James invests $20,000 in an account that offers a compound interest rate of 8.3% per year. Which of the following is the correct equation for how much James will have in year 6?

The general formula for compound interest is given by:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $ A $ is the amount of money accumulated after $ t $ years (including interest),
- $ P $ is the principal amount (the initial money invested),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time the money is invested for, in years.

For this problem:
- $ P = 20,000 $,
- $ r = 0.083 $ (since 8.3% is expressed as 0.083 in decimal),
- $ t = 6 $ years,
- $ n = 1 $ (since it's not specified, we assume the interest is compounded annually).

Substituting these values into the compound interest formula:

\[
A = 20,000 \left( 1 + \frac{0.083}{1} \right)^{1 \times 6}
\]

Simplifying the expression:

\[
A = 20,000 \left( 1 + 0.083 \right)^6
\]

\[
A = 20,000 \times (1.083)^6
\]

Thus, the equation for how much James will have after 6 years is:

\[
A = 20,000 \times (1.083)^6
\]

Would you like me to calculate the exact amount, or explain further details?

Here are 5 relative questions to expand on this:
1. How does the compounding frequency affect the final amount?
2. What would be the equation if the interest were compounded monthly?
3. How much would James have after 10 years with the same interest rate?
4. How does simple interest differ from compound interest in terms of growth?
5. What happens if the interest rate changes over the investment period?

**Tip:** The more frequent the compounding, the greater the amount of interest accumulated over time!

Learn how to calculate the compound interest for a $20,000 investment at an 8.3% interest rate over 6 years. This detailed explanation uses the compound interest formula, ideal for students in Grades 10-12 studying exponential growth.

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