Sure, let's work through this problem step-by-step.
1. Understand the given information:
- Initial loan amount (Principal, [tex]\( P \)[/tex]): [tex]$1280
- Time to repay the loan: 4 months
- Annual interest rate: 3.8%
2. Convert the annual interest rate to a monthly interest rate:
- The annual interest rate is 3.8%, which in decimal form is 0.038.
- To find the monthly interest rate, you use the formula for converting annual interest rate to monthly interest rate:
\[
\text{monthly interest rate} = \left(1 + \text{annual interest rate}\right)^{\frac{1}{12}} - 1
\]
- Plugging in the values:
\[
\text{monthly interest rate} = \left(1 + 0.038\right)^{\frac{1}{12}} - 1
\]
- The calculated monthly interest rate is approximately 0.0031128 or 0.31128%.
3. Calculate the total amount to be paid after 4 months:
- Using the formula for the accumulated amount with compound interest, we have:
\[
A = P \left(1 + r\right)^n
\]
Where:
- \( A \) is the amount of money accumulated after n months, including interest.
- \( P \) is the principal amount $[/tex]\left(1280\right)[tex]$.
- \( r \) is the monthly interest rate $[/tex]\left(0.0031128\right)[tex]$.
- \( n \) is the number of months $[/tex]\left(4\right)[tex]$.
- Plugging in the values:
\[
A = 1280 \left(1 + 0.0031128\right)^4
\]
- The total amount to be paid after 4 months is approximately $[/tex]1296.012193148505.
4. Calculate the monthly payment:
- To find out how much Nancy has to pay each month, divide the total amount by the number of months:
[tex]\[
\text{monthly payment} = \frac{A}{\text{number of months}}
\][/tex]
- Plugging in the values:
[tex]\[
\text{monthly payment} = \frac{1296.012193148505}{4}
\][/tex]
- The monthly payment is approximately [tex]$324.00304828712626.
5. Conclusion:
- Nancy would have to pay approximately $[/tex]324.00 each month for 4 months to clear the loan if the interest charged is 3.8% compounded annually.
