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To solve the problem, we need to find the **annual rate of return** that represents the growth of your investment from $16,041 to $31,271 over 3 years.

This type of problem can be solved using the formula for compound annual growth rate (CAGR), which is given by:

\[
\text{CAGR} = \left( \frac{{\text{Ending Value}}}{{\text{Beginning Value}}} \right)^{\frac{1}{\text{Number of Years}}} - 1
\]

Here:
- The **Beginning Value** is $16,041.
- The **Ending Value** is $31,271.
- The **Number of Years** is 3.

Let’s compute the annual rate of return:

\[
\text{CAGR} = \left( \frac{31,271}{16,041} \right)^{\frac{1}{3}} - 1
\]

Now, I'll calculate that.

\[
\frac{31,271}{16,041} = 1.9496
\]

Now, take the cube root of 1.9496:

\[
1.9496^{\frac{1}{3}} \approx 1.263
\]

Now, subtract 1:

\[
1.263 - 1 = 0.263
\]

Convert this to a percentage by multiplying by 100:

\[
0.263 \times 100 = 26.3
\]

Thus, the annual rate of return is **26.30%**.

Would you like further details on any of the steps? Here are five questions related to this problem:

1. What is the significance of using the compound annual growth rate (CAGR) formula?
2. How does the rate of return differ from simple interest in this scenario?
3. Could the formula change if the investment had different compounding periods (e.g., quarterly)?
4. How would you calculate the return if the investment duration was not 3 years but another time period?
5. What are the practical implications of such a high return on investment?

**Tip:** When solving for annual growth rates, always check if the return is compounded or simple, as it affects the calculation method.

Your roommate has a great business idea, but needs a little more money to get it off the ground. You offer your entire savings of $16,041 as a silent partner in his business. You think that over the first 3 years, the value of your investment will grow to be worth $31,271. What annual rate of return does that represent?

Learn how to calculate the annual rate of return for an investment that grows from $16,041 to $31,271 over 3 years using the Compound Annual Growth Rate (CAGR) formula. This problem introduces the concept of exponential growth and the importance of rate of return calculations in financial planning.

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