Determine the annual rate of interest, to the nearest tenth of a percent, given the initial investment, the final amount, the compounding period, and the length of the investment:

An investment of $7 500 compounded semi-annually for five years to produce a final amount of 12500

To determine the annual rate of interest for the investment compounded semi-annually, we can use the compound interest formula:

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

Where:
- $A$ is the final amount.
- $P$ is the initial principal (investment).
- $r$ is the annual interest rate (which we want to find).
- $n$ is the number of compounding periods per year.
- $t$ is the number of years.
- $A = 12,500$, $P = 7,500$, $n = 2$ (since it is compounded semi-annually), and $t = 5$.

Now, let's substitute the values and solve for $r$:

\[
12,500 = 7,500 \left(1 + \frac{r}{2}\right)^{2 \times 5}
\]

This simplifies to:

\[
12,500 = 7,500 \left(1 + \frac{r}{2}\right)^{10}
\]

Next, we'll solve for $r$. I'll calculate this now.It seems there was an issue with the calculation. Let me recheck the steps and solve it correctly.

We are solving for the annual interest rate $ r $ from:

\[
12,500 = 7,500 \left(1 + \frac{r}{2}\right)^{10}
\]

I will correct the approach and recalculate the interest rate.The annual interest rate is approximately **10.5%**, rounded to the nearest tenth of a percent.

Would you like further details on the steps or any clarifications?

Here are 5 related questions to consider:
1. How does compounding frequency affect the final amount in an investment?
2. What would the interest rate be if the investment was compounded quarterly?
3. How does increasing the initial investment affect the final amount?
4. What is the difference between simple and compound interest calculations?
5. How would changing the investment period from 5 to 10 years change the final amount?

**Tip**: Always pay attention to the compounding period, as it significantly impacts the total interest earned over time.

Determine the annual rate of interest, to the nearest tenth of a percent, given the initial investment, the final amount, the compounding period, and the length of the investment: An investment of $7,500 compounded semi-annually for five years to produce a final amount of $12,500.

Learn how to calculate the annual interest rate for an investment of $7,500 compounded semi-annually over five years, resulting in a final amount of $12,500. This guide covers key mathematical concepts, including the compound interest formula and step-by-step calculations. Suitable for Grades 10-12.

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