document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186531>Question 1186531</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three years ago Mr Smith borrowed 4000 from Sarah for a duration of five years at 12% interest rate per annum and compounded monthly. One year ago He borrowed 8000 at 16% interest per annum and compounded quarterly,also for five years. He agrees to pay his debt in two equal instalments,one now and another one five years later. If Sarah charges 20% interest rate per annum compounded half yearly, how much will each payment be?.\r<BR>\n" );
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document.write( " Please help with the homework.thanks in advance </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849433 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to break down this compound interest problem:\r<BR>\n" );
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document.write( "**Loan 1:**\r<BR>\n" );
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document.write( "*   Principal: $4000<BR>\n" );
document.write( "*   Interest Rate: 12% per annum (0.12/year)<BR>\n" );
document.write( "*   Compounding: Monthly (12 times/year)<BR>\n" );
document.write( "*   Loan Duration: 5 years\r<BR>\n" );
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document.write( "Since the loan was taken three years ago, it has been accruing interest for three years already. We need to calculate the accumulated amount after three years.\r<BR>\n" );
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document.write( "1.  **Monthly interest rate:** 0.12 / 12 = 0.01<BR>\n" );
document.write( "2.  **Number of compounding periods:** 3 years * 12 months/year = 36 months<BR>\n" );
document.write( "3.  **Accumulated amount:** $4000 * (1 + 0.01)^36 = $4000 * 1.43076878 ≈ $5723.08\r<BR>\n" );
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document.write( "**Loan 2:**\r<BR>\n" );
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document.write( "*   Principal: $8000<BR>\n" );
document.write( "*   Interest Rate: 16% per annum (0.16/year)<BR>\n" );
document.write( "*   Compounding: Quarterly (4 times/year)<BR>\n" );
document.write( "*   Loan Duration: 5 years\r<BR>\n" );
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document.write( "Since the loan was taken one year ago, it has been accruing interest for one year.\r<BR>\n" );
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document.write( "1.  **Quarterly interest rate:** 0.16 / 4 = 0.04<BR>\n" );
document.write( "2.  **Number of compounding periods:** 1 year * 4 quarters/year = 4 quarters<BR>\n" );
document.write( "3.  **Accumulated amount:** $8000 * (1 + 0.04)^4 = $8000 * 1.16985856 ≈ $9358.87\r<BR>\n" );
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document.write( "**Total Debt:**\r<BR>\n" );
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document.write( "Mr. Smith's total debt now is $5723.08 + $9358.87 = $15081.95\r<BR>\n" );
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document.write( "**Repayment:**\r<BR>\n" );
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document.write( "Mr. Smith will make two equal payments: one now and one five years from now. Sarah charges 20% interest compounded semi-annually.\r<BR>\n" );
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document.write( "Let 'x' be the amount of each payment. The present value of these two payments must equal the total debt.\r<BR>\n" );
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document.write( "1.  **Semi-annual interest rate:** 0.20 / 2 = 0.10<BR>\n" );
document.write( "2.  **Number of compounding periods for the future payment:** 5 years * 2 periods/year = 10 periods\r<BR>\n" );
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document.write( "The present value equation is:\r<BR>\n" );
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document.write( "$15081.95 = x + \frac{x}{(1 + 0.10)^{10}}$\r<BR>\n" );
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document.write( "$15081.95 = x + \frac{x}{2.59374246}$\r<BR>\n" );
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document.write( "$15081.95 = x + 0.38554329x$\r<BR>\n" );
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document.write( "$15081.95 = 1.38554329x$\r<BR>\n" );
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document.write( "$x = \frac{15081.95}{1.38554329}$\r<BR>\n" );
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document.write( "$x ≈ 10885.56$\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "Each payment will be approximately $10885.56.<BR>\n" );
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