Elena invested $2300 in an account for 12 years.  The investment is now worth $8046.44.  What was the rate of interest as a percent

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

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

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

Since the problem does not specify how often the interest is compounded, we will assume it is compounded annually, so $ n = 1 $.

Given:
- $ A = 8046.44 $
- $ P = 2300 $
- $ t = 12 $
- $ n = 1 $

We substitute these values into the formula and solve for $ r $:

\[
8046.44 = 2300 \left(1 + \frac{r}{1}\right)^{12}
\]

First, divide both sides by 2300:

\[
\frac{8046.44}{2300} = \left(1 + r\right)^{12}
\]

\[
3.5 = \left(1 + r\right)^{12}
\]

Now, take the 12th root of both sides:

\[
1 + r = \sqrt[12]{3.5}
\]

\[
1 + r = 1.1108
\]

Finally, subtract 1 from both sides:

\[
r = 0.1108
\]

Convert this to a percentage:

\[
r = 11.08\%
\]

So, the rate of interest is approximately **11.08%**.

Would you like more details or have any questions about this solution?

Here are five related questions to further your understanding:
1. What if the interest was compounded quarterly instead of annually?
2. How would the formula change if Elena invested the same amount but for only 8 years?
3. Can you calculate the interest rate if the final value of the investment was $9000 instead of $8046.44?
4. How much would the investment be worth if the rate of interest was 10% instead of 11.08%?
5. How does the interest rate affect the total return over a long-term investment?

**Tip**: Always clarify the compounding frequency (annually, quarterly, etc.) for a more accurate calculation of compound interest.

Elena invested $2300 in an account for 12 years. The investment is now worth $8046.44. What was the rate of interest as a percent?

Learn how to calculate the interest rate for an investment of $2300 growing to $8046.44 over 12 years using the compound interest formula. This problem demonstrates key financial and mathematical concepts, suitable for students in Grades 9-12.

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