To solve this problem using the United States rule, we need to follow these steps:
1. Identify Key Dates and Values:
- Principal: [tex]$2000
- Annual Interest Rate: 5% (0.05 as a decimal)
- Effective Date: April 1 (91st day of the year)
- Partial Payment Date: May 1 (121st day of the year)
- Maturity Date: June 1 (152nd day of the year)
- Partial Payment Amount: $[/tex]1000
2. Calculate the Number of Days from the Effective Date to the Partial Payment Date:
- Partial Payment Date (May 1) - Effective Date (April 1) = 121 - 91 = 30 days
3. Calculate Accrued Interest from the Effective Date to the Partial Payment Date:
- Using the formula for simple interest:
[tex]\[
\text{Interest} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{365}
\][/tex]
- Here, Principal = [tex]$2000, Rate = 0.05, and Time = 30 days:
\[
\text{Interest} = \frac{2000 \times 0.05 \times 30}{365} = 8.219178082191782
\]
4. Calculate the Remaining Principal after the Partial Payment:
- Add the accrued interest to the principal:
\[
2000 + 8.219178082191782 = 2008.2191780821918
\]
- Subtract the partial payment:
\[
2008.2191780821918 - 1000 = 1008.2191780821918
\]
5. Calculate the Number of Days from the Partial Payment Date to the Maturity Date:
- Maturity Date (June 1) - Partial Payment Date (May 1) = 152 - 121 = 31 days
6. Calculate Accrued Interest on Remaining Principal till Maturity Date:
- Using the formula for simple interest again:
\[
\text{Interest} = \frac{\text{Remaining Principal} \times \text{Rate} \times \text{Time}}{365}
\]
- Here, Remaining Principal = $[/tex]1008.2191780821918, Rate = 0.05, and Time = 31 days:
[tex]\[
\text{Interest} = \frac{1008.2191780821918 \times 0.05 \times 31}{365} = 4.281478701444924
\][/tex]
7. Calculate the Balance Due at the Date of Maturity:
- Add the accrued interest on the remaining principal to the remaining principal:
[tex]\[
1008.2191780821918 + 4.281478701444924 = 1012.5006567836367
\][/tex]
Thus, the balance due on the maturity date is $1012.50 (rounded to two decimal places).
