Questions: Nadia needs to borrow 6000 to pay for ocean cruise tickets for her family. She borrows from the credit union with 36 monthly payments of 187.16 each with an APR of 6.5%. What would Nadia save in interest if she paid in full at the time of the twenty-fourth payment and the credit union used the actuarial method for computing unearned interest? Nadia would save in interest. (Round to the nearest cent as needed.)

Nadia needs to borrow 6000 to pay for ocean cruise tickets for her family. She borrows from the credit union with 36 monthly payments of 187.16 each with an APR of 6.5%. What would Nadia save in interest if she paid in full at the time of the twenty-fourth payment and the credit union used the actuarial method for computing unearned interest?

Nadia would save in interest. (Round to the nearest cent as needed.)

Transcript text: Nadia needs to borrow $\$ 6000$ to pay for ocean cruise tickets for her family. She borrows from the credit union with 36 monthly payments of $\$ 187.16$ each with an APR of $6.5 \%$. What would Nadia save in interest if she paid in full at the time of the twenty-fourth payment and the credit union used the actuarial method for computing unearned interest? Nadia would save $\$$ $\square$ in interest. (Round to the nearest cent as needed.)

Solution

Solution Steps

To determine how much Nadia would save in interest if she paid off her loan at the time of the twenty-fourth payment, we need to calculate the total interest paid over the full term and compare it to the interest paid up to the twenty-fourth payment. The actuarial method will be used to compute the unearned interest.

Calculate the total amount paid over the full term of the loan.
Calculate the total interest paid over the full term.
Calculate the remaining balance after the twenty-fourth payment.
Calculate the interest paid up to the twenty-fourth payment.
Determine the unearned interest and the savings.

Step 1: Total Amount Paid Over the Full Term

The total amount paid over the full term of the loan is calculated as: \[ \text{Total Amount Paid} = \text{Monthly Payment} \times \text{Total Payments} = 187.16 \times 36 = 6737.76 \]

Step 2: Total Interest Paid Over the Full Term

The total interest paid over the full term is given by: \[ \text{Total Interest Paid} = \text{Total Amount Paid} - \text{Loan Amount} = 6737.76 - 6000 = 737.76 \]

Step 3: Remaining Balance After 24 Payments

To find the remaining balance after 24 payments, we calculate the interest and principal paid for each month. After 24 payments, the remaining balance is: \[ \text{Remaining Balance} \approx 2047.4919 \]

Step 4: Total Paid Up to the 24th Payment

The total amount paid up to the 24th payment is: \[ \text{Total Paid Up to 24} = \text{Monthly Payment} \times \text{Payments Made} = 187.16 \times 24 = 4491.84 \]

Step 5: Interest Paid Up to the 24th Payment

The interest paid up to the 24th payment is calculated as: \[ \text{Interest Paid Up to 24} = \text{Total Paid Up to 24} - (\text{Loan Amount} - \text{Remaining Balance}) = 4491.84 - (6000 - 2047.4919) \approx 539.3319 \]

Step 6: Unearned Interest

The unearned interest, which represents the interest that would not be paid if the loan is paid off early, is: \[ \text{Unearned Interest} = \text{Total Interest Paid} - \text{Interest Paid Up to 24} = 737.76 - 539.3319 \approx 198.4281 \]

Step 7: Savings in Interest

Thus, the savings in interest if Nadia pays off the loan at the time of the twenty-fourth payment is: \[ \text{Savings} \approx 198.43 \]