Liam deposited $2,000.00 into a new savings account that earns interest compounded quarterly. After 8 years, the balance in the account was $4,700.00. What was the interest rate on the account?
Round your answer to the nearest tenth of a percent.

We can solve for the interest rate using the formula for compound interest:

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

Where:
- $A$ is the final amount in the account.
- $P$ is the principal amount (initial deposit).
- $r$ is the annual interest rate (as a decimal).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the time in years.

Given values:
- $A = 4700.00$
- $P = 2000.00$
- $n = 4$ (since the interest is compounded quarterly)
- $t = 8$ years.

Now, we will plug these values into the formula and solve for $r$.

\[
4700 = 2000 \left(1 + \frac{r}{4}\right)^{4 \times 8}
\]

\[
4700 = 2000 \left(1 + \frac{r}{4}\right)^{32}
\]

First, divide both sides by 2000:

\[
\frac{4700}{2000} = \left(1 + \frac{r}{4}\right)^{32}
\]

\[
2.35 = \left(1 + \frac{r}{4}\right)^{32}
\]

Next, take the 32nd root of both sides to remove the exponent:

\[
\left(2.35\right)^{\frac{1}{32}} = 1 + \frac{r}{4}
\]

Now, subtract 1 from both sides:

\[
\left(2.35\right)^{\frac{1}{32}} - 1 = \frac{r}{4}
\]

Finally, multiply both sides by 4 to solve for $r$:

\[
r = 4 \left[\left(2.35\right)^{\frac{1}{32}} - 1\right]
\]

Let me calculate that for you.The interest rate on the account is approximately **10.8%** per year, compounded quarterly.

Would you like further clarification or details on any part of this solution?

Here are five related questions you might find useful:
1. What is the formula for compound interest when compounded continuously?
2. How would the interest rate change if the interest was compounded monthly instead of quarterly?
3. Can you calculate the total interest earned over the 8 years?
4. How would the future value change if the principal was doubled?
5. What would happen to the final amount if the time period was reduced to 5 years?

**Tip:** When working with compounded interest, it's crucial to note how often the interest is compounded, as it significantly affects the final amount.

Liam deposited $2,000.00 into a new savings account that earns interest compounded quarterly. After 8 years, the balance in the account was $4,700.00. What was the interest rate on the account? Round your answer to the nearest tenth of a percent.

Calculate the interest rate for an account with an initial deposit of $2,000.00, compounded quarterly, with a final balance of $4,700.00 after 8 years. Learn how to use the compound interest formula to determine the rate of return. Suitable for high school students (Grades 10-12).

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