Ms. Jasmine invested RM X into an account that paid 3% interest compounded monthly. After 8 years, the investment will be used to pay the RM100,000 down payment for a condominium that cost RM500,000. The balance of the payment will be financed by a bank that charges 2% interest compounded monthly for 15 years. She plans to pay equal monthly payments to settle the debt. 

(a)	Determine the value of X.
(4 marks)
(b)	Calculate the monthly payment that she has to pay for the loan. 


Let's solve the problem step by step.

### Part (a): Determine the value of \(X\)

Given:
- Annual interest rate: \(3\%\) compounded monthly
- Time period: \(8\) years
- Future value needed: \(RM 100,000\)

To determine the initial investment \(X\), we use the future value formula for compound interest:

\[
FV = X \times \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \(FV\) is the future value (\(RM 100,000\)),
- \(r\) is the annual interest rate (\(0.03\)),
- \(n\) is the number of times the interest is compounded per year (\(12\) for monthly compounding),
- \(t\) is the number of years the money is invested (\(8\) years),
- \(X\) is the present value or initial investment.

Substituting the values:

\[
100,000 = X \times \left(1 + \frac{0.03}{12}\right)^{12 \times 8}
\]

Simplifying the equation:

\[
100,000 = X \times \left(1 + 0.0025\right)^{96}
\]
\[
100,000 = X \times (1.0025)^{96}
\]

Using the formula:

\[
(1.0025)^{96} \approx 1.26824
\]

So,

\[
100,000 = X \times 1.26824
\]

Finally, solving for \(X\):

\[
X = \frac{100,000}{1.26824} \approx RM 78,842.58
\]

So, the initial investment \(X\) is approximately \(RM 78,842.58\).

### Part (b): Calculate the monthly payment for the loan

Given:
- Remaining balance to be financed: \(RM 400,000\) (since the total cost is \(RM 500,000\) and the down payment is \(RM 100,000\))
- Annual interest rate: \(2\%\) compounded monthly
- Loan term: \(15\) years

The monthly payment can be calculated using the amortization formula:

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

Where:
- \(M\) is the monthly payment,
- \(P\) is the principal amount (\(RM 400,000\)),
- \(r\) is the monthly interest rate (\(\frac{2\%}{12} = 0.0016667\)),
- \(n\) is the total number of payments (\(15 \times 12 = 180\)).

Substituting the values:

\[
M = 400,000 \times \frac{0.0016667 \times (1.0016667)^{180}}{(1.0016667)^{180} - 1}
\]

First, calculate \((1.0016667)^{180}\):

\[
(1.0016667)^{180} \approx 1.34935
\]

So,

\[
M = 400,000 \times \frac{0.0016667 \times 1.34935}{1.34935 - 1}
\]
\[
M = 400,000 \times \frac{0.002249}{0.34935}
\]
\[
M \approx 400,000 \times 0.00644
\]
\[
M \approx RM 2,576.19
\]

So, the monthly payment that Ms. Jasmine has to pay for the loan is approximately \(RM 2,576.19\).

---

Do you have any questions or need further details?

**Related Questions:**

1. How would the initial investment \(X\) change if the interest rate was compounded quarterly instead of monthly?
2. What is the total amount Ms. Jasmine will pay over the 15 years for the loan?
3. How would a change in the interest rate to 4% affect the monthly payment?
4. What would happen if Ms. Jasmine made a lump-sum payment after 10 years? How would this affect her remaining payments?
5. Can Ms. Jasmine negotiate a lower interest rate on the loan? How would that impact her monthly payment?

**Tip:** When dealing with loans or investments, always double-check the interest compounding frequency as it significantly affects the overall amount paid or earned!

Learn how to calculate the initial investment and monthly loan payments using compound interest formulas. Solve for the initial investment required and the monthly payments needed to finance a condominium purchase with detailed step-by-step solutions.

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