document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166216>Question 1166216</A>: <I><SPAN STYLE=\"word-break: break-all;\"> LH should have paid a loan company $2,700 3 months ago and should also pay $\1,900 today. He agrees to pay $ 2,500 in 2 months and the rest in 6 months, and agrees to include interest at 11%. What would be the size of his final payment? Use 6 months as the focal date. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852865 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.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=852865\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a time-value of money problem involving simple interest and a rescheduled set of payments. To find the final payment, we will use the **focal date** method, moving all debts (old payments) and all new payments to the agreed-upon focal date of **6 months from today**.\r<BR>\n" );
document.write( "\n" );
document.write( "The simple interest rate is $r = 11\%$. Time $t$ must be expressed in years.\r<BR>\n" );
document.write( "\n" );
document.write( "## 📅 Timeline and Time Factors\r<BR>\n" );
document.write( "\n" );
document.write( "Let \"Today\" be $t=0$. The focal date is $t=6$ months.\r<BR>\n" );
document.write( "\n" );
document.write( "| Payment/Debt | Original Date | Time to Focal Date ($t$ in months) | Time Factor ($t$ in years) |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| **Debt 1** | 3 months ago | $6 - (-3) = 9$ months | $t_{D1} = 9/12$ |<BR>\n" );
document.write( "| **Debt 2** | Today | $6 - 0 = 6$ months | $t_{D2} = 6/12$ |<BR>\n" );
document.write( "| **Payment 1** | 2 months from today | $6 - 2 = 4$ months | $t_{P1} = 4/12$ |<BR>\n" );
document.write( "| **Payment 2 (Final)** | 6 months from today | $6 - 6 = 0$ months | $t_{P2} = 0/12$ |\r<BR>\n" );
document.write( "\n" );
document.write( "## 💰 Equation of Value\r<BR>\n" );
document.write( "\n" );
document.write( "The fundamental equation of value is:<BR>\n" );
document.write( "$$\text{Sum of (Debts + Interest) at Focal Date} = \text{Sum of (Payments + Interest) at Focal Date}$$\r<BR>\n" );
document.write( "\n" );
document.write( "The formula for the future value (FV) with simple interest is $FV = P(1 + rt)$.\r<BR>\n" );
document.write( "\n" );
document.write( "### 1. Future Value of the Debts\r<BR>\n" );
document.write( "\n" );
document.write( "| Debt | Amount ($P$) | Time ($t$) | Calculation $P(1 + 0.11t)$ | FV at 6 Months |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| **D1** | $\$2,700$ | $9/12 = 0.75$ | $2,700 (1 + 0.11 \times 0.75)$ | $\$2,922.75$ |<BR>\n" );
document.write( "| **D2** | $\$1,900$ | $6/12 = 0.50$ | $1,900 (1 + 0.11 \times 0.50)$ | $\$2,004.50$ |<BR>\n" );
document.write( "| **Total Debts** | | | | **\$4,927.25** |\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Total Debts at Focal Date} = \$2,922.75 + \$2,004.50 = \mathbf{\$4,927.25}$$\r<BR>\n" );
document.write( "\n" );
document.write( "### 2. Future Value of the Payments\r<BR>\n" );
document.write( "\n" );
document.write( "Let $X$ be the final payment.\r<BR>\n" );
document.write( "\n" );
document.write( "| Payment | Amount ($P$) | Time ($t$) | Calculation $P(1 + 0.11t)$ | FV at 6 Months |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| **P1** | $\$2,500$ | $4/12 \approx 0.3333$ | $2,500 (1 + 0.11 \times 4/12)$ | $\$2,591.67$ |<BR>\n" );
document.write( "| **P2 (Final)** | $X$ | $0/12 = 0$ | $X (1 + 0.11 \times 0)$ | $X$ |<BR>\n" );
document.write( "| **Total Payments** | | | | **\$2,591.67 + X** |\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Total Payments at Focal Date} = \$2,591.67 + X$$\r<BR>\n" );
document.write( "\n" );
document.write( "## ⚖️ Solving for the Final Payment ($X$)\r<BR>\n" );
document.write( "\n" );
document.write( "Set the total future value of the debts equal to the total future value of the payments:<BR>\n" );
document.write( "$$\text{Total Debts} = \text{Total Payments}$$<BR>\n" );
document.write( "$$4,927.25 = 2,591.67 + X$$\r<BR>\n" );
document.write( "\n" );
document.write( "Solve for $X$:<BR>\n" );
document.write( "$$X = 4,927.25 - 2,591.67$$<BR>\n" );
document.write( "$$X = \mathbf{\$2,335.58}$$\r<BR>\n" );
document.write( "\n" );
document.write( "The size of the final payment will be **\$2,335.58**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );