To calculate the amount of interest earned on a principal compounded quarterly, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is invested or borrowed for.
In this case:
- \(P = 675\)
- \(r = 0.03\) (3% as a decimal)
- \(n = 4\) (quarterly compounding)
- \(t = 6\)
Now, we can substitute these values into the formula:
\[ A = 675 \left(1 + \frac{0.03}{4}\right)^{4 \cdot 6} \]
First, calculate \( \frac{0.03}{4}\):
\[ \frac{0.03}{4} = 0.0075 \]
Next, substitute this back into the equation:
\[ A = 675 \left(1 + 0.0075\right)^{24} \]
This simplifies to:
\[ A = 675 \left(1.0075\right)^{24} \]
Now calculate \( \left(1.0075\right)^{24} \):
\[ 1.0075^{24} \approx 1.194052 \]
Now, substitute this back into the amount equation:
\[ A \approx 675 \times 1.194052 \approx 805.24 \]
To find the interest earned, subtract the principal from the total amount:
\[ \text{Interest} = A - P = 805.24 - 675 = 130.24 \]
Therefore, the estimated interest earned on the principal of $675 after 6 years at a 3% interest rate compounded quarterly is approximately $130.24.