To calculate the amount of interest earned on a principal amount with compound interest, we can use the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
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.
Given:
- Principal \( P = 675 \)
- Annual interest rate \( r = 3% = 0.03 \)
- Compounded quarterly, so \( n = 4 \)
- Time period \( t = 6 \) years
Now plug these values into the formula:
\[ A = 675 \left(1 + \frac{0.03}{4}\right)^{4 \times 6} \] \[ A = 675 \left(1 + 0.0075\right)^{24} \] \[ A = 675 \left(1.0075\right)^{24} \]
Now calculate \( (1.0075)^{24} \):
\[ (1.0075)^{24} \approx 1.19786 \]
Now, plug this back into the equation for \( A \):
\[ A \approx 675 \times 1.19786 \approx 808.14 \]
Now, to find the interest earned, subtract the principal from the amount:
\[ \text{Interest} = A - P = 808.14 - 675 = 133.14 \]
Thus, the approximate interest earned is about $133.14. Since that isn't exactly one of the options provided, the closest response would be $132.58.
So the answer is:
$132.58
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