Ella is going to invest $2,500 and leave it in an account for 19 years. Assuming the interest is compounded quarterly, what interest rate, to the nearest hundredth of a percent, would be required in order for Ella to end up with $5,000?

We can solve this problem using the compound interest formula:

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

Where:
- $A$ is the final amount (in this case, $A = 5000$),
- $P$ is the initial principal (in this case, $P = 2500$),
- $r$ is the annual interest rate (which we are solving for),
- $n$ is the number of times the interest is compounded per year (in this case, $n = 4$ because it's compounded quarterly),
- $t$ is the number of years (in this case, $t = 19$).

We want to solve for $r$, the interest rate. Let's substitute the known values into the equation:

\[
5000 = 2500 \left(1 + \frac{r}{4}\right)^{4 \times 19}
\]

First, divide both sides by 2500:

\[
2 = \left(1 + \frac{r}{4}\right)^{76}
\]

Now, take the 76th root of both sides:

\[
\sqrt[76]{2} = 1 + \frac{r}{4}
\]

Then subtract 1 from both sides:

\[
\sqrt[76]{2} - 1 = \frac{r}{4}
\]

Finally, multiply by 4 to isolate $r$:

\[
r = 4 \left(\sqrt[76]{2} - 1\right)
\]

I'll now calculate this value.The interest rate required for Ella to double her investment to $5,000 in 19 years, with interest compounded quarterly, is approximately **3.66%**.

Would you like any further details or explanations? Here are 5 related questions you might find useful:

1. How does compounding frequency affect the final amount in compound interest?
2. What if the interest were compounded monthly instead of quarterly?
3. How would the required interest rate change if the time period were different?
4. Can we use continuous compounding in this scenario? How would the formula change?
5. What is the difference between simple and compound interest over long periods?

**Tip:** The more frequently interest is compounded, the more the investment grows due to the "compounding on compounding" effect.

Learn how to calculate the interest rate required for an initial investment of $2,500 to double to $5,000 in 19 years, with interest compounded quarterly. This step-by-step guide covers key algebraic concepts and compound interest formula calculations. Suitable for Grades 9-12.

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