Let's go through a detailed, step-by-step solution to find the size of the monthly payment for Hunan's car loan. We'll break down the calculations as follows:
1. Identify the given information:
- Car price: [tex]$15,800
- Down payment percentage: 15% (or 0.15 in decimal form)
- Annual interest rate: 7% (or 0.07 in decimal form)
- Loan term: 5 years
2. Calculate the down payment:
The down payment is a percentage of the car price.
\[
\text{Down payment} = \text{Car price} \times \text{Down payment percentage}
\]
Substituting the values:
\[
\text{Down payment} = 15800 \times 0.15 = 2370
\]
3. Calculate the loan amount:
The loan amount is the car price minus the down payment.
\[
\text{Loan amount} = \text{Car price} - \text{Down payment}
\]
Substituting the values:
\[
\text{Loan amount} = 15800 - 2370 = 13430
\]
4. Converting the annual interest rate to a monthly interest rate:
Since the interest is compounded monthly, we need to convert the annual interest rate to a monthly rate.
\[
\text{Monthly interest rate} = \frac{\text{Annual interest rate}}{12}
\]
Substituting the values:
\[
\text{Monthly interest rate} = \frac{0.07}{12} = 0.005833 \quad (\text{rounded to six decimal places})
\]
5. Calculate the total number of monthly payments:
Multiplying the number of years by the number of months in a year gives the total number of monthly payments.
\[
\text{Total payments} = \text{Years} \times 12
\]
Substituting the values:
\[
\text{Total payments} = 5 \times 12 = 60
\]
6. Calculate the monthly payment using the annuity formula:
The annuity formula to calculate the monthly payment is given by:
\[
\text{Monthly payment} = \text{Loan amount} \times \frac{\text{Monthly interest rate} \times (1 + \text{Monthly interest rate})^{\text{Total payments}}}{(1 + \text{Monthly interest rate})^{\text{Total payments}} - 1}
\]
Substituting the values:
\[
\text{Monthly payment} = 13430 \times \frac{0.005833 \times (1 + 0.005833)^{60}}{(1 + 0.005833)^{60} - 1}
\]
7. Simplify the calculation:
First, calculate \((1 + 0.005833)^{60}\):
\[
(1 + 0.005833)^{60} \approx 1.416599
\]
Now, calculate the numerator and the denominator:
\[
\text{Numerator} = 0.005833 \times 1.416599 = 0.008265
\]
\[
\text{Denominator} = 1.416599 - 1 = 0.416599
\]
Finally, calculate the monthly payment:
\[
\text{Monthly payment} = 13430 \times \frac{0.008265}{0.416599} \approx 13430 \times 0.01984 = 265.93
\]
8. Final answer:
The size of the monthly payment is \(\$[/tex]265.93\).
So, Hunan's monthly payment for the car loan is $265.93.
