Find the interest rate for a ​$7500 deposit accumulating to ​$12 comma 265​, compounded annually for 9 years.
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Part 1
The interest rate is 
  
enter your response here​%.

To find the interest rate for the given problem, we can use the formula for compound interest:

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

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

Given:
- $A = 12,265$
- $P = 7,500$
- $t = 9$ years
- $n = 1$ (compounded annually)

We need to find $r$. First, we will rearrange the formula to solve for $r$:

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

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

\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]

Now, substitute the given values:

\[
r = \left(\frac{12,265}{7,500}\right)^{\frac{1}{9}} - 1
\]

Let's calculate that.The interest rate is approximately **5.62%**.

Would you like a detailed explanation of the process, or do you have any other questions?

Here are 5 related questions to explore further:
1. What is the formula for compound interest when interest is compounded more than once a year?
2. How does the interest rate change if the time period is extended?
3. What would the final amount be if the interest rate were increased to 7%?
4. How would the calculation change if interest were compounded quarterly instead of annually?
5. What is the difference between compound and simple interest?

**Tip:** When solving for the interest rate, always ensure the equation is rearranged correctly before calculating.

Learn how to calculate compound interest for a $7500 deposit accumulating to $12,265 over 9 years with annual compounding. Understand the formula and steps involved in finding the interest rate. Suitable for high school students and anyone interested in financial mathematics.

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