document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193755>Question 1193755</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me with the homework.\r<BR>\n" );
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document.write( "1. Compare the amounts accumulated on a principal of R10000 is invested from 10 march 20.3 to 1 July 20.5 at 16 and half% per annum compounded semi-annually,and credited on 1 January and 1 July if,<BR>\n" );
document.write( "1.1. simple interest is used for the odd period and compound interest for the rest of the term;<BR>\n" );
document.write( "1.2. fractional compounding is used for the term<BR>\n" );
document.write( "Note ignore the slight differences between the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848506 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848506\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1.1 Simple Interest for Odd Period, Compound Interest for Rest**\r<BR>\n" );
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document.write( "* **Calculate the odd period:** <BR>\n" );
document.write( "    * From March 10, 2020, to July 1, 2020, is approximately 3 months.\r<BR>\n" );
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document.write( "* **Calculate simple interest for the odd period:**<BR>\n" );
document.write( "    * Principal (P) = R10,000<BR>\n" );
document.write( "    * Rate (R) = 16.5% per annum = 0.165<BR>\n" );
document.write( "    * Time (T) = 3 months = 3/12 years = 0.25 years<BR>\n" );
document.write( "    * Simple Interest (SI) = P * R * T = R10,000 * 0.165 * 0.25 = R412.50\r<BR>\n" );
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document.write( "* **Calculate the amount after the odd period:**<BR>\n" );
document.write( "    * Amount = Principal + Simple Interest = R10,000 + R412.50 = R10,412.50\r<BR>\n" );
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document.write( "* **Calculate the remaining period:**<BR>\n" );
document.write( "    * From July 1, 2020, to July 1, 2025, is 5 years.\r<BR>\n" );
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document.write( "* **Calculate the amount after 5 years with compound interest:**<BR>\n" );
document.write( "    * Principal (P) = R10,412.50<BR>\n" );
document.write( "    * Rate (R) = 16.5% per annum compounded semi-annually = 0.165 / 2 = 0.0825 per half-year<BR>\n" );
document.write( "    * Number of periods (n) = 5 years * 2 periods/year = 10 periods<BR>\n" );
document.write( "    * Amount = P * (1 + R)^n = R10,412.50 * (1 + 0.0825)^10 <BR>\n" );
document.write( "    * Amount ≈ R23,967.47\r<BR>\n" );
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document.write( "**1.2 Fractional Compounding**\r<BR>\n" );
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document.write( "* **Calculate the total number of days:**<BR>\n" );
document.write( "    * From March 10, 2020, to July 1, 2025, is approximately 1826 days.\r<BR>\n" );
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document.write( "* **Calculate the daily interest rate:**<BR>\n" );
document.write( "    * Annual interest rate = 16.5% = 0.165<BR>\n" );
document.write( "    * Daily interest rate = 0.165 / 365 \r<BR>\n" );
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document.write( "* **Calculate the amount with daily compounding:**<BR>\n" );
document.write( "    * Amount = P * (1 + (R/365))^N <BR>\n" );
document.write( "    * Amount = R10,000 * (1 + (0.165/365))^1826<BR>\n" );
document.write( "    * Amount ≈ R24,178.12\r<BR>\n" );
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document.write( "**Comparison**\r<BR>\n" );
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document.write( "* **Method 1 (Simple + Compound):** R23,967.47<BR>\n" );
document.write( "* **Method 2 (Fractional Compounding):** R24,178.12\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "Fractional compounding (daily compounding in this case) results in a slightly higher accumulated amount compared to using simple interest for the odd period and then switching to semi-annual compounding. This is because fractional compounding applies interest more frequently, leading to slightly higher returns.\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This calculation assumes a year has 365 days.<BR>\n" );
document.write( "* Slight variations in the exact number of days and rounding may result in minor differences in the final amounts.<BR>\n" );
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