Cindy borrowed money from a bank and she is repaying the loan by making monthly payments of R1664 at the end of every month. The loan will be paid off in 8 years and interest is calculated at a rate of 10.202% per year, compounded monthly. How much will Cindy still owe the bank immediately after making payment number 68?

We can solve this using the formula for the remaining balance (outstanding loan) on an amortized loan. The key elements here are the loan payment amount, the interest rate, the total number of payments, and the number of payments made so far.

### Step 1: Formula for the Remaining Balance
The formula for the remaining balance \(B_n\) after \(n\) payments is:

\[
B_n = P \cdot \frac{(1 + r)^N - (1 + r)^n}{(1 + r)^N - 1}
\]

Where:
- \(B_n\) is the remaining balance after \(n\) payments.
- \(P\) is the monthly payment (R1664).
- \(r\) is the monthly interest rate (annual rate divided by 12).
- \(N\) is the total number of payments.
- \(n\) is the number of payments made so far.

### Step 2: Gather the Data
- Monthly payment: \(P = 1664\)
- Annual interest rate: \(10.202\%\) (monthly interest rate \(r = \frac{10.202\%}{12} = 0.00850167\))
- Loan term: 8 years (total number of payments \(N = 8 \times 12 = 96\))
- Number of payments made so far: \(n = 68\)

### Step 3: Calculate the Remaining Balance
Now we substitute the known values into the formula and calculate the remaining balance after 68 payments. I will calculate that for you.After Cindy makes her 68th payment, she will still owe the bank approximately **R631.21**.

Would you like more details on the steps involved, or do you have any further questions?

Here are five related questions to consider:
1. How would the remaining balance change if the interest rate increased?
2. What if Cindy decided to make larger monthly payments? How would that affect the loan term?
3. How much total interest will Cindy have paid by the time she makes her 68th payment?
4. How can Cindy pay off the loan faster, and what would be the financial impact?
5. What happens if the loan is repaid earlier than scheduled?

**Tip:** The higher the interest rate, the more interest accumulates over time, which increases the total cost of the loan.

Learn how to calculate the remaining balance on Cindy's loan after making 68 payments of R1664 per month at an interest rate of 10.202% compounded monthly. This problem demonstrates the application of loan amortization formulas and is suitable for students in Grades 11-12.

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