document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138828>Question 1138828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You owe your parents $16,000 (in present day dollars) and want to repay them in equal amounts the first to occur in 2 years from today and the other in 6 years from today. If the interest rate is 4.4% per annum compounding monthly, what will be the amount of each repayment?  \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 #756622 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=756622\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you owe them 16,000 to be paid in two equal installments; the first 2 years from today, and the second 6 years from today.\r<BR>\n" );
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document.write( "the interest rate is 4.4% compounded monthly.\r<BR>\n" );
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document.write( "your payment is going to be equal to x.\r<BR>\n" );
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document.write( "the loan is paid off when the present value of the payments is equal to the present value of the loan.\r<BR>\n" );
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document.write( "the present value of the loan is 16,000.\r<BR>\n" );
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document.write( "the annual interest rate is 4.4%.\r<BR>\n" );
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document.write( "divide that by 100 to get an annual interest rate of .044.\r<BR>\n" );
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document.write( "divide that by 12 to get a monthly interest rate of .044/12.\r<BR>\n" );
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document.write( "the first payment is 2 years out which is 24 months out.\r<BR>\n" );
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document.write( "the second payment is 6 years out which is 72 months out.\r<BR>\n" );
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document.write( "the present value of the first payment is therefore x / (1 + .044/12)^24.\r<BR>\n" );
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document.write( "the present value of the second payment is therefore x / (1 + .044/12)^72.\r<BR>\n" );
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document.write( "for now, we'll let (1 + .044/12)^24 be equal to a, and we'll let (1 + .044/12)^72  be equal to b so we don't have to write them down so many times.\r<BR>\n" );
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document.write( "the present value of the payments are therefore x / a and x / b.\r<BR>\n" );
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document.write( "the present value must be equal to 16,000, so we get 16,000 = x/a + x/b.\r<BR>\n" );
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document.write( "multiply both sides of this equation by a * b and we get:\r<BR>\n" );
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document.write( "16,000 * a * b = x * b + x * a\r<BR>\n" );
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document.write( "factor out the x to get 16,000 * a * b = x * (b + a)\r<BR>\n" );
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document.write( "divide both sides of this equation by (b + a) to get:\r<BR>\n" );
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document.write( "(16,000 * a * b) / (b + a) = x\r<BR>\n" );
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document.write( "since a = (1 + .044/12)^24 and b = (1 + .044/12)^72, we can solve for x to get:\r<BR>\n" );
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document.write( "x = 9499.762157.\r<BR>\n" );
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document.write( "that's your payment in 24 months and in 72 months.\r<BR>\n" );
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document.write( "the present value of the first payment is 9499.762157 / (1 + .044/12)^24 = 8700.910734.\r<BR>\n" );
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document.write( "the present value of the second payment is 9499.762157 / (1 + .044/12)^72 = 7299.089266.\r<BR>\n" );
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document.write( "the sum of the present value of the payments is equal to 16000 which means the loan is paid off.\r<BR>\n" );
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document.write( "this can be seen in more detail in the following excel printout.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/041111.jpg\" alt=\"$$$\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/041112.jpg\" alt=\"$$$\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/041113.jpg\" alt=\"$$$\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/041114.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the coments after the first payment is made should say the calculations for timer period 24 are, rather than the calculations for time period 4 are.\r<BR>\n" );
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document.write( "that was a typo.\r<BR>\n" );
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document.write( "the general form of the equation for future value is:\r<BR>\n" );
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document.write( "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 period.\r<BR>\n" );
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document.write( "in your problem, the interest rate was 4.4% per year.<BR>\n" );
document.write( "that was divided by 100 to get an interest rate of .044 per year.<BR>\n" );
document.write( "that was divided by 12 to get an interest rate of .044/12 per month.\r<BR>\n" );
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document.write( "since it was a repeating decimal, i left it as is rather than showing .044/12 as .036666666666666666666.......\r<BR>\n" );
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document.write( "to find the present value, you would use the same formula solved for p rather than f.\r<BR>\n" );
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document.write( "that would be p = f / (1 + r) ^ n\r<BR>\n" );
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document.write( "this is the formula used to bet the present value of the payments.\r<BR>\n" );
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document.write( "the payments were x.\r<BR>\n" );
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document.write( "the present value of the first payment was x / (1 + .044/12)^24.\r<BR>\n" );
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document.write( "the present value of the second payment was x / (1 + .044/12)^72.\r<BR>\n" );
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