You are the loan department supervisor for a bank. This installment loan is being paid off early, and it is your task to calculate the rebate fraction, the finance charge rebate (in $), and the payoff for the loan (in $). (Round dollars to the nearest cent.)
Amount
Financed	Number of
Payments	Monthly
Payment	Payments
Made	Rebate
Fraction	Finance
Charge
Rebate	Loan
Payoff
$1,700	18	$121.89	13
Incorrect: Your answer is incorrect.
$
Incorrect: Your answer is incorrect.
$
Incorrect: Your answer is incorrect.

To solve this, we need to calculate the **rebate fraction**, **finance charge rebate**, and **loan payoff** for a loan that is being paid off early. Here's the step-by-step method to approach it:

### 1. Calculate Total Loan Amount
First, we need to calculate the **total payments** and the **finance charge**:
- Amount Financed = $1,700
- Number of Payments = 18
- Monthly Payment = $121.89

The total amount to be paid over the life of the loan:
\[
\text{Total Payment Amount} = \text{Monthly Payment} \times \text{Number of Payments}
= 121.89 \times 18 = 2,194.02
\]
The finance charge is the difference between the total payment and the amount financed:
\[
\text{Finance Charge} = \text{Total Payment Amount} - \text{Amount Financed}
= 2,194.02 - 1,700 = 494.02
\]

### 2. Rebate Fraction
The **Rule of 78** is often used to calculate the rebate when paying off a loan early. The Rule of 78 weights earlier payments more heavily because more interest is paid in the beginning. It works like this:

The sum of the digits for 18 months is:
\[
S = 1 + 2 + 3 + \cdots + 18 = \frac{18 \times 19}{2} = 171
\]

Since 13 payments have already been made, the sum of the digits for the remaining 5 payments is:
\[
S_{\text{remaining}} = 1 + 2 + 3 + 4 + 5 = \frac{5 \times 6}{2} = 15
\]

The **rebate fraction** is the ratio of the sum of the remaining payments to the total sum of digits:
\[
\text{Rebate Fraction} = \frac{S_{\text{remaining}}}{S} = \frac{15}{171} = 0.0877
\]

### 3. Finance Charge Rebate
The finance charge rebate is calculated by multiplying the finance charge by the rebate fraction:
\[
\text{Finance Charge Rebate} = \text{Finance Charge} \times \text{Rebate Fraction} 
= 494.02 \times 0.0877 \approx 43.32
\]

### 4. Loan Payoff
The loan payoff is the amount financed, plus the remaining unpaid finance charge, minus the finance charge rebate. The unpaid finance charge is the total finance charge minus the amount already paid off:
\[
\text{Loan Payoff} = \text{Amount Financed} + \text{Unpaid Finance Charge} - \text{Finance Charge Rebate}
\]
The unpaid finance charge is:
\[
\text{Unpaid Finance Charge} = 494.02 - \left( \frac{13}{18} \times 494.02 \right) 
= 494.02 - 356.35 = 137.67
\]

Now we can calculate the loan payoff:
\[
\text{Loan Payoff} = 1,700 + 137.67 - 43.32 = 1,794.35
\]

### Final Answer:
- **Rebate Fraction**: 0.0877
- **Finance Charge Rebate**: $43.32
- **Loan Payoff**: $1,794.35

---

Would you like further details on this process? Let me know if any part needs more clarification.

### Related Questions:
1. How would the calculations change if 15 payments were made instead of 13?
2. What impact does the number of payments have on the rebate fraction?
3. Can you explain the Rule of 78 and its relevance to loan rebates?
4. How does paying off a loan early affect the total amount of interest paid?
5. What is the difference between simple interest and the Rule of 78 when calculating loan payoffs?

### Tip:
Always review the loan terms to understand how early repayments affect interest rebates and the total amount due.

You are the loan department supervisor for a bank. This installment loan is being paid off early, and it is your task to calculate the rebate fraction, the finance charge rebate (in $), and the payoff for the loan (in $). (Round dollars to the nearest cent.) Amount Financed: $1,700, Number of Payments: 18, Monthly Payment: $121.89, Payments Made: 13.

Learn how to calculate the rebate fraction, finance charge rebate, and loan payoff for a $1,700 installment loan being paid off early after 13 out of 18 payments. This problem uses the Rule of 78 and involves essential financial math concepts. Detailed formulas and step-by-step calculations provided.

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