(Round the final answer to four decimal places as needed. Round all intermediate values to six decimal places as needed.) The nominal annual rate of interest is % per annum compounded monthly Yin Li deposited $300.00 into a savings account that compounded interest monthly. What nominal annual rate compounded monthly was earned on the investment if the balance was $597.10 in eight years?

To solve this problem, we need to find the nominal annual rate of interest $ r $ (compounded monthly) that would result in Yin Li's investment of $300 growing to $597.10 in 8 years. We'll use the future value formula for compound interest:

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

Where:
- $ A $ is the final amount (future value), which is $597.10.
- $ P $ is the initial principal (present value), which is $300.00.
- $ r $ is the nominal annual interest rate.
- $ n $ is the number of compounding periods per year (12 for monthly compounding).
- $ t $ is the number of years the money is invested, which is 8 years.

Let's plug in the known values and solve for $ r $:

\[
597.10 = 300 \left(1 + \frac{r}{12}\right)^{12 \times 8}
\]

Simplifying:

\[
597.10 = 300 \left(1 + \frac{r}{12}\right)^{96}
\]

Now, divide both sides by 300:

\[
\frac{597.10}{300} = \left(1 + \frac{r}{12}\right)^{96}
\]

\[
1.990333 = \left(1 + \frac{r}{12}\right)^{96}
\]

Next, take the 96th root of both sides to solve for $ 1 + \frac{r}{12} $:

\[
\left(1.990333\right)^{\frac{1}{96}} = 1 + \frac{r}{12}
\]

Then subtract 1 from both sides:

\[
\left(1.990333\right)^{\frac{1}{96}} - 1 = \frac{r}{12}
\]

Finally, multiply by 12 to solve for $ r $:

\[
r = 12 \times \left(\left(1.990333\right)^{\frac{1}{96}} - 1\right)
\]

Let's calculate this.The nominal annual rate of interest compounded monthly is approximately **0.0863** or **8.6347%** when rounded to four decimal places.

Would you like further details on this calculation or have any other questions?

Here are some related questions you might find useful:
1. How would the final amount change if the interest rate were different?
2. What would the balance be after 5 years instead of 8 years with the same interest rate?
3. How does monthly compounding compare to quarterly compounding?
4. What is the formula for continuous compounding?
5. How can you calculate the effective annual rate given the nominal rate?
6. How does inflation affect the real value of this investment?
7. What if additional deposits were made during the 8 years? How would that affect the balance?
8. What happens if the investment period is extended to 10 years?

**Tip:** When solving compound interest problems, always make sure to correctly identify and use the compounding frequency to accurately calculate the interest.

Learn how to calculate the nominal annual interest rate compounded monthly using compound interest formulas. Solve the problem where Yin Li deposited $300.00 into a savings account that compounded monthly, growing to $597.10 in 8 years.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation