To find the equivalent monthly interest rate for an annual interest rate of 23%, follow these steps:
1. Convert the annual interest rate from a percentage to a decimal:
[tex]\[
I_{\text{annual}} = \frac{23}{100} = 0.23
\][/tex]
2. Use the formula for converting the annual interest rate to a monthly interest rate:
[tex]\[
I_{\text{monthly}} = \left(1 + I_{\text{annual}}\right)^{\frac{1}{12}} - 1
\][/tex]
3. Substitute the annual interest rate into the formula:
[tex]\[
I_{\text{monthly}} = \left(1 + 0.23\right)^{\frac{1}{12}} - 1
\][/tex]
4. Calculate the value inside the parentheses first:
[tex]\[
1 + 0.23 = 1.23
\][/tex]
5. Next, raise 1.23 to the power of [tex]\(\frac{1}{12}\)[/tex]:
[tex]\[
1.23^{\frac{1}{12}}
\][/tex]
6. After finding the 12th root of 1.23, subtract 1 from the result to get the equivalent monthly interest rate:
[tex]\[
I_{\text{monthly}} = 1.017400841772181597 - 1
\][/tex]
7. Finally, the result is:
[tex]\[
I_{\text{monthly}} \approx 0.017400841772181597
\][/tex]
So, the equivalent monthly interest rate for an annual interest rate of 23% is approximately [tex]\(0.0174\)[/tex] or [tex]\(1.74\%\)[/tex].
