

A debt of RM7000 matures at the end of 2nd year and another RM8000  matures at the end  of 6th year. John Rambo decides to settle by making 2 equal payments one on the 4th year and another on the 7th year. The payments were RM8171.05 with the assumption money is worth 6% compounded semi-annually with the focal date at the 7th year.


True

False

This problem involves determining whether the proposed payment plan (two payments of RM8171.05) is equivalent to the original debt amounts of RM7000 and RM8000, given a 6% interest rate compounded semi-annually, with the focal date at the 7th year.

Let’s break down the problem:

### Step 1: Understanding the terms
- **Debts:**
  - RM7000 is due at the end of the 2nd year.
  - RM8000 is due at the end of the 6th year.
  
- **Payments:**
  - Two equal payments of RM8171.05 are made: one at the end of the 4th year, and another at the end of the 7th year.
  
- **Interest rate:**
  - 6% per annum compounded semi-annually, which gives an interest rate per half-year of \( \frac{6\%}{2} = 3\% \) per half-year.

- **Focal date:**
  - The 7th year.

### Step 2: Finding the value of debts at the 7th year
We need to calculate the value of the debts as of the 7th year.

1. **RM7000 at the end of 2nd year (compounded forward to 7th year):**
   - The time difference from the 2nd year to the 7th year is 5 years or 10 half-years.
   - Using the compound interest formula:  
     \[
     A = P \left( 1 + \frac{i}{n} \right)^{nt}
     \]
     where \( P \) is the principal, \( i \) is the annual interest rate, \( n \) is the number of compounding periods per year, and \( t \) is the number of years.
     \[
     A = 7000 \left( 1 + 0.03 \right)^{10}
     \]
     \[
     A = 7000 \times (1.03)^{10} = 7000 \times 1.3439 \approx 9407.52
     \]

2. **RM8000 at the end of 6th year (compounded forward to 7th year):**
   - The time difference from the 6th year to the 7th year is 1 year or 2 half-years.
   - Using the same formula:
     \[
     A = 8000 \left( 1 + 0.03 \right)^2
     \]
     \[
     A = 8000 \times (1.03)^2 = 8000 \times 1.0609 \approx 8487.20
     \]

   So, the total value of the debts at the 7th year is:
   \[
   9407.52 + 8487.20 = 17894.72
   \]

### Step 3: Calculating the value of payments at the 7th year
Now, we’ll calculate the value of the two equal payments (RM8171.05) made at the 4th year and the 7th year.

1. **RM8171.05 paid at the end of 4th year (compounded forward to 7th year):**
   - The time difference from the 4th year to the 7th year is 3 years or 6 half-years.
   \[
   A = 8171.05 \left( 1 + 0.03 \right)^6
   \]
   \[
   A = 8171.05 \times (1.03)^6 = 8171.05 \times 1.19405 \approx 9759.24
   \]

2. **RM8171.05 paid at the end of 7th year (no compounding needed since it’s the focal date):**
   \[
   A = 8171.05
   \]

   The total value of the payments at the 7th year is:
   \[
   9759.24 + 8171.05 = 17930.29
   \]

### Step 4: Comparison
- **Debts at the 7th year:** RM17894.72
- **Payments at the 7th year:** RM17930.29

The payments (RM17930.29) exceed the total debt (RM17894.72) by a small amount. Since the payments cover the debt, **the statement is true**.

---

Would you like more details or have any questions?

Here are five related questions you can explore:
1. How would the solution change if the interest rate was compounded quarterly instead of semi-annually?
2. What if the focal date was the 4th year instead of the 7th year?
3. How would you solve this problem if John made only one payment at the 7th year?
4. How does the principle of time value of money apply to this problem?
5. How would the payment change if the second debt matured in the 8th year?

**Tip:** Always ensure to calculate interest for every period correctly, as minor errors in exponentiation can lead to significant differences in results.

A debt of RM7000 matures at the end of 2nd year and another RM8000 matures at the end of 6th year. John Rambo decides to settle by making 2 equal payments one on the 4th year and another on the 7th year. The payments were RM8171.05 with the assumption money is worth 6% compounded semi-annually with the focal date at the 7th year.

Explore how to settle two debts of RM7000 and RM8000 with two equal payments of RM8171.05, using a 6% semi-annual compound interest rate. This solution covers key concepts like the time value of money and the compound interest formula, calculating future value at the 7th year. Suitable for college-level finance or economics studies.

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