Questions: Find the monthly payment and estimate the remaining balance at the given time. Assume interest is on the unpaid balance. Fifteen-year student loan for 27,800 at 2.64%; remaining balance after 5 years. The monthly payment is . (Round to the nearest cent as needed.)

Solution Steps

To solve this problem, we need to calculate the monthly payment for a loan using the loan amount, interest rate, and loan term. Then, we will estimate the remaining balance after a specified period.

1. Calculate the monthly interest rate from the annual interest rate.
2. Use the loan payment formula to find the monthly payment.
3. Use the amortization formula to find the remaining balance after the specified period.

The annual interest rate is \(2.64\%\). To find the monthly interest rate, we divide the annual interest rate by 12: \[ \text{Monthly Interest Rate} = \frac{0.0264}{12} = 0.0022 \]

The loan term is 15 years. To find the total number of monthly payments, we multiply the loan term by 12: \[ \text{Number of Payments} = 15 \times 12 = 180 \]

Using the loan payment formula: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where:

• \(M\) is the monthly payment,
• \(P\) is the loan amount (\$27,800),
• \(r\) is the monthly interest rate (0.0022),
• \(n\) is the number of payments (180).

Substituting the values: \[ M = 27800 \frac{0.0022(1+0.0022)^{180}}{(1+0.0022)^{180} - 1} \approx 187.21 \]

To find the remaining balance after 5 years (60 months), we use the amortization formula: \[ B = P \frac{(1+r)^n - (1+r)^p}{(1+r)^n - 1} \] where:

• \(B\) is the remaining balance,
• \(P\) is the loan amount (\$27,800),
• \(r\) is the monthly interest rate (0.0022),
• \(n\) is the number of payments (180),
• \(p\) is the number of payments elapsed (60).

Substituting the values: \[ B = 27800 \frac{(1+0.0022)^{180} - (1+0.0022)^{60}}{(1+0.0022)^{180} - 1} \approx 19724.93 \]

Final Answer