To solve this problem, let's break it down based on the details provided:

1. Initial Deposit: Clare's aunt deposited an initial amount of $300.
2. Growth Factor: The balance grows each year with a growth factor of $x = 1 + r$, where $r$ is the annual interest rate.
3. Function of Growth Factor: The total amount $y$, after four years, is a function of the growth factor $x$.
4. Goal: After four years, the amount Clare receives is $1,580.

Setting up the Equation

Since the growth follows a compounding pattern, the equation for the balance after four years, given an initial deposit $P = 300$, can be written as: $y = P \cdot x^4$ where $y$ is the final amount after 4 years, $x = 1 + r$, and $P = 300$.

Given: $y = 1580$ Substitute $y$ and $P$ into the equation: $1580 = 300 \cdot x^4$

Solving for $x$

1. Divide both sides by 300: $x^4 = \frac{1580}{300}$ $x^4 = 5.2667$

2. Take the fourth root of both sides to solve for $x$: $x = \sqrt[4]{5.2667}$ Calculating this value will give the approximate growth factor.

3. Once you find $x$, use the formula $r = x - 1$ to find the interest rate.

Would you like me to perform these calculations for an exact result?

Here are 5 follow-up questions you might consider:

1. How would the result change if the initial deposit was different?
2. What if the interest was compounded monthly instead of yearly?
3. How would the equation look if Clare made additional deposits each year?
4. Can you predict the balance after 10 years with the same interest rate?
5. How does compounding frequency affect the total amount after 4 years?

Tip: When working with exponential growth, remember that each time period multiplies the principal by a growth factor, making the initial amount grow faster over time.