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 computig unearned interest, how much will she save by paying off the loan with the thirtieth payment?

To determine how much Laura Kennedy saves by paying off her credit union loan with the thirtieth payment using the Rule of 78 for computing unearned interest, we need to follow these steps:

1. **Calculate the total interest paid over the life of the loan.**
2. **Compute the amount of interest she has already paid by the time she makes her thirtieth payment.**
3. **Determine the amount of unearned interest and thus how much she saves by paying off early.**

### Step 1: Calculate the total interest paid over the life of the loan

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

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:
\[
\text{Total interest} = \text{Total amount paid} - \text{Principal}
\]
\[
\text{Total interest} = 5793.48 - 5009 = 784.48 \text{ dollars}
\]

### Step 2: Compute the amount of interest she has already paid by the thirtieth payment using the Rule of 78

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

The interest paid each month is proportional to the reverse of the remaining months. By the 30th month, she has made 30 payments, so there are 6 months remaining.

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

The interest she has paid in the first 30 months is:
\[
\text{Interest paid in first 30 months} = \left(\frac{465}{666}\right) \times 784.48 = 547.36 \text{ dollars}
\]

### Step 3: Determine the amount of unearned interest and savings

The amount of unearned interest is the interest for the remaining 6 months:
\[
\text{Unearned interest} = \text{Total interest} - \text{Interest paid in first 30 months}
\]
\[
\text{Unearned interest} = 784.48 - 547.36 = 237.12 \text{ dollars}
\]

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.

Calculate how Laura Kennedy saves money by paying off her credit union loan early using the Rule of 78. Understand the mechanics of interest calculation, monthly payments, and total savings. Suitable for advanced high school students and anyone interested in financial calculations.

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