document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173889>Question 1173889</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Aya and Harumi would like to buy a house and their dream house costs $500,000.  They have $50,000 saved up for a down payment but would still need to take out a mortgage loan for the remaining $450,000 and they’re not sure whether they could afford the monthly loan payments.  The bank has offered them an interest rate of 4.25%, compounded monthly. \r<BR>\n" );
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document.write( "Q1 How much would they have to be able to afford to pay each month in order to pay off their mortgage in 25 years?\r<BR>\n" );
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document.write( "Q2 What is the total amount that would be paid to the lender after 25 years of payments? \r<BR>\n" );
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document.write( "Q3 What if Aya and Harumi could only afford a monthly payment of $2,000? What would be the maximum mortgage amount they could afford to borrow from the bank, if all the other conditions were the same?\r<BR>\n" );
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document.write( "Q4 What is the total amount that would be paid to the lender over 25 years? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850715 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=850715\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this mortgage calculation problem step-by-step.\r<BR>\n" );
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document.write( "**Q1: Monthly Payment for a $450,000 Loan (25 years, 4.25%)**\r<BR>\n" );
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document.write( "* **Loan Amount (P):** $450,000<BR>\n" );
document.write( "* **Annual Interest Rate (r):** 4.25% or 0.0425<BR>\n" );
document.write( "* **Monthly Interest Rate (i):** 0.0425 / 12 ≈ 0.00354167<BR>\n" );
document.write( "* **Number of Payments (n):** 25 years * 12 months/year = 300 months\r<BR>\n" );
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document.write( "We'll use the mortgage payment formula:\r<BR>\n" );
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document.write( "M = P [ i(1 + i)^n ] / [ (1 + i)^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 = Principal loan amount<BR>\n" );
document.write( "* i = Monthly interest rate<BR>\n" );
document.write( "* n = Total number of payments\r<BR>\n" );
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document.write( "Let's calculate:\r<BR>\n" );
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document.write( "M = 450000 [ 0.00354167(1 + 0.00354167)^300 ] / [ (1 + 0.00354167)^300  -  1]\r<BR>\n" );
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document.write( "M = 450000 [ 0.00354167(2.871036) ] / [ 2.871036  -  1 ]\r<BR>\n" );
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document.write( "M = 450000 [ 0.010196 ] / [ 1.871036 ]\r<BR>\n" );
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document.write( "M = 450000 * 0.00545\r<BR>\n" );
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document.write( "M ≈ $2,452.50\r<BR>\n" );
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document.write( "Therefore, Aya and Harumi would have to afford to pay approximately **$2,452.50** each month.\r<BR>\n" );
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document.write( "**Q2: Total Amount Paid to the Lender**\r<BR>\n" );
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document.write( "* Total Paid = Monthly Payment * Number of Payments<BR>\n" );
document.write( "* Total Paid = $2,452.50 * 300<BR>\n" );
document.write( "* Total Paid = $735,750\r<BR>\n" );
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document.write( "The total amount paid to the lender would be **$735,750**.\r<BR>\n" );
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document.write( "**Q3: Maximum Mortgage Amount with $2,000 Monthly Payment**\r<BR>\n" );
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document.write( "* **Monthly Payment (M):** $2,000<BR>\n" );
document.write( "* **Monthly Interest Rate (i):** 0.00354167<BR>\n" );
document.write( "* **Number of Payments (n):** 300\r<BR>\n" );
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document.write( "We'll rearrange the mortgage payment formula to solve for P:\r<BR>\n" );
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document.write( "P = M [ (1 + i)^n  -  1 ] / [ i(1 + i)^n ]\r<BR>\n" );
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document.write( "P = 2000 [ (1 + 0.00354167)^300  -  1 ] / [ 0.00354167(1 + 0.00354167)^300 ]\r<BR>\n" );
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document.write( "P = 2000 [ 1.871036 ] / [ 0.010196 ]\r<BR>\n" );
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document.write( "P = 2000 * 183.506\r<BR>\n" );
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document.write( "P ≈ $367,012\r<BR>\n" );
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document.write( "The maximum mortgage amount they could afford to borrow would be approximately **$367,012**.\r<BR>\n" );
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document.write( "**Q4: Total Amount Paid to the Lender (with $2,000 Payment)**\r<BR>\n" );
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document.write( "* Total Paid = Monthly Payment * Number of Payments<BR>\n" );
document.write( "* Total Paid = $2,000 * 300<BR>\n" );
document.write( "* Total Paid = $600,000\r<BR>\n" );
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document.write( "The total amount paid to the lender would be **$600,000**.<BR>\n" );
document.write( " </SPAN>\n" );
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