Cindy borrowed money from a bank and she is repaying the loan by making monthly payments of R621 at the beginning of every month, the first payment is on the day that she received the loan. The loan will be paid off after 5 payments and interest is calculated at a rate of 9.903% per year, compounded monthly. How much will Cindy still owe the bank immediately before making payment number 4?

To determine how much Cindy still owes the bank immediately before making payment number 4, we can follow these steps:

### 1. **Determine the Effective Monthly Interest Rate**
The annual interest rate is given as 9.903%, compounded monthly. To find the monthly interest rate, we divide the annual rate by 12:
\[
i = \frac{9.903\%}{12} = \frac{0.09903}{12} = 0.0082525 \text{ (per month)}
\]

### 2. **Calculate the Present Value of the Loan**
Since Cindy makes her payments at the beginning of each period, the loan can be modeled as an annuity due. The present value \( PV \) of an annuity due can be calculated using the formula:
\[
PV = P \times \frac{1 - (1 + i)^{-n}}{i} \times (1 + i)
\]
where:
- \( P = R621 \) (the payment amount),
- \( i = 0.0082525 \) (the monthly interest rate),
- \( n = 5 \) (the total number of payments).

Substitute the values into the formula:
\[
PV = 621 \times \frac{1 - (1 + 0.0082525)^{-5}}{0.0082525} \times (1 + 0.0082525)
\]

### 3. **Calculate the Outstanding Balance Before Payment 4**
The outstanding balance before the 4th payment is made can be calculated by considering that 3 payments have already been made. We need to subtract the present value of the remaining 2 payments from the original present value of the loan.

To find the balance after 3 payments, we can calculate the present value of the remaining 2 payments:
\[
PV_{remaining} = 621 \times \frac{1 - (1 + 0.0082525)^{-2}}{0.0082525} \times (1 + 0.0082525)
\]

### 4. **Compute the Exact Values**

Let's compute these calculations.

#### Present Value of Loan:
\[
PV = 621 \times \frac{1 - (1 + 0.0082525)^{-5}}{0.0082525} \times (1 + 0.0082525)
\]
#### Present Value of Remaining Payments:
\[
PV_{remaining} = 621 \times \frac{1 - (1 + 0.0082525)^{-2}}{0.0082525} \times (1 + 0.0082525)
\]

Let me compute these values.### 5. **Results**

- The **original loan amount** (present value when the loan was first taken) was approximately **R3054.59**.
- The **outstanding balance** immediately before making the 4th payment, considering the remaining 2 payments, is approximately **R1236.92**.

### Summary
Before making payment number 4, Cindy will still owe the bank **R1236.92**.

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Do you want more details or have any questions? Here are some related questions you might consider:

1. How does the calculation change if the payments were made at the end of each month?
2. What is the total interest paid over the 5-month period?
3. How would the loan balance change with a different interest rate?
4. Can you compute the outstanding balance after each payment?
5. What if Cindy decided to make a lump sum payment instead of monthly payments?

**Tip:** When calculating loans or annuities, always check if payments are made at the beginning or end of the period, as this affects the calculation method.

Learn how to calculate Cindy's loan repayment before making payment number 4. This problem involves compound interest and annuity due calculations at a specific interest rate and payment schedule. Suitable for finance students or those studying compound interest calculations.

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