To calculate the interest earned on a principal amount 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 time in years.
In your case:
- \(P = 675\)
- \(r = 0.03\) (3% expressed as a decimal)
- \(n = 4\) (quarterly compounding)
- \(t = 6\)
Substituting these values into the formula gives:
\[ A = 675 \left(1 + \frac{0.03}{4}\right)^{4 \times 6} \]
Calculating the values step-by-step:
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Calculate \( \frac{r}{n} \): \[ \frac{0.03}{4} = 0.0075 \]
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Calculate \( nt \): \[ 4 \times 6 = 24 \]
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Substitute into the formula: \[ A = 675 \left(1 + 0.0075\right)^{24} \] \[ A = 675 \left(1.0075\right)^{24} \]
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Calculate \( (1.0075)^{24} \): \[ (1.0075)^{24} \approx 1.194052 \]
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Now calculate \( A \): \[ A \approx 675 \times 1.194052 \approx 805.24 \]
Now, to find the interest earned, we subtract the principal from the total amount:
\[ \text{Interest} = A - P = 805.24 - 675 = 130.24 \]
So, the interest earned over 6 years is approximately $130.24.
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