Zandile borrowed money from a bank and she is repaying the loan by making monthly payments of R1144 at the end of every month, each payment includes interest and repayment of the principle. The loan will be paid off in 8 years and interest is calculated at a rate of 8.542% per year, compounded monthly. How much will Zandile still owe the bank immediately after making the 6th payment of the 4th year? 

To calculate how much Zandile will still owe the bank after the 6th payment of the 4th year, we will break down the problem into the following steps:

### Step 1: General Loan Formula
The loan can be modeled as an annuity, where the monthly payment \( P \), the interest rate, and the loan term are known. The formula for calculating the present value \( PV \) of an annuity (the initial loan amount) is:

\[
PV = P \times \frac{1 - (1 + i)^{-n}}{i}
\]

Where:
- \( P \) is the monthly payment amount (R1144).
- \( i \) is the monthly interest rate, calculated as the annual interest rate divided by 12.
- \( n \) is the total number of payments (8 years × 12 months = 96 months).

### Step 2: Calculate the Monthly Interest Rate
The annual interest rate is given as 8.542%, so the monthly interest rate is:

\[
i = \frac{8.542\%}{12} = \frac{0.08542}{12} = 0.00711833
\]

### Step 3: Calculate the Total Number of Payments
The loan term is 8 years, so the total number of payments over the life of the loan is:

\[
n = 8 \times 12 = 96 \text{ payments}
\]

### Step 4: Calculate the Loan Amount
Using the annuity formula for the present value, we calculate the loan amount \( PV \):

\[
PV = 1144 \times \frac{1 - (1 + 0.00711833)^{-96}}{0.00711833}
\]

Let's calculate the value of \( PV \):

\[
PV = 1144 \times \frac{1 - (1 + 0.00711833)^{-96}}{0.00711833}
\]

\[
PV = 1144 \times \frac{1 - (1.00711833)^{-96}}{0.00711833}
\]

### Step 5: Find Remaining Balance After 42 Payments
The loan is to be paid off in 96 payments. After the 6th payment of the 4th year, Zandile has made 42 payments (3 years × 12 + 6 payments = 42 payments).

The remaining balance of a loan after \( t \) payments can be found using the formula for the outstanding balance of an annuity:

\[
\text{Remaining Balance} = P \times \frac{1 - (1 + i)^{-(n - t)}}{i}
\]

Here:
- \( P = 1144 \)
- \( i = 0.00711833 \)
- \( n = 96 \) (total payments)
- \( t = 42 \) (payments made)

Substitute the values into the formula:

\[
\text{Remaining Balance} = 1144 \times \frac{1 - (1 + 0.00711833)^{-(96 - 42)}}{0.00711833}
\]

\[
\text{Remaining Balance} = 1144 \times \frac{1 - (1.00711833)^{-54}}{0.00711833}
\]

Let's calculate this expression to find out how much Zandile still owes after the 6th payment of the 4th year.

Zandile will still owe approximately **R51,139.53** to the bank immediately after making the 6th payment of the 4th year.

Would you like more details or have any questions about this process?

Here are five related questions to explore:
1. How is the monthly payment amount determined for a loan with compound interest?
2. How would Zandile's loan amount change if the interest rate increased to 10%?
3. What would be the total interest paid over the entire 8 years of the loan?
4. How would the repayment schedule differ if Zandile made additional payments each month?
5. What happens if Zandile decides to pay off the loan early after the 4th year?

**Tip:** Always make sure you understand the difference between interest compounded monthly and annually, as it affects the total cost of the loan.

Zandile borrowed money from a bank and she is repaying the loan by making monthly payments of R1144 at the end of every month, each payment includes interest and repayment of the principle. The loan will be paid off in 8 years and interest is calculated at a rate of 8.542% per year, compounded monthly. How much will Zandile still owe the bank immediately after making the 6th payment of the 4th year?

Learn how to calculate the remaining balance on Zandile's loan after 42 payments. The loan is repaid in monthly installments of R1144 with 8.542% interest, compounded monthly, over 8 years. Discover key concepts in annuities, loan amortization, and compound interest in this step-by-step solution.

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