To find the Annual Percentage Rate (APR) from the effective annual rate (EAR) with monthly compounding, we can use the formula that relates the two:
\[ \text{EAR} = \left(1 + \frac{\text{APR}}{n}\right)^n - 1 \]
where \( n \) is the number of compounding periods per year. In this case, since the bank compounds monthly, \( n = 12 \).
Given that the EAR is 7.78%, or 0.0778 in decimal form, we can rearrange the formula to solve for APR:
\[ 1 + \frac{\text{APR}}{12} = (1 + \text{EAR})^{\frac{1}{n}} \]
Substituting in the values we know:
\[ 1 + \frac{\text{APR}}{12} = (1 + 0.0778)^{\frac{1}{12}} \]
First, we calculate \( (1 + 0.0778)^{\frac{1}{12}} \):
\[ (1.0778)^{\frac{1}{12}} \approx 1.006217 \]
Now, we set up the equation:
\[ 1 + \frac{\text{APR}}{12} = 1.006217 \]
Subtracting 1 from both sides:
\[ \frac{\text{APR}}{12} = 1.006217 - 1 \] \[ \frac{\text{APR}}{12} \approx 0.006217 \]
Multiplying both sides by 12 to find the APR:
\[ \text{APR} \approx 0.006217 \times 12 \approx 0.074604 \]
Converting this back to a percentage:
\[ \text{APR} \approx 7.46% \]
Therefore, the APR is approximately 7.46%.