Calculate The Monthly Payment For A \$14,790 Auto Loan Over Four Years At $6.9\%$ Annual Interest.Use The Formula: $ P = A \left(\frac{r}{n}\right) \frac{\left(1+\frac{r}{n}\right)^{nt}}{\left(1+\frac{r}{n}\right)^{nt} - 1} $Where: -

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Understanding Auto Loan Calculations

Calculating the monthly payment for an auto loan can be a daunting task, especially for those who are not familiar with financial mathematics. However, with the right formula and a clear understanding of the variables involved, anyone can calculate their monthly payments with ease. In this article, we will use the formula for calculating monthly payments on a fixed-rate loan, which is given by:

$ P = A \left(\frac{r}{n}\right) \frac{\left(1+\frac{r}{n}\right){nt}}{\left(1+\frac{r}{n}\right){nt} - 1} $

Where:

  • PP is the monthly payment
  • AA is the principal amount (the initial amount borrowed)
  • rr is the annual interest rate (in decimal form)
  • nn is the number of payments per year
  • tt is the number of years the money is borrowed for

Breaking Down the Variables

Before we dive into the calculation, let's break down the variables involved:

  • Principal Amount (A): The principal amount is the initial amount borrowed, which in this case is $14,790.
  • Annual Interest Rate (r): The annual interest rate is the rate at which interest is charged on the loan. In this case, the annual interest rate is 6.9%, which is equivalent to 0.069 in decimal form.
  • Number of Payments per Year (n): The number of payments per year is the number of times interest is compounded per year. In this case, we will assume that interest is compounded monthly, so n = 12.
  • Number of Years (t): The number of years is the length of time the money is borrowed for. In this case, we will assume that the loan is for 4 years.

Calculating the Monthly Payment

Now that we have broken down the variables, let's plug them into the formula:

$ P = 14790 \left(\frac{0.069}{12}\right) \frac{\left(1+\frac{0.069}{12}\right)^{12 \cdot 4}}{\left(1+\frac{0.069}{12}\right)^{12 \cdot 4} - 1} $

To calculate the monthly payment, we need to follow the order of operations (PEMDAS):

  1. Calculate the value inside the parentheses: (1+0.06912)12â‹…4\left(1+\frac{0.069}{12}\right)^{12 \cdot 4}
  2. Calculate the value of the denominator: (1+0.06912)12⋅4−1\left(1+\frac{0.069}{12}\right)^{12 \cdot 4} - 1
  3. Divide the value of the numerator by the value of the denominator
  4. Multiply the result by the principal amount

Performing the Calculation

Let's perform the calculation step by step:

  1. Calculate the value inside the parentheses: (1+0.06912)12â‹…4\left(1+\frac{0.069}{12}\right)^{12 \cdot 4}

(1+0.06912)12⋅4=(1+0.00575)48=1.0057548≈1.3433\left(1+\frac{0.069}{12}\right)^{12 \cdot 4} = \left(1+0.00575\right)^{48} = 1.00575^{48} \approx 1.3433

  1. Calculate the value of the denominator: (1+0.06912)12⋅4−1\left(1+\frac{0.069}{12}\right)^{12 \cdot 4} - 1

(1+0.06912)12⋅4−1=1.3433−1=0.3433\left(1+\frac{0.069}{12}\right)^{12 \cdot 4} - 1 = 1.3433 - 1 = 0.3433

  1. Divide the value of the numerator by the value of the denominator

1.34330.3433≈3.915\frac{1.3433}{0.3433} \approx 3.915

  1. Multiply the result by the principal amount

14790⋅3.915≈578.1914790 \cdot 3.915 \approx 578.19

Conclusion

In conclusion, the monthly payment for a $14,790 auto loan over four years at 6.9% annual interest is approximately $578.19. This calculation assumes that interest is compounded monthly and that the loan is for 4 years. By using the formula for calculating monthly payments on a fixed-rate loan, anyone can calculate their monthly payments with ease.

Tips and Variations

  • Compounding Frequency: The calculation assumes that interest is compounded monthly. However, interest can be compounded daily, weekly, or quarterly, depending on the loan terms.
  • Loan Terms: The calculation assumes that the loan is for 4 years. However, the loan terms can vary, and the calculation can be adjusted accordingly.
  • Interest Rate: The calculation assumes that the annual interest rate is 6.9%. However, the interest rate can vary, and the calculation can be adjusted accordingly.

Q: What is the formula for calculating monthly payments on a fixed-rate loan?

A: The formula for calculating monthly payments on a fixed-rate loan is given by:

$ P = A \left(\frac{r}{n}\right) \frac{\left(1+\frac{r}{n}\right){nt}}{\left(1+\frac{r}{n}\right){nt} - 1} $

Where:

  • PP is the monthly payment
  • AA is the principal amount (the initial amount borrowed)
  • rr is the annual interest rate (in decimal form)
  • nn is the number of payments per year
  • tt is the number of years the money is borrowed for

Q: What is the principal amount (A) in the formula?

A: The principal amount (A) is the initial amount borrowed. For example, if you borrow $14,790 to buy a car, the principal amount is $14,790.

Q: What is the annual interest rate (r) in the formula?

A: The annual interest rate (r) is the rate at which interest is charged on the loan. For example, if the annual interest rate is 6.9%, the value of r is 0.069.

Q: What is the number of payments per year (n) in the formula?

A: The number of payments per year (n) is the number of times interest is compounded per year. For example, if interest is compounded monthly, the value of n is 12.

Q: What is the number of years (t) in the formula?

A: The number of years (t) is the length of time the money is borrowed for. For example, if the loan is for 4 years, the value of t is 4.

Q: How do I calculate the monthly payment using the formula?

A: To calculate the monthly payment using the formula, you need to follow the order of operations (PEMDAS):

  1. Calculate the value inside the parentheses: (1+rn)nt\left(1+\frac{r}{n}\right)^{nt}
  2. Calculate the value of the denominator: (1+rn)nt−1\left(1+\frac{r}{n}\right)^{nt} - 1
  3. Divide the value of the numerator by the value of the denominator
  4. Multiply the result by the principal amount

Q: What is the monthly payment for a $14,790 auto loan over four years at 6.9% annual interest?

A: The monthly payment for a $14,790 auto loan over four years at 6.9% annual interest is approximately $578.19.

Q: Can I use the formula to calculate the monthly payment for a loan with a different interest rate or loan term?

A: Yes, you can use the formula to calculate the monthly payment for a loan with a different interest rate or loan term. Simply plug in the new values for the principal amount, annual interest rate, number of payments per year, and number of years, and follow the order of operations (PEMDAS).

Q: What are some common mistakes to avoid when calculating monthly payments?

A: Some common mistakes to avoid when calculating monthly payments include:

  • Not using the correct formula
  • Not plugging in the correct values for the principal amount, annual interest rate, number of payments per year, and number of years
  • Not following the order of operations (PEMDAS)
  • Not rounding the result to the nearest cent

By understanding the formula and avoiding common mistakes, you can accurately calculate your monthly payments and make informed decisions about your finances.