To find the adjusted balance (principal) using the U.S. Rule, we need to follow a series of steps:
1. Calculate the interest for the first 90 days:
The formula for calculating simple interest is:
[tex]\[
I = P \times r \times \frac{t}{360}
\][/tex]
Given:
- Principal ([tex]\(P\)[/tex]) = \[tex]$7,000
- Annual rate (\(r\)) = 11\% (or 0.11 as a decimal)
- Time (\(t\)) = 90 days
\[
I_{90} = 7000 \times 0.11 \times \frac{90}{360}
\]
Performing the calculation:
\[
I_{90} = 192.5
\]
Therefore, the interest for the first 90 days is \$[/tex]192.5.
2. Calculate the new principal after the partial payment:
We need to determine the principal after accounting for the accumulated interest and subtracting the partial payment.
- Partial payment = \[tex]$1,200
Adding the interest to the principal:
\[
P_{\text{new}} = 7000 + 192.5 - 1200
\]
Performing the calculation:
\[
P_{\text{new}} = 5992.5
\]
The new principal after the partial payment is \$[/tex]5,992.5.
3. Calculate the interest for the remaining period (30 days):
The remaining period after day 90 is [tex]\(120 - 90 = 30\)[/tex] days.
Using the interest formula again with the new principal:
[tex]\[
I_{30} = P_{\text{new}} \times r \times \frac{t_{\text{remaining}}}{360}
\][/tex]
Given:
- New principal ([tex]\(P_{\text{new}}\)[/tex]) = \[tex]$5,992.5
- Annual rate (\(r\)) = 11\% (or 0.11 as a decimal)
- Remaining time (\(t_{\text{remaining}}\)) = 30 days
\[
I_{30} = 5992.5 \times 0.11 \times \frac{30}{360}
\]
Performing the calculation:
\[
I_{30} = 54.93125
\]
Therefore, the interest for the remaining 30 days is \$[/tex]54.93125.
4. Calculate the final adjusted balance:
The final adjusted balance is the new principal plus the interest accumulated over the remaining period.
[tex]\[
P_{\text{adjusted}} = P_{\text{new}} + I_{30}
\][/tex]
Performing the calculation:
[tex]\[
P_{\text{adjusted}} = 5992.5 + 54.93125
\][/tex]
Therefore, the adjusted balance is:
[tex]\[
P_{\text{adjusted}} = 6047.43125
\][/tex]
So, the adjusted balance after the first payment on day 90 is \$6,047.43125.
