To calculate the total interest earned by the end of the third month in a savings account with an APR of 3.55% compounded monthly, we can use the compound interest 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 time the money is invested or borrowed for in years.
Given:
- \(P = 3260.00\)
- \(r = 3.55% = 0.0355\)
- \(n = 12\) (since it's compounded monthly)
- \(t = \frac{3}{12} = 0.25\) years (for 3 months)
Now, substituting these values into the formula:
\[ A = 3260 \left(1 + \frac{0.0355}{12}\right)^{12 \cdot 0.25} \]
Calculating \(\frac{0.0355}{12}\):
\[ \frac{0.0355}{12} \approx 0.00295833 \]
Now calculate \(A\):
\[ A = 3260 \left(1 + 0.00295833\right)^{3} \]
Calculating \((1 + 0.00295833)^3\):
\[ (1.00295833)^3 \approx 1.008895 \]
Now, substituting this back into the formula for \(A\):
\[ A \approx 3260 \times 1.008895 \approx 3284.06 \]
Now, to find the interest earned, subtract the initial principal from the total amount:
\[ \text{Interest} = A - P = 3284.06 - 3260.00 = 24.06 \]
Thus, the total interest earned by the end of the third month, rounded to the nearest penny, is $24.06.