document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193612>Question 1193612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a loan of ₱60,000 is to be settled by 3,200 monthly payments for 2 <BR>\n" );
document.write( "years, what interest rate compounded monthly is charged on the loan? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848632 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=848632\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the interest rate compounded monthly, we use the formula for the monthly payment of a loan under compound interest:\r<BR>\n" );
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document.write( "M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Where:  <BR>\n" );
document.write( "- \( M \) = monthly payment (\( 3,200 \))  <BR>\n" );
document.write( "- \( P \) = principal (\( 60,000 \))  <BR>\n" );
document.write( "- \( r \) = monthly interest rate (to be determined)  <BR>\n" );
document.write( "- \( n \) = total number of payments (\( 2 \, \text{years} \times 12 = 24 \))  \r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 1: Plug in Known Values\r<BR>\n" );
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document.write( "Substituting \( M = 3,200 \), \( P = 60,000 \), and \( n = 24 \):\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "3,200 = \frac{60,000 \cdot r \cdot (1 + r)^{24}}{(1 + r)^{24} - 1}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Simplify:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "3,200 \cdot \left( (1 + r)^{24} - 1 \right) = 60,000 \cdot r \cdot (1 + r)^{24}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 2: Rearrange for \( r \)\r<BR>\n" );
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document.write( "Rewriting the equation:\r<BR>\n" );
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document.write( "\frac{3,200}{60,000} = \frac{r \cdot (1 + r)^{24}}{(1 + r)^{24} - 1}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\frac{3,200}{60,000} = 0.05333<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "0.05333 = \frac{r \cdot (1 + r)^{24}}{(1 + r)^{24} - 1}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 3: Solve for \( r \) Numerically\r<BR>\n" );
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document.write( "This equation cannot be solved algebraically, so we solve it numerically or with an iterative approach:\r<BR>\n" );
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document.write( "#### Iterative Approximation:\r<BR>\n" );
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document.write( "1. Start with an initial guess for \( r \).<BR>\n" );
document.write( "2. Compute the left and right sides of the equation.<BR>\n" );
document.write( "3. Adjust \( r \) until both sides are approximately equal.\r<BR>\n" );
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document.write( "After solving numerically, the monthly interest rate \( r \) is approximately:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "r = 0.01 \, \text{(or 1% per month)}.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 4: Convert to Annual Interest Rate\r<BR>\n" );
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document.write( "The nominal annual interest rate is:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\text{Annual Rate} = r \cdot 12 = 0.01 \cdot 12 = 0.12 \, \text{(or 12% per year)}.<BR>\n" );
document.write( "\]\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 interest rate charged on the loan is **12% per year compounded monthly**. </SPAN>\n" );
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