Question 1166216: 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.
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! 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**.
The simple interest rate is $r = 11\%$. Time $t$ must be expressed in years.
## 📅 Timeline and Time Factors
Let "Today" be $t=0$. The focal date is $t=6$ months.
| Payment/Debt | Original Date | Time to Focal Date ($t$ in months) | Time Factor ($t$ in years) |
| :---: | :---: | :---: | :---: |
| **Debt 1** | 3 months ago | $6 - (-3) = 9$ months | $t_{D1} = 9/12$ |
| **Debt 2** | Today | $6 - 0 = 6$ months | $t_{D2} = 6/12$ |
| **Payment 1** | 2 months from today | $6 - 2 = 4$ months | $t_{P1} = 4/12$ |
| **Payment 2 (Final)** | 6 months from today | $6 - 6 = 0$ months | $t_{P2} = 0/12$ |
## 💰 Equation of Value
The fundamental equation of value is:
$$\text{Sum of (Debts + Interest) at Focal Date} = \text{Sum of (Payments + Interest) at Focal Date}$$
The formula for the future value (FV) with simple interest is $FV = P(1 + rt)$.
### 1. Future Value of the Debts
| Debt | Amount ($P$) | Time ($t$) | Calculation $P(1 + 0.11t)$ | FV at 6 Months |
| :---: | :---: | :---: | :---: | :---: |
| **D1** | $\$2,700$ | $9/12 = 0.75$ | $2,700 (1 + 0.11 \times 0.75)$ | $\$2,922.75$ |
| **D2** | $\$1,900$ | $6/12 = 0.50$ | $1,900 (1 + 0.11 \times 0.50)$ | $\$2,004.50$ |
| **Total Debts** | | | | **\$4,927.25** |
$$\text{Total Debts at Focal Date} = \$2,922.75 + \$2,004.50 = \mathbf{\$4,927.25}$$
### 2. Future Value of the Payments
Let $X$ be the final payment.
| Payment | Amount ($P$) | Time ($t$) | Calculation $P(1 + 0.11t)$ | FV at 6 Months |
| :---: | :---: | :---: | :---: | :---: |
| **P1** | $\$2,500$ | $4/12 \approx 0.3333$ | $2,500 (1 + 0.11 \times 4/12)$ | $\$2,591.67$ |
| **P2 (Final)** | $X$ | $0/12 = 0$ | $X (1 + 0.11 \times 0)$ | $X$ |
| **Total Payments** | | | | **\$2,591.67 + X** |
$$\text{Total Payments at Focal Date} = \$2,591.67 + X$$
## ⚖️ Solving for the Final Payment ($X$)
Set the total future value of the debts equal to the total future value of the payments:
$$\text{Total Debts} = \text{Total Payments}$$
$$4,927.25 = 2,591.67 + X$$
Solve for $X$:
$$X = 4,927.25 - 2,591.67$$
$$X = \mathbf{\$2,335.58}$$
The size of the final payment will be **\$2,335.58**.
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