 1.05.\r<BR>\n" );
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