Determine the total​ first-year cost of car ownership for Milagros. She just purchased a vehicle valued for ​$20 comma 660 with the following​ costs:Auto​ Loan: Amountlong dash​$20 comma 660​, Durationlong dash4 ​years, APRlong dash8.50 percent
Property​ Taxes: 1.88 percent of vehicle​ value/year
Sales​ Taxes: 3.44 percent of the sales price
Title and​ Tags: ​$47.45​/year
Maintenance and Usage​ Costs: ​$1 comma 992.56​/year
​Insurance: ​$1 comma 551.65​/year

To determine the total first-year cost of car ownership for Milagros, we need to add up all the costs associated with owning the car in the first year. Here’s the breakdown:

### 1. **Sales Taxes**
   - Sales Tax Rate: $ 3.44\% $
   - Sales Tax Amount: 
     \[
     \text{Sales Tax} = 20,660 \times \frac{3.44}{100} = 710.70
     \]

### 2. **Auto Loan Payments (First Year)**
   - Principal: $ 20,660 $
   - Duration: $ 4 $ years
   - APR: $ 8.5\% $
   
   Since this is a fixed-term loan, we can calculate the monthly payment using the formula for a loan payment:
   \[
   M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1}
   \]
   where:
   - $ P = 20,660 $ (principal amount)
   - $ r = \frac{8.5\%}{12} = \frac{0.085}{12} \approx 0.0070833 $ (monthly interest rate)
   - $ n = 4 \times 12 = 48 $ (number of payments over 4 years)
   
   Plugging these values in:
   \[
   M \approx \frac{20,660 \times 0.0070833 \times (1 + 0.0070833)^{48}}{(1 + 0.0070833)^{48} - 1} \approx 508.73
   \]
   
   - Total Loan Payments for the First Year:
     \[
     508.73 \times 12 = 6,104.76
     \]

### 3. **Property Taxes**
   - Property Tax Rate: $ 1.88\% $ of vehicle value per year
   - Property Tax Amount:
     \[
     20,660 \times \frac{1.88}{100} = 388.41
     \]

### 4. **Title and Tags**
   - Cost: $ 47.45 $ per year

### 5. **Maintenance and Usage Costs**
   - Cost: $ 1,992.56 $ per year

### 6. **Insurance**
   - Cost: $ 1,551.65 $ per year

### Total First-Year Cost
Adding all these components together:
\[
710.70 + 6,104.76 + 388.41 + 47.45 + 1,992.56 + 1,551.65 = 10,795.53
\]

So, the **total first-year cost of car ownership** for Milagros is **$10,795.53**.

Would you like to explore any details on the calculations or have any questions?

---

Here are a few related questions for further exploration:
1. How would the first-year cost change if the loan APR was reduced to 5%?
2. What would be the impact of higher maintenance costs on the total first-year cost?
3. How does the total ownership cost change if the duration of the loan is extended to 5 years?
4. How much would Milagros save annually if she paid off the loan within two years?
5. What would be the impact on the cost if the sales tax rate increased by 1%?

**Tip:** Breaking down large financial calculations step-by-step can make it easier to understand and avoid errors.

Determine the total first-year cost of car ownership for Milagros. She just purchased a vehicle valued at $20,660 with the following costs: Auto Loan: Amount—$20,660, Duration—4 years, APR—8.50 percent; Property Taxes: 1.88 percent of vehicle value/year; Sales Taxes: 3.44 percent of the sales price; Title and Tags: $47.45/year; Maintenance and Usage Costs: $1,992.56/year; Insurance: $1,551.65/year.

Calculate the total first-year cost of car ownership for Milagros, including her auto loan, sales taxes, property taxes, and maintenance. Learn how to break down these calculations using formulas for sales tax, property tax, and loan payments. Ideal for high school students learning financial math.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation