Quinn invested  into an account that grew to  over a period of  years. Assuming that the interest in the account is compounded monthly, what was the annual interest rate of the account? Give your answer as a percent rounded to two decimal places.

To determine the annual interest rate for Quinn's investment, we can use the formula for compound interest:

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

Where:
- \(A\) is the final amount (the value of the account after the investment grows),
- \(P\) is the initial investment (the principal),
- \(r\) is the annual interest rate (what we need to find),
- \(n\) is the number of times the interest is compounded per year (since the interest is compounded monthly, \(n = 12\)),
- \(t\) is the time the money is invested for, in years.

### Rearranging the formula:
We need to solve for \(r\). First, isolate \( \left(1 + \frac{r}{n} \right)^{nt} \):

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

Now, take the natural logarithm on both sides:

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

The exponent \(nt\) can come out in front:

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

Next, divide both sides by \(nt\):

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

Finally, solve for \(r\):

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

Subtract 1 from both sides:

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

Multiply by \(n\) to solve for \(r\):

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

Now, plug in the values for \(A\), \(P\), \(t\), and \(n\). Could you please provide the values of the initial investment \(P\), the final amount \(A\), and the number of years \(t\)?

Quinn invested into an account that grew to over a period of years. Assuming that the interest in the account is compounded monthly, what was the annual interest rate of the account? Give your answer as a percent rounded to two decimal places.

Learn how to calculate the annual interest rate for an account where interest is compounded monthly. This step-by-step guide uses the compound interest formula and logarithms to find the interest rate, rounded to two decimal places. Suitable for students in Grades 11-12 or early college.

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