Question 1166217
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.