document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169507>Question 1169507</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a firm will need 300000 at the end of 3 years to repay a loan. the firm decided that it would deposit 20000 at the start of each quarter during these 3 years into the account. the account yield 12% per annum compounded quarterly during the first year. what rate of interest should it earn during the remaining 2 years to accumulate enough amount into this account to pay the loan at the end of 3 years.(Answer:13.8%) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794261 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=794261\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Consider the first year for now only. One year is 4 quarters.<BR>\n" );
document.write( "The 12% annual interest rate converts to the quarterly rate of (12%)/4 = 3%<BR>\n" );
document.write( "We'll let i = 0.03 to reflect this.\r<BR>\n" );
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document.write( "Compute the future value (FV) of an annuity due with <BR>\n" );
document.write( "<ul><li>deposit of P = 20000</li><li>quarterly interest rate i = 0.03</li><li>n = 4 quarters</li></ul> <BR>\n" );
document.write( "We get the following<BR>\n" );
document.write( "FV = (1+i)*P*( (1+i)^n - 1)/i<BR>\n" );
document.write( "FV = (1+0.03)*20000*( (1+0.03)^4 - 1)/0.03<BR>\n" );
document.write( "FV = 86182.7162<BR>\n" );
document.write( "FV = 86182.72\r<BR>\n" );
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document.write( "After four quarters pass by, we have 86182.72 in the account.\r<BR>\n" );
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document.write( "Let x be the quarterly interest rate for the next 8 quarters (2 years). <BR>\n" );
document.write( "We'll deposit P = 86182.72 dollars at this rate and time. <BR>\n" );
document.write( "At the end of 8 quarters, we'll be left with 86182.72*(1+x)^8 dollars<BR>\n" );
document.write( "Note I'm using the compound interest formula here.\r<BR>\n" );
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document.write( "Let S = 86182.72*(1+x)^8<BR>\n" );
document.write( "We'll come back to this later.\r<BR>\n" );
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document.write( "Again keeping x the same, whatever it is, we'll compute the FV of an annuity due like we did earlier. However, this time we don't know the interest rate and the value of n is doubled (n = 8 instead of n = 4). \r<BR>\n" );
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document.write( "So we have<BR>\n" );
document.write( "FV = (1+i)*P*( (1+i)^n - 1)/i<BR>\n" );
document.write( "FV = (1+x)*20000*( (1+x)^8 - 1)/x<BR>\n" );
document.write( "Let's call this R<BR>\n" );
document.write( "R = (1+x)*20000*( (1+x)^8 - 1)/x\r<BR>\n" );
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document.write( "Add the expressions of S and R<BR>\n" );
document.write( "S+R = 86182.72*(1+x)^8+(1+x)*20000*( (1+x)^8 - 1)/x<BR>\n" );
document.write( "call this sum T. We'll make it a function of x<BR>\n" );
document.write( "T(x) = 86182.72*(1+x)^8+(1+x)*20000*( (1+x)^8 - 1)/x\r<BR>\n" );
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document.write( "The goal is to solve T(x) = 300000 which is the same as finding the roots of T(x)-300000 = 0\r<BR>\n" );
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document.write( "The variable x is buried under an exponent of 8, which suggests that solving this by hand is going to be daunting. Luckily we can use technology. \r<BR>\n" );
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document.write( "You can use any technology you like. For me, I prefer GeoGebra since it's a very handy program in many ways. You should find the root on the interval 0 < x < 1 is approximately <BR>\n" );
document.write( "x = 0.0345028581\r<BR>\n" );
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document.write( "This represents the decimal form of the quarterly interest rate.<BR>\n" );
document.write( "So the quarterly rate is about 3.45%\r<BR>\n" );
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document.write( "Multiplying this by 4 leads to the annual rate needed<BR>\n" );
document.write( "4*x = 4*0.0345 = 0.138 = 13.8%\r<BR>\n" );
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document.write( "Answer: 13.8%<BR>\n" );
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