Answer:
Let's first calculate the number of compounding periods in a year, given that the interest is compounded quarterly:
Number of compounding periods in a year = 4 (since there are 4 quarters in a year)
Next, let's use the formula for compound interest to find the annual interest rate:
A = P (1 + r/n)^(nt)
Where:
A is the balance after t years
P is the principal (initial balance)
r is the annual interest rate (what we are looking for)
n is the number of compounding periods in a year
t is the time in years
We know that P = $1250, A = $1406.08, n = 4, and t = 9/12 years (since the interest is earned for 9 months).
Plugging these values into the formula, we get:
$1406.08 = $1250 (1 + r/4)^(4/3)
Dividing both sides by $1250 and taking the cube root of both sides, we get:
1 + r/4 = (1406.08/1250)^(3/4)
1 + r/4 = 1.038
Subtracting 1 from both sides, we get:
r/4 = 0.038
Multiplying both sides by 4, we get:
r = 0.152
Therefore, the annual interest rate is 15.2%.