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We have a debt situation involving two amounts and three payments. The problem involves calculating the final payment with interest compounded semi-annually at 7% p.a. Let's break down the problem step by step:

### Given:
1. A debt of ₹ 8,000 is due 2 years from now.
2. Another debt of ₹ 10,000 is due 5 years from now.
3. Payments:
   - ₹ 2,000 in 3 years,
   - ₹ 5,000 in 6 years,
   - A final payment at the end of 4 years (this is what we need to find).
4. Interest rate: 7% p.a. compounded semi-annually.

### Step-by-Step Solution:

We'll approach the problem by calculating the present value (PV) of the payments and setting them equal to the present value of the debts. The idea is to bring all amounts to the same time (present value, typically at time = 0) and solve for the unknown final payment.

#### 1. Interest Rate Conversion:
The nominal rate is 7% p.a. compounded semi-annually, so we convert it:
- Semi-annual rate \( = \frac{7\%}{2} = 3.5\% \) or 0.035 per half-year.

#### 2. Present Value of Debts:
We need to calculate the present value (at time = 0) of the two debts:

- **Debt 1 (₹ 8,000 due in 2 years)**:
  - 2 years = 4 semi-annual periods.
  \[
  PV_1 = \frac{8,000}{(1 + 0.035)^4}
  \]

- **Debt 2 (₹ 10,000 due in 5 years)**:
  - 5 years = 10 semi-annual periods.
  \[
  PV_2 = \frac{10,000}{(1 + 0.035)^{10}}
  \]

#### 3. Present Value of Payments:
Now, calculate the present value of the known payments:

- **Payment 1 (₹ 2,000 in 3 years)**:
  - 3 years = 6 semi-annual periods.
  \[
  PV_{P1} = \frac{2,000}{(1 + 0.035)^6}
  \]

- **Payment 2 (₹ 5,000 in 6 years)**:
  - 6 years = 12 semi-annual periods.
  \[
  PV_{P2} = \frac{5,000}{(1 + 0.035)^{12}}
  \]

- **Final Payment (unknown amount at the end of 4 years)**:
  - 4 years = 8 semi-annual periods.
  \[
  PV_{F} = \frac{X}{(1 + 0.035)^8}
  \]

#### 4. Equation Setup:
The total present value of the debts must equal the total present value of the payments. Thus, the equation becomes:

\[
PV_1 + PV_2 = PV_{P1} + PV_{P2} + PV_{F}
\]

Substituting the values:

\[
\frac{8,000}{(1 + 0.035)^4} + \frac{10,000}{(1 + 0.035)^{10}} = \frac{2,000}{(1 + 0.035)^6} + \frac{5,000}{(1 + 0.035)^{12}} + \frac{X}{(1 + 0.035)^8}
\]

#### 5. Solve for X (Final Payment):
We will now calculate the individual present values and solve for \( X \).

Let me calculate the values and find \( X \).The final payment required to fully repay the debt is approximately ₹ 12,015.63.

Would you like a detailed breakdown or any clarifications?

Here are 5 related questions to expand on this:
1. How does semi-annual compounding differ from annual compounding?
2. What if the interest rate was compounded quarterly instead of semi-annually?
3. How would the payments change if they were made annually instead of semi-annually?
4. What would happen if an additional payment were introduced earlier, say in year 1?
5. How would inflation impact the calculation of these debts and payments?

**Tip**: Always check the compounding period (semi-annual, quarterly, etc.) carefully when solving financial problems, as it significantly affects the outcome.

A debt of ₹ 8,000 due 2 years from now and ₹ 10,000 due 5 years from now is to be repaid by a payment of ₹ 2,000 in 3 years, a payment of ₹ 5,000 in 6 years, and a final payment at the end of 4 years. If the interest rate is 7% p.a. compounded semi-annually, how much is the final payment?

Learn how to calculate the final payment for a debt of ₹ 8,000 and ₹ 10,000 using compound interest at a 7% semi-annual rate. Step-by-step breakdown includes present value calculations and interest rate conversions. Suitable for university finance or accounting students.

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