Laura Kennedy needs to borrow $5009 to pay for NBA season tickets for her family. She can borrow the amount from a finance company (at 7.8% add-on interest for 3 years) or from the credit union (36 monthly payments of $160.93 each). Laura  takes the credit union loan. At the time of her thirtieth payment she pays it off, If the credit union uses the rule of 78 for computing unearned interest, how much will she save by paying off the loan with the thirtieth payment? Round to the nearest cent if needed.

Let's revisit the problem and perform detailed calculations to ensure accuracy.

### Step 1: Calculate the Total Interest for the Credit Union Loan

Given:
- Monthly payment: $160.93
- Total number of payments: 36
- Principal amount: $5009

First, calculate the total amount paid over the life of the loan:
\[
\text{Total amount paid} = 36 \times 160.93 = 5793.48 \text{ dollars}
\]

Next, find the total interest paid:
\[
\text{Total interest} = \text{Total amount paid} - \text{Principal} = 5793.48 - 5009 = 784.48 \text{ dollars}
\]

### Step 2: Compute the Interest Paid by the Thirtieth Payment Using the Rule of 78

The Rule of 78 formula weights earlier payments more heavily. The sum of the digits for a 36-month loan is:
\[
\text{Sum of the digits} = \frac{36 \times (36 + 1)}{2} = 666
\]

Calculate the sum of the digits for the first 30 months:
\[
\text{Sum of the digits for the first 30 months} = \frac{30 \times (30 + 1)}{2} = 465
\]

The interest paid by the thirtieth payment is proportional to this sum:
\[
\text{Interest paid by the thirtieth payment} = \left(\frac{465}{666}\right) \times 784.48
\]

Calculate this amount:
\[
\text{Interest paid by the thirtieth payment} = \frac{465}{666} \times 784.48 = 547.36 \text{ dollars}
\]

### Step 3: Determine the Unearned Interest and Savings

The unearned interest for the remaining 6 months is:
\[
\text{Unearned interest} = \text{Total interest} - \text{Interest paid by the thirtieth payment} = 784.48 - 547.36 = 237.12 \text{ dollars}
\]

Therefore, Laura Kennedy saves $\$237.12$ by paying off the loan with the thirtieth payment.

### Summary
Laura Kennedy saves **$237.12** by paying off her loan early at the thirtieth payment using the Rule of 78.

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Do you want details or have any questions?

Here are some additional questions you might be interested in:
1. What is the Rule of 78 and how is it applied in loan calculations?
2. How does the Rule of 78 compare to simple interest calculations for loans?
3. What are the advantages and disadvantages of the Rule of 78 for borrowers?
4. How would the savings change if the loan term were different, e.g., 24 months instead of 36 months?
5. How would the interest calculation change if Laura made a larger principal payment halfway through the loan?
6. Can the Rule of 78 be applied to any type of loan, or is it specific to certain loan agreements?
7. What would happen if Laura missed a payment and then decided to pay off the loan early?
8. How does paying off a loan early affect a borrower's credit score?

**Tip:** Always check the terms and conditions of your loan agreement for any penalties or fees associated with early repayment.

Learn how Laura Kennedy saves $237.12 by paying off her credit union loan early at the thirtieth payment using the Rule of 78. This problem involves detailed calculations of total payments, interest, and unearned interest, demonstrating advanced concepts in finance and interest calculations.

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