document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1096173>Question 1096173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You want to have $6,000 saved up for a new car in 4 years. How much should you deposit each quarter into an account paying 8% compounded quarterly? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #710638 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=710638\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "We will use this formula<BR>\n" );
document.write( "FV = P*( (1+i)^n - 1 )/i<BR>\n" );
document.write( "where generally,<BR>\n" );
document.write( "FV = future value of annuity<BR>\n" );
document.write( "i = interest rate per period<BR>\n" );
document.write( "n = number of periods\r<BR>\n" );
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document.write( "Specifically we can say<BR>\n" );
document.write( "FV = target amount of money we want four years into the future<BR>\n" );
document.write( "i = interest rate per quarter<BR>\n" );
document.write( "n = number of quarters\r<BR>\n" );
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document.write( "In this case, we are given<BR>\n" );
document.write( "FV = 6000<BR>\n" );
document.write( "i = (interest rate in decimal form)/(compounding frequency) = 0.08/4 = 0.02<BR>\n" );
document.write( "n = (compounding frequency)*(number of years) = 4*4 = 16\r<BR>\n" );
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document.write( "Plug FV = 6000, i = 0.02, and n = 16 into the formula. Then solve for P\r<BR>\n" );
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document.write( "FV = P*( (1+i)^n - 1 )/i<BR>\n" );
document.write( "6000 = P*( (1+0.02)^16 - 1 )/0.02<BR>\n" );
document.write( "6000 = P*( (1.02)^16 - 1 )/0.02<BR>\n" );
document.write( "6000 = P*(1.372786 - 1 )/0.02<BR>\n" );
document.write( "6000 = P*(0.372786/0.02)<BR>\n" );
document.write( "6000 = P*18.6393<BR>\n" );
document.write( "6000 = 18.6393*P<BR>\n" );
document.write( "18.6393*P = 6000<BR>\n" );
document.write( "18.6393*P/18.6393 = 6000/18.6393<BR>\n" );
document.write( "P = 321.900501<BR>\n" );
document.write( "P = 321.91\r<BR>\n" );
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document.write( "So the answer is <font color=red size=4>$321.91</font>\r<BR>\n" );
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document.write( "I rounded up to the nearest penny so we could clear the hurdle. Notice how if P = 321.90, then we have FV equal to...<BR>\n" );
document.write( "FV = P*( (1+i)^n - 1 )/i<BR>\n" );
document.write( "FV = 321.90*( (1+0.02)^16 - 1 )/0.02<BR>\n" );
document.write( "FV = 321.90*( (1.02)^16 - 1 )/0.02<BR>\n" );
document.write( "FV = 321.90*(1.372786 - 1 )/0.02<BR>\n" );
document.write( "FV = 321.90*(0.372786/0.02)<BR>\n" );
document.write( "FV = 321.90*18.6393<BR>\n" );
document.write( "FV = 5999.99067<BR>\n" );
document.write( "FV = 5999.99<BR>\n" );
document.write( "Showing that we come up 1 cent short of our goal\r<BR>\n" );
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document.write( "On the other hand, if we have P = 321.91, then FV is...<BR>\n" );
document.write( "FV = P*( (1+i)^n - 1 )/i<BR>\n" );
document.write( "FV = 321.91*( (1+0.02)^16 - 1 )/0.02<BR>\n" );
document.write( "FV = 321.91*( (1.02)^16 - 1 )/0.02<BR>\n" );
document.write( "FV = 321.91*(1.372786 - 1 )/0.02<BR>\n" );
document.write( "FV = 321.91*(0.372786/0.02)<BR>\n" );
document.write( "FV = 321.91*18.6393<BR>\n" );
document.write( "FV = 6000.177063<BR>\n" );
document.write( "FV = 6000.18<BR>\n" );
document.write( "we haven't landed on the exact value of $6000 but overshot it (which is better than coming up short). So this confirms we have the correct answer of <font color=red size=4>$321.91</font><BR>\n" );
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