To calculate the total interest earned by the end of the third month in a savings account with a principal amount of $4,500.00, an APR of 3.67%, and monthly compounding, we can use the formula for compound interest.
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n months, including interest.
- \( P \) is the principal amount ($4,500.00).
- \( r \) is the annual interest rate (decimal) (3.67% = 0.0367).
- \( n \) is the number of times that interest is compounded per year (12 for monthly).
- \( t \) is the time the money is invested or borrowed for in years.
To find the total interest earned at the end of the third month, we set \( t = \frac{3}{12} = 0.25 \).
Now, substituting in the values:
- \( P = 4500 \)
- \( r = 0.0367 \)
- \( n = 12 \)
- \( t = 0.25 \)
The amount \( A \) at the end of 3 months can be calculated as follows:
\[ A = 4500 \left(1 + \frac{0.0367}{12}\right)^{12 \times 0.25} \]
First, calculate \( \frac{r}{n} \):
\[ \frac{0.0367}{12} = 0.0030583333 \]
Next, calculate \( nt \):
\[ 12 \times 0.25 = 3 \]
Now, substituting these values into the formula:
\[ A = 4500 \left(1 + 0.0030583333\right)^{3} \]
Calculating \( 1 + 0.0030583333 \):
\[ 1 + 0.0030583333 \approx 1.0030583333 \]
Now raise it to the power of 3:
\[ (1.0030583333)^{3} \approx 1.009177 \]
Now we calculate \( A \):
\[ A \approx 4500 \times 1.009177 \approx 4538.29 \]
Now, to find the total interest earned, we subtract the principal from \( A \):
\[ \text{Interest} = A - P \approx 4538.29 - 4500 = 38.29 \]
Therefore, the total interest earned by the end of the third month is approximately $38.29.