A person owes $2200 at the end of 4 years and $5000 at the end of 9 years. Due to changes in their financial situation, the person was allowed to pay $1800 at the end of 2 year(s) and a final payment at the end of 12 years using 17.5% compounded annually for the settlement. Determine the amount required to settle the debt at the end of 12 years.

To solve this, we need to adjust both the payments due and the one made using the 17.5% compounded annually interest rate. The idea is to bring all payments to the common time point at the end of 12 years and calculate the final amount required to settle the debt.

### Step 1: Define the relevant interest formula
We use the compound interest formula to move the values forward in time:

\[
A = P(1 + i)^n
\]
where:
- $A$ is the future value,
- $P$ is the present value (initial amount),
- $i$ is the interest rate per period,
- $n$ is the number of periods.

The interest rate $i = 0.175$ (17.5% per year).

### Step 2: Future value of the debt of $2200 at the end of 4 years
This debt needs to be moved forward to the end of 12 years. The number of periods from 4 years to 12 years is 8 years, so:

\[
FV_{2200} = 2200(1 + 0.175)^8
\]

### Step 3: Future value of the debt of $5000 at the end of 9 years
This debt also needs to be moved to the end of 12 years, which is 3 years into the future:

\[
FV_{5000} = 5000(1 + 0.175)^3
\]

### Step 4: Future value of the payment of $1800 at the end of 2 years
This payment reduces the debt. Since it was made at the end of 2 years, it needs to be moved forward 10 years to the end of 12 years:

\[
FV_{1800} = 1800(1 + 0.175)^{10}
\]

### Step 5: Calculate the total future value of the debt
The total future debt at the end of 12 years is the sum of the future values of the two debts minus the future value of the payment made:

\[
FV_{\text{debt}} = FV_{2200} + FV_{5000} - FV_{1800}
\]

### Step 6: Determine the final payment
The amount required to settle the debt at the end of 12 years will be the total future value calculated in Step 5. Let’s compute this:

#### Step-by-step calculations:
1. $FV_{2200} = 2200(1 + 0.175)^8 = 2200 \times (1.175)^8 = 2200 \times 3.6122 = 7946.84$
2. $FV_{5000} = 5000(1 + 0.175)^3 = 5000 \times (1.175)^3 = 5000 \times 1.6112 = 8056.00$
3. $FV_{1800} = 1800(1 + 0.175)^{10} = 1800 \times (1.175)^{10} = 1800 \times 5.048 = 9086.40$

Now we can find the total future value of the debt:

\[
FV_{\text{debt}} = 7946.84 + 8056.00 - 9086.40 = 6916.44
\]

So, the amount required to settle the debt at the end of 12 years is **$6916.44**.

Would you like more details or have any questions?

Here are some related questions to explore:
1. What if the interest rate was compounded quarterly instead of annually?
2. How would the final payment change if an additional payment was made after 5 years?
3. How does the formula change if the interest is compounded continuously?
4. What if the person could pay $1000 at the end of each year for the first 4 years?
5. What is the present value of the debt if no interest were applied?

**Tip:** When calculating future values of debts and payments, always bring all amounts to the same point in time to ensure accuracy.

Determine the final payment needed to settle a debt using compound interest at 17.5% annually. This problem includes future value calculations for multiple debts and payments over 12 years, suitable for students in Grades 10-12.

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