Juanita Juarez bought a computer system for $$ \$ 1200 $$. She made a down payment of $$ \$ 120 $$ and arranged to make 26 monthly payments of $$ \$ 48.46 $$. After making 17 payments she decided to pay the loan in full. Use the Rule of 78 to determine the unearned interest and the amount necessary to pay the loan in full. The unearned interest is $$ \$ \square $$. (Round to the nearest cent.) The amount required to pay the loan in full is $$ \$ \square $$

Question

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To solve this problem, we will follow these steps:

1. **Calculate the total interest on the loan.**
2. **Determine the unearned interest using the Rule of 78.**
3. **Calculate the amount necessary to pay the loan in full after 17 payments.**

### Step 1: Calculate the Total Interest on the Loan

The total cost of the computer system is $$ \$1200 $$. The down payment is $$ \$120 $$, so the amount financed is:

$$
P = 1200 - 120 = 1080
$$

The total amount paid over the 26 monthly payments is:

$$
\text{Total Payments} = 26 \times 48.46
$$

Now, let's calculate the total payments:

$$
\text{Total Payments} = 26 \times 48.46
$$

### Step 2: Calculate the Total Interest

The total interest paid over the life of the loan is:

$$
\text{Total Interest} = \text{Total Payments} - P
$$

### Step 3: Determine the Unearned Interest Using the Rule of 78

The Rule of 78 allocates interest based on the sum of the digits of the number of payments. For 26 payments, the sum of the digits is:

$$
S = 1 + 2 + 3 + \ldots + 26 = \frac{26 \times (26 + 1)}{2} = 351
$$

After 17 payments, the remaining payments are $$ 26 - 17 = 9 $$. The sum of the digits for the remaining payments is:

$$
S_{\text{remaining}} = 1 + 2 + 3 + \ldots + 9 = \frac{9 \times (9 + 1)}{2} = 45
$$

The unearned interest is calculated as follows:

$$
\text{Unearned Interest} = \text{Total Interest} \times \frac{S_{\text{remaining}}}{S}
$$

### Step 4: Calculate the Amount Necessary to Pay the Loan in Full

The amount necessary to pay the loan in full is:

$$
\text{Amount to Pay Off} = P + \text{Unearned Interest}
$$

Now, let's perform the calculations.

1. Calculate the total payments.
2. Calculate the total interest.
3. Calculate the unearned interest.
4. Calculate the amount necessary to pay the loan in full.

Let's execute these calculations.
Numbers by following steps:
- step0: Solution:
  $$351$$
Calculate or simplify the expression $$ 26 * 48.46 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$26\times 48.46$$
- step1: Multiply the numbers:
  $$1259.96$$
Calculate or simplify the expression $$ 1259.96 - 1080 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1259.96-1080$$
- step1: Subtract the numbers:
  $$179.96$$
Calculate or simplify the expression $$ 179.96 * (45 / 351) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$179.96\left(\frac{45}{351}\right)$$
- step1: Reduce the fraction:
  $$179.96\times \frac{5}{39}$$
- step2: Convert the expressions:
  $$\frac{4499}{25}\times \frac{5}{39}$$
- step3: Multiply the fractions:
  $$\frac{4499\times 5}{25\times 39}$$
- step4: Multiply the terms:
  $$\frac{22495}{25\times 39}$$
- step5: Multiply the terms:
  $$\frac{22495}{975}$$
- step6: Reduce the fraction:
  $$\frac{4499}{195}$$
Calculate or simplify the expression $$ 1080 + (179.96 * (45 / 351)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1080+\left(179.96\left(\frac{45}{351}\right)\right)$$
- step1: Reduce the fraction:
  $$1080+\left(179.96\times \frac{5}{39}\right)$$
- step2: Multiply the numbers:
  $$1080+\frac{4499}{195}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{1080\times 195}{195}+\frac{4499}{195}$$
- step4: Transform the expression:
  $$\frac{1080\times 195+4499}{195}$$
- step5: Multiply the numbers:
  $$\frac{210600+4499}{195}$$
- step6: Add the numbers:
  $$\frac{215099}{195}$$
Now, let's summarize the calculations step by step:

1. **Total Payments:**
   $$
   \text{Total Payments} = 26 \times 48.46 = 1259.96
   $$

2. **Amount Financed:**
   $$
   P = 1200 - 120 = 1080
   $$

3. **Total Interest:**
   $$
   \text{Total Interest} = 1259.96 - 1080 = 179.96
   $$

4. **Unearned Interest Calculation:**
   - The sum of the digits for the remaining payments is $$ S_{\text{remaining}} = 45 $$.
   - The total sum of the digits for all payments is $$ S = 351 $$.
   $$
   \text{Unearned Interest} = 179.96 \times \frac{45}{351} \approx 23.07
   $$

5. **Amount Necessary to Pay the Loan in Full:**
   $$
   \text{Amount to Pay Off} = 1080 + 23.07 \approx 1103.07
   $$

### Final Answers:
- The unearned interest is $$ \$23.07 $$.
- The amount required to pay the loan in full is $$ \$1103.07 $$.

Thus, we have:
- The unearned interest is $$ \$ \text{23.07} $$.
- The amount required to pay the loan in full is $$ \$ \text{1103.07} $$.
The unearned interest is \$23.07, and the amount required to pay the loan in full is \$1103.07.

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