To find the principal that will earn $55.99 interest at an annual interest rate of 9.75% from February 4, 2017, to July 6, 2017, we need to follow these steps:
1. Calculate the number of days between the two dates.
- Days in February after the 4th: \( 28 - 4 = 24 \)
- Days in March: 31
- Days in April: 30
- Days in May: 31
- Days in June: 30
- Days in July until the 6th: 6
Adding these up gives:
[tex]\[
24 + 31 + 30 + 31 + 30 + 6 = 152 \text{ days}
\][/tex]
2. Convert the number of days to a fraction of a year.
Since there are 365 days in a year:
[tex]\[
\text{Fraction of the year} = \frac{152}{365}
\][/tex]
3. Set up the interest formula:
The formula for simple interest is:
[tex]\[
I = P \times r \times t
\][/tex]
Where:
- \( I \) is the interest earned ($55.99)
- \( P \) is the principal (the amount we need to find)
- \( r \) is the annual interest rate (9.75% or 0.0975 as a decimal)
- \( t \) is the time in years (fraction of the year we calculated)
4. Substitute the known values into the formula and solve for \( P \):
Substitute \( I = 55.99 \), \( r = 0.0975 \), and \( t = \frac{152}{365} \):
[tex]\[
55.99 = P \times 0.0975 \times \frac{152}{365}
\][/tex]
5. Simplify the equation to isolate \( P \):
First, calculate \( t = \frac{152}{365} \):
[tex]\[
t \approx 0.4164
\][/tex]
Then,
[tex]\[
55.99 = P \times 0.0975 \times 0.4164
\][/tex]
[tex]\[
55.99 = P \times 0.04057
\][/tex]
Finally, divide both sides by 0.04057 to solve for \( P \):
[tex]\[
P = \frac{55.99}{0.04057}
\][/tex]
6. Calculate the principal:
[tex]\[
P \approx \frac{55.99}{0.04057} \approx 1379.93
\][/tex]
So, the principal that will earn [tex]$55.99 in interest at an annual interest rate of 9.75% from February 4, 2017, to July 6, 2017, is approximately $[/tex]1379.93.
