You have an investment account that started with 

​$3 comma 0003,000 

1515 

years ago and which now has grown to 

​$18 comma 00018,000.

**a.** What annual rate of return have you earned​ (you have made no additional contributions to the​ account)?

**b.** If the investment account earns 

14 %14% 

per year from now​ on, what will the​ account's value be 

1515 

years from​ now?

Question content area bottom

Part 1

**a.** What annual rate of return have you earned​ (you have made no additional contributions to the​ account)?

Your annual rate of return is 

enter your response here​%. 

 ​(Round to two decimal​ places.

### Part 1: Finding the Annual Rate of Return (a)

To calculate the annual rate of return on an investment, we use the **compound interest formula**:

\[
A = P \times (1 + r)^t
\]

Where:
- $A$ is the final amount ($18,000),
- $P$ is the initial investment ($3,000),
- $r$ is the annual rate of return,
- $t$ is the time in years (15 years).

We want to solve for $r$, so the formula becomes:

\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]

Substituting the values:

\[
r = \left(\frac{18,000}{3,000}\right)^{\frac{1}{15}} - 1
\]

Let's calculate the value of $r$.The annual rate of return earned is approximately **12.69%**.

### Part 2: Future Value of the Investment (b)

To calculate the future value of the account after 15 years with a 14% annual return, we use the same compound interest formula:

\[
A = P \times (1 + r)^t
\]

Where:
- $P = 18,000$,
- $r = 0.14$ (14%),
- $t = 15$ years.

Now, let's calculate the future value.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to attempt the calculation manually, or I can guide you through the steps to solve it yourself?

### Follow-up Questions:
1. What happens to your investment if the rate of return changes after a few years?
2. How does compound interest differ from simple interest in long-term investments?
3. How would your investment grow if you contributed more to it annually?
4. Can the rate of return vary with different types of investments?
5. How is risk related to the potential rate of return in investment decisions?

**Tip:** When calculating future values, always account for the compounding frequency (annually, monthly, etc.) to ensure accurate results.

You have an investment account that started with $3,000 15 years ago and which now has grown to $18,000. 
a. What annual rate of return have you earned​ (you have made no additional contributions to the account)?
b. If the investment account earns 14% per year from now on, what will the account's value be 15 years from now?

Learn how to calculate the annual rate of return on an investment account that started with $3,000 and grew to $18,000 over 15 years. Also, explore how to determine the future value of the account with a 14% annual return over the next 15 years. This problem involves the compound interest formula and exponential growth.

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