To find the effective annual rate of return on Juan's investment, we use the formula for compound interest growth:

$A = P(1 + r)^t$

Where:

• $A$ is the final amount ($32,735),
• $P$ is the initial investment ($24,000),
• $r$ is the annual rate of return (what we want to find),
• $t$ is the time period in years (3 years).

First, we plug the known values into the formula:

$32,735 = 24,000(1 + r)^3$

Step 1: Isolate $(1 + r)$

$\frac{32,735}{24,000} = (1 + r)^3$

$1.36479 = (1 + r)^3$

Step 2: Solve for $1 + r$ by taking the cube root

$1 + r = \sqrt[3]{1.36479}$

Step 3: Calculate the cube root

$1 + r \approx 1.1079$

Step 4: Solve for $r$

$r = 1.1079 - 1$

$r \approx 0.1079$

Thus, the effective annual rate of return is approximately 10.79%.

Let me know if you'd like further details on this, or have any questions! Here are some related questions you can explore:

1. How would the rate change if the investment was for 5 years instead of 3?
2. What would the future value be if the initial investment was $30,000 instead?
3. How does effective annual rate differ from nominal rate in compounding interest?
4. What would the rate be if the final amount was $40,000 instead of $32,735?
5. How is continuous compounding different from annual compounding?

Tip: When working with growth formulas, it's important to distinguish between compounding annually, quarterly, or continuously, as it impacts the rate significantly.