What principle will amount to $\$ 8500$ if invested at $5 \%$ compounded quarterly for 12 years?

The Principle was $\$$

Step 1 ：We are given that the final amount (A) is $8500, the annual interest rate (r) is 5% or 0.05, the interest is compounded quarterly so n = 4, and the time (t) is 12 years.

Step 2 ：We can use the formula for compound interest, rearranged to solve for the principal amount (P): \(P = \frac{A}{(1 + \frac{r}{n})^{nt}}\)

Step 3 ：Substituting the given values into the formula, we get: \(P = \frac{8500}{(1 + \frac{0.05}{4})^{4*12}}\)

Step 4 ：Solving the equation, we find that \(P \approx 4682.28\)

Step 5 ：Final Answer: The principle amount that will grow to $8500 if invested at a 5% interest rate compounded quarterly for 12 years is approximately \(\boxed{4682.28}\)