To find the principal amount that will earn [tex]$24.87 in interest over 156 days at an annual interest rate of 4.75%, we'll break down the process into several steps:
1. Convert the annual interest rate to a daily interest rate:
The annual interest rate (4.75%) needs to be converted to a daily rate because the interest is earned over a period of days. Since there are 365 days in a year, we divide the annual rate by 365.
\( \text{Daily Rate} = \frac{4.75\%}{365} = \frac{0.0475}{365} \approx 0.000130 \)
2. Calculate the time period in years:
The interest is earned over 156 days. To express this in years, divide the number of days by 365.
\( \text{Time Period in Years} = \frac{156}{365} \approx 0.427397 \)
3. Use the formula for simple interest to find the principal:
The formula for calculating simple interest is:
\[
\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}
\]
To solve for the principal, we rearrange the formula:
\[
\text{Principal} = \frac{\text{Interest}}{\text{Rate} \times \text{Time}}
\]
Substitute the known values into the equation:
\[
\text{Principal} = \frac{24.87}{0.000130 \times 0.427397} \approx 447139.78
\]
4. Round the principal to the nearest cent:
Since we need to round the final answer to the nearest cent, the principal amount is:
\[
\text{Principal} \approx \$[/tex]447139.78
\]
Thus, the principal that will earn [tex]$24.87 at an annual interest rate of 4.75% in 156 days is approximately $[/tex]447,139.78.
