document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179081>Question 1179081</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Nick has been depositing $ 190 in a savings account every three months for the past three years. This account paid 3 % convertible quarterly. Nick has just made the last deposit. Nick is buying a car for $ 14000. He is taking out a car loan. He will use the accumulated value of his savings account as the downpayment on the car. The loan is at 3 % convertible semiannually and has a term of 4 years. Find the size of Nick's monthly car loan payment. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850265 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850265\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break this problem into two parts:\r<BR>\n" );
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document.write( "**Part 1: Calculate the Accumulated Value of Nick's Savings Account**\r<BR>\n" );
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document.write( "1.  **Deposits:** $190 every three months (quarterly)<BR>\n" );
document.write( "2.  **Time Period:** 3 years<BR>\n" );
document.write( "3.  **Interest Rate:** 3% convertible quarterly (0.03/4 = 0.0075 per quarter)<BR>\n" );
document.write( "4.  **Number of Deposits:** 3 years * 4 quarters/year = 12 deposits\r<BR>\n" );
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document.write( "We can use the future value of an ordinary annuity formula:\r<BR>\n" );
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document.write( "FV = P * [((1 + r)^n - 1) / r]\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* FV = Future Value<BR>\n" );
document.write( "* P = Periodic Payment ($190)<BR>\n" );
document.write( "* r = Interest Rate per Period (0.0075)<BR>\n" );
document.write( "* n = Number of Periods (12)\r<BR>\n" );
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document.write( "FV = 190 * [((1 + 0.0075)^12 - 1) / 0.0075]<BR>\n" );
document.write( "FV = 190 * [(1.0075^12 - 1) / 0.0075]<BR>\n" );
document.write( "FV = 190 * [(1.093806897 - 1) / 0.0075]<BR>\n" );
document.write( "FV = 190 * [0.093806897 / 0.0075]<BR>\n" );
document.write( "FV = 190 * 12.50758627<BR>\n" );
document.write( "FV ≈ $2376.44\r<BR>\n" );
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document.write( "**Part 2: Calculate Nick's Monthly Car Loan Payment**\r<BR>\n" );
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document.write( "1.  **Car Price:** $14,000<BR>\n" );
document.write( "2.  **Down Payment:** $2376.44 (from the savings account)<BR>\n" );
document.write( "3.  **Loan Amount:** $14,000 - $2376.44 = $11,623.56<BR>\n" );
document.write( "4.  **Loan Term:** 4 years<BR>\n" );
document.write( "5.  **Interest Rate:** 3% convertible semiannually (0.03/2 = 0.015 per 6 months)<BR>\n" );
document.write( "6.  **Number of Semiannual Periods:** 4 years * 2 = 8<BR>\n" );
document.write( "7.  **Number of Monthly Payments:** 4 years * 12 months/year = 48\r<BR>\n" );
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document.write( "First, we need to find the equivalent monthly interest rate.\r<BR>\n" );
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document.write( "* (1 + 0.015)^2 = 1.030225. This is the effective annual rate.<BR>\n" );
document.write( "* (1.030225)^(1/12) = 1.002470126. This is the monthly multiplier.<BR>\n" );
document.write( "* Monthly Interest Rate = 1.002470126 - 1 = 0.002470126\r<BR>\n" );
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document.write( "Now, we can use the loan payment formula:\r<BR>\n" );
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document.write( "M = P * [r(1 + r)^n] / [(1 + r)^n - 1]\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* M = Monthly Payment<BR>\n" );
document.write( "* P = Loan Amount ($11,623.56)<BR>\n" );
document.write( "* r = Monthly Interest Rate (0.002470126)<BR>\n" );
document.write( "* n = Number of Monthly Payments (48)\r<BR>\n" );
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document.write( "M = 11623.56 * [0.002470126 * (1.002470126)^48] / [(1.002470126)^48 - 1]<BR>\n" );
document.write( "M = 11623.56 * [0.002470126 * 1.127493457] / [1.127493457 - 1]<BR>\n" );
document.write( "M = 11623.56 * [0.002785721] / [0.127493457]<BR>\n" );
document.write( "M = 11623.56 * 0.021850327<BR>\n" );
document.write( "M ≈ $253.97\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "Nick's monthly car loan payment will be approximately $253.97.<BR>\n" );
document.write( " </SPAN>\n" );
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