deposited

​$300.00300.00

into a savings account that compounded interest

quarterlyquarterly.

What nominal annual rate compounded

quarterlyquarterly

was earned on the investment if the balance was

​$589.20589.20

in

threethree

​years?

Question content area bottom

Part 1

The nominal annual rate of interest is

enter your response here​%

per annum compounded

quarterlyquarterly.

​(Round the final answer to four decimal places as needed. Round all intermediate values to six decimal places as​ needed.)

To solve this problem, we need to calculate the nominal annual interest rate compounded quarterly, based on the given information. Here's the step-by-step process:

### Step 1: Identify the given values
- **Principal (P)**: $300.00
- **Future Value (A)**: $589.20
- **Time (t)**: 3 years
- **Number of compounding periods per year (n)**: 4 (since the interest is compounded quarterly)

### Step 2: Use the compound interest formula
The formula for compound interest is:

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

Where:
- $A$ = Future value of the investment ($589.20)
- $P$ = Principal amount ($300.00)
- $r$ = Nominal annual interest rate (which we need to find)
- $n$ = Number of times the interest is compounded per year (4 for quarterly)
- $t$ = Number of years (3)

### Step 3: Plug in the known values
\[
589.20 = 300.00 \left(1 + \frac{r}{4}\right)^{4 \times 3}
\]

Simplifying:
\[
589.20 = 300.00 \left(1 + \frac{r}{4}\right)^{12}
\]

### Step 4: Solve for $r$

1. Divide both sides by 300.00:

\[
\frac{589.20}{300.00} = \left(1 + \frac{r}{4}\right)^{12}
\]

\[
1.964 = \left(1 + \frac{r}{4}\right)^{12}
\]

2. Take the 12th root of both sides to eliminate the exponent:

\[
\sqrt[12]{1.964} = 1 + \frac{r}{4}
\]

3. Subtract 1 from both sides:

\[
\sqrt[12]{1.964} - 1 = \frac{r}{4}
\]

4. Multiply both sides by 4 to solve for $r$:

\[
r = 4 \times \left(\sqrt[12]{1.964} - 1\right)
\]

### Step 5: Calculate the value of $r$

Let's calculate the numerical value using a calculator:

\[
r = 4 \times \left(1.059947 - 1\right)
\]

\[
r = 4 \times 0.059947
\]

\[
r \approx 0.239788
\]

### Step 6: Convert to percentage

Finally, convert the decimal rate to a percentage:

\[
r \approx 0.2398 \times 100 = 23.9788\%
\]

### Answer:
The nominal annual interest rate is approximately **23.9788%** per annum, compounded quarterly.

Would you like more details or have any questions?

### Related Questions:
1. How would the calculation change if the interest were compounded monthly instead of quarterly?
2. What would be the future value if the interest rate was 20% compounded quarterly?
3. How much interest is earned after 3 years if the interest rate is 15% compounded annually?
4. What is the effective annual rate corresponding to the nominal rate found?
5. How do you calculate the nominal rate if the future value is known but not the time period?
6. How does compounding frequency affect the amount of interest earned?
7. How do different compounding periods (daily, monthly, annually) compare in terms of future value?
8. If the investment had been $500 instead of $300, what would the future value be at the same rate?

### Tip:
Always ensure that the interest rate and the compounding period match when using the compound interest formula. If you mix them up, the results will be inaccurate.

Learn how to calculate the nominal annual interest rate compounded quarterly using a step-by-step solution. This problem involves compound interest with quarterly compounding over 3 years. Suitable for advanced high school students (Grades 10-12).

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