document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193609>Question 1193609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A second-hand car was bought for ₱100,000 downpayment and <BR>\n" );
document.write( "monthly installments of ₱10,000 to be paid for 4 years. What is the <BR>\n" );
document.write( "cash price of the car if money is worth 12% compounded monthly? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848633 by </B><A HREF=/tutors/aboutme.mpl?userid=yurtman><B>yurtman(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=yurtman><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=848633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To calculate the cash price of the car, we need to find the present value of the monthly installment payments and add the downpayment.\r<BR>\n" );
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document.write( "### Step 1: Formula for Present Value of an Annuity<BR>\n" );
document.write( "The formula for the present value of an annuity is:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "PV = M \cdot \frac{1 - (1 + r)^{-n}}{r}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Where:  <BR>\n" );
document.write( "- \( PV \) = present value of the monthly installments (the amount equivalent to these payments today)  <BR>\n" );
document.write( "- \( M \) = monthly payment (\( 10,000 \))  <BR>\n" );
document.write( "- \( r \) = monthly interest rate (\( \frac{12\%}{12} = 0.01 \))  <BR>\n" );
document.write( "- \( n \) = total number of payments (\( 4 \, \text{years} \times 12 = 48 \))  \r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 2: Calculate the Present Value of Installments<BR>\n" );
document.write( "Substitute the known values:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "PV = 10,000 \cdot \frac{1 - (1 + 0.01)^{-48}}{0.01}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Calculate Step-by-Step:<BR>\n" );
document.write( "1. \( (1 + 0.01)^{-48} = 1.01^{-48} \approx 0.608437 \)<BR>\n" );
document.write( "2. \( 1 - 0.608437 = 0.391563 \)<BR>\n" );
document.write( "3. \( \frac{0.391563}{0.01} = 39,156.30 \)\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "PV \approx 39,156.30<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 3: Add the Downpayment<BR>\n" );
document.write( "The total cash price of the car is:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\text{Cash Price} = \text{Downpayment} + PV<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Substitute the values:\r<BR>\n" );
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document.write( "\text{Cash Price} = 100,000 + 39,156.30 = 139,156.30<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Final Answer:<BR>\n" );
document.write( "The cash price of the car is approximately **₱139,156.30**. </SPAN>\n" );
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