Question 1166217: AW barrowed $9,000 on January 30, 2022 and agreed to pay 14% simple interest on the balance outstanding at any time. She paid $5,000 on March 9, 2022 and $2,500 on May 25, 2022. How much will she have to pay on June 30, 2022 in order to pay off the debt? Use June 30, 2022 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 partial payments. To find the final payment, we will use the **focal date** method, moving all transactions (the loan and the payments) to the focal date of **June 30, 2022**.
The interest rate is $r = 14\%$ simple interest.
## 📅 Calculating the Time Periods
We need to find the number of days between each transaction date and the focal date (June 30, 2022).
| Month | Days in Month |
| :---: | :---: |
| January | 31 |
| February (2022) | 28 |
| March | 31 |
| April | 30 |
| May | 31 |
| June | 30 |
| Date | Calculation (Days to June 30) | Time ($t$) in years |
| :---: | :---: | :---: |
| **Loan:** Jan 30 | 1 in Jan + 28 in Feb + 31 in Mar + 30 in Apr + 31 in May + 30 in June = **151 days** | $t_L = 151/365$ |
| **Payment 1:** Mar 9 | 22 in Mar + 30 in Apr + 31 in May + 30 in June = **113 days** | $t_1 = 113/365$ |
| **Payment 2:** May 25 | 6 in May + 30 in June = **36 days** | $t_2 = 36/365$ |
| **Final Payment:** Jun 30 | **0 days** | $t_F = 0/365$ |
## 💰 Moving Transactions to the Focal Date
The fundamental equation of value states:
$$\text{Sum of (Loan + Interest) at Focal Date} = \text{Sum of (Payments + Interest) at Focal Date}$$
The formula for the future value (FV) with simple interest is $FV = P(1 + rt)$.
### 1. Loan Value (Amount Owed)
The loan amount is $L = \$9,000$.
$$FV_{\text{Loan}} = L(1 + r t_L)$$
$$FV_{\text{Loan}} = 9,000 \left(1 + 0.14 \times \frac{151}{365}\right)$$
$$FV_{\text{Loan}} = 9,000 (1 + 0.0577534...) \approx 9,000 (1.0577534)$$
$$FV_{\text{Loan}} \approx \mathbf{\$9,519.78}$$
### 2. Payment 1 Value
The first payment is $P_1 = \$5,000$.
$$FV_{\text{P1}} = P_1(1 + r t_1)$$
$$FV_{\text{P1}} = 5,000 \left(1 + 0.14 \times \frac{113}{365}\right)$$
$$FV_{\text{P1}} = 5,000 (1 + 0.0432328...) \approx 5,000 (1.0432328)$$
$$FV_{\text{P1}} \approx \mathbf{\$5,216.16}$$
### 3. Payment 2 Value
The second payment is $P_2 = \$2,500$.
$$FV_{\text{P2}} = P_2(1 + r t_2)$$
$$FV_{\text{P2}} = 2,500 \left(1 + 0.14 \times \frac{36}{365}\right)$$
$$FV_{\text{P2}} = 2,500 (1 + 0.0137534...) \approx 2,500 (1.0137534)$$
$$FV_{\text{P2}} \approx \mathbf{\$2,534.38}$$
### 4. Final Payment ($X$)
The final payment is $X$, and since it occurs on the focal date, its future value is just $X$.
$$FV_{\text{Final}} = X$$
## ⚖️ Solving for the Final Payment
$$\text{Loan Value} = \text{Payment 1 Value} + \text{Payment 2 Value} + \text{Final Payment}$$
$$FV_{\text{Loan}} = FV_{\text{P1}} + FV_{\text{P2}} + X$$
$$9,519.78 = 5,216.16 + 2,534.38 + X$$
$$9,519.78 = 7,750.54 + X$$
$$X = 9,519.78 - 7,750.54$$
$$X = \mathbf{\$1,769.24}$$
AW will have to pay **\$1,769.24** on June 30, 2022 to pay off the debt.
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