Certainly! Here's the step-by-step process to solve the problem, including the formulas used.

### Step 1: Understand the Problem
You need to find the amount owed after 14 years on a loan of $24,000, with an interest rate of 8.25%, compounded annually. No payments are made during this time.

### Step 2: Identify the Given Values
- **Principal (P)**: $24,000
- **Annual interest rate (r)**: 8.25% or 0.0825 in decimal form
- **Time (t)**: 14 years
- **Compounding frequency (n)**: 1 time per year (since it's compounded annually)

### Step 3: Write Down the Compound Interest Formula
The compound interest formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ is the amount owed after $ t $ years.
- $ P $ is the initial principal (amount borrowed).
- $ r $ is the annual interest rate (as a decimal).
- $ n $ is the number of times the interest is compounded per year.
- $ t $ is the number of years the money is borrowed.

### Step 4: Substitute the Values
Substitute the given values into the formula:
\[
A = 24000 \left(1 + \frac{0.0825}{1}\right)^{1 \times 14}
\]

### Step 5: Simplify the Expression
Calculate the amount inside the parentheses:
\[
1 + \frac{0.0825}{1} = 1.0825
\]
Raise it to the power of 14:
\[
A = 24000 \times 1.0825^{14}
\]

### Step 6: Calculate the Final Amount
Using a calculator or computational tool:
\[
A \approx 24000 \times 3.034
\]
\[
A \approx 72,812
\]

### Step 7: Round the Answer
The final amount owed after 14 years is **$72,812**.

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Would you like to explore this further or have any questions?

Here are 8 related questions:
1. How would the amount change if the loan was compounded semi-annually?
2. Can you explain how continuous compounding works?
3. How would the result change with different interest rates?
4. What is the difference between nominal and effective interest rates?
5. How does increasing the principal affect the total owed?
6. How would monthly compounding affect the final amount?
7. How much interest would be paid if the interest rate was reduced by 1%?
8. How can you calculate the total interest paid over the 14 years?

**Tip:** To avoid errors, double-check the conversion of percentage rates to decimals and ensure the compounding frequency matches the problem's context.

Learn how to calculate compound interest on a $24,000 loan after 14 years with an 8.25% annual interest rate. This problem involves using the compound interest formula and understanding how interest accumulates over time. Suitable for advanced high school students and anyone interested in financial mathematics.

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