Sure, let's solve the problem step by step using the United States rule to determine the balance due on the note at the date of maturity.
### Given Information
- Principal Amount (P): [tex]$3000
- Annual Interest Rate (r): 4% (or 0.04 as a decimal)
- Effective Date: April 1
- Partial Payment Amount: $[/tex]1000
- Partial Payment Date: May 1
- Maturity Date: June 1
### Step 1: Calculate the Day of the Year for Each Date
To compute the number of days between the dates, we need to convert the dates to the day of the year:
- April 1: 91st day (January has 31 days, February has 28 days, March has 31 days, hence 31 + 28 + 31 + 1 = 91)
- May 1: 121st day (April has 30 days, so 91 + 30 = 121)
- June 1: 152nd day (May has 31 days, so 121 + 31 = 152)
### Step 2: Calculate Interest to the Partial Payment Date (May 1)
Number of days from the effective date (April 1) to the partial payment date (May 1) is:
[tex]\[ 121 - 91 = 30 \text{ days} \][/tex]
Interest accrued in these 30 days:
[tex]\[ \text{Interest} = P \times \left( \frac{r}{365} \right) \times \text{number of days} \][/tex]
[tex]\[ \text{Interest} = 3000 \times \left( \frac{0.04}{365} \right) \times 30 \][/tex]
[tex]\[ \text{Interest} \approx 9.86 \][/tex]
So, the interest accrued is approximately [tex]$9.86.
### Step 3: Update Principal After Partial Payment
New principal after the payment on May 1:
\[ \text{New Principal} = \text{Initial Principal} + \text{Interest Accrued} - \text{Partial Payment} \]
\[ \text{New Principal} = 3000 + 9.86 - 1000 \]
\[ \text{New Principal} \approx 2009.86 \]
### Step 4: Calculate Interest from Payment Date to Maturity Date (June 1)
Number of days from May 1 to June 1:
\[ 152 - 121 = 31 \text{ days} \]
Interest accrued on the new principal in these 31 days:
\[ \text{Interest} = \text{New Principal} \times \left( \frac{r}{365} \right) \times 31 \]
\[ \text{Interest} \approx 2009.86 \times \left( \frac{0.04}{365} \right) \times 31 \]
\[ \text{Interest} \approx 6.83 \]
### Step 5: Calculate the Total Amount Due at Maturity
Total amount due at maturity:
\[ \text{Balance Due} = \text{New Principal} + \text{Interest Accrued} \]
\[ \text{Balance Due} = 2009.86 + 6.83 \]
\[ \text{Balance Due} \approx 2016.69 \]
### Final Answer
The balance due on the note at the date of maturity is $[/tex]2016.69.
