Certainly! Let's go through the process of finding the annual interest rate [tex]\( r \)[/tex] given a quarterly rate of 2.775%.
### Step-by-Step Solution:
1. Identify the given quarterly rate:
The rate per compounding period (quarter) is 2.775%.
2. Convert the quarterly rate from a percentage to a decimal:
[tex]\[
\text{Rate per quarter (decimal form)} = \frac{2.775}{100} = 0.02775
\][/tex]
3. Determine the number of compounding periods per year:
Since we are dealing with quarterly compounding, there are 4 quarters in a year.
[tex]\[
n = 4
\][/tex]
4. Apply the compound interest formula to find the annual rate:
The formula to find the equivalent annual rate (EAR) given the rate per compounding period is:
[tex]\[
(1 + \text{Rate per quarter})^n - 1
\][/tex]
Plugging in the values:
[tex]\[
(1 + 0.02775)^4 - 1
\][/tex]
5. Calculate the annual rate:
Evaluating the expression inside the parentheses first, then raising it to the power of 4, and lastly, subtracting 1:
[tex]\[
(1 + 0.02775)^4 - 1 = 1.02775^4 - 1 \approx 1.115712 - 1 = 0.11571
\][/tex]
6. Convert the annual rate back to a percentage:
[tex]\[
0.11571 \times 100 = 11.571\%
\][/tex]
7. Round to three decimal places:
The final answer should be rounded to three decimal places. Hence, the annual rate is:
[tex]\[
r = 11.571\%
\][/tex]
So, the annual rate [tex]\( r \)[/tex] that corresponds to a quarterly rate of 2.775% is [tex]\( 11.571\% \)[/tex].
