Robert bought a piece of land for RM100,000. He made an initial down payment of RM20,000 and the balance was financed through a bank. For the repayment of the bank loan, he has to pay RM1,779.56 a month.
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iii.
If the bank's interest rate is 12% compounded monthly, find the number of monthly payments he will make to fully settle the loan.
(9 marks)

To find the number of monthly payments Robert will make to fully settle the loan, we will use the formula for the present value of an annuity. The formula is:

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

where:
- \( PV \) is the present value of the loan (the amount borrowed),
- \( PMT \) is the monthly payment,
- \( r \) is the monthly interest rate,
- \( n \) is the number of monthly payments.

First, we need to determine the principal amount borrowed. Robert made an initial down payment of RM20,000, so the amount financed through the bank is:

\[ 100,000 - 20,000 = 80,000 \text{ RM} \]

Given:
- \( PV = 80,000 \text{ RM} \)
- \( PMT = 1,779.56 \text{ RM} \)
- Annual interest rate = 12%, so the monthly interest rate \( r \) is:

\[ r = \frac{12\%}{12} = 1\% = 0.01 \]

Now, we can plug these values into the formula and solve for \( n \):

\[ 80,000 = 1,779.56 \times \frac{1 - (1 + 0.01)^{-n}}{0.01} \]

Rearranging to solve for \( n \):

\[ 80,000 \times 0.01 = 1,779.56 \times (1 - (1.01)^{-n}) \]

\[ 800 = 1,779.56 \times (1 - (1.01)^{-n}) \]

\[ \frac{800}{1,779.56} = 1 - (1.01)^{-n} \]

\[ 0.44963 = 1 - (1.01)^{-n} \]

\[ (1.01)^{-n} = 1 - 0.44963 \]

\[ (1.01)^{-n} = 0.55037 \]

Taking the natural logarithm on both sides:

\[ \ln((1.01)^{-n}) = \ln(0.55037) \]

\[ -n \ln(1.01) = \ln(0.55037) \]

\[ n = \frac{\ln(0.55037)}{-\ln(1.01)} \]

Calculating the values:

\[ n = \frac{\ln(0.55037)}{-\ln(1.01)} \approx \frac{-0.59784}{-0.00995} \approx 60.1 \]

Since the number of payments must be a whole number, Robert will make 61 monthly payments to fully settle the loan.

If you have any questions or need further details, feel free to ask. Here are 5 additional questions you might consider:

1. How do you calculate the effective annual interest rate from a monthly interest rate?
2. What is the difference between nominal and effective interest rates?
3. How does compound interest differ from simple interest?
4. Can you provide an example of calculating the monthly payment for a different loan amount and interest rate?
5. How does an amortization schedule work in loan repayments?

**Tip:** When dealing with loan repayments, always ensure that you convert the interest rate to match the payment frequency (e.g., monthly, quarterly) to get accurate results.

Learn how to calculate the number of monthly payments needed to fully settle a loan with compound interest. This problem involves a loan amount, down payment, monthly payments, and a specified interest rate. Get step-by-step guidance on solving similar financial problems.

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