Question

Justin wants to borrow $19,864 to buy a used car. He examined his budget and decided that he can afford a payment of $400 per month. If his bank offers him an APR of 9%, how long should he borrow the money so he can afford his monthly payment? Round to the nearest tenth of a year.

Answer 1

Justin should borrow the money for approximately 4.9 years so he can afford his monthly payment of $400.

Explanation:

To determine how long Justin should borrow the money so he can afford his monthly payment, we need to calculate the loan term.

Justin wants to borrow $19,864 and can afford a monthly payment of $400. The annual percentage rate (APR) offered by his bank is 9%.

First, we need to calculate the monthly interest rate. We divide the APR by 12 to get the monthly interest rate:

Monthly interest rate = 9% / 12 = 0.09 / 12 = 0.0075

Next, we can use the loan payment formula to find the loan term. The loan payment formula is:

Loan Payment = (Loan Amount * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-n))

Where:

Loan Payment = $400 (Justin's monthly payment)

Loan Amount = $19,864

Monthly interest rate = 0.0075 (calculated above)

n = loan term in months (what we're trying to find)

Substituting the known values into the loan payment formula:

$400 = (19864 * 0.0075) / (1 - (1 + 0.0075)^(-n))

To solve for n, we can rearrange the equation and solve for the loan term using logarithms or trial and error. In this case, we can use trial and error or a financial calculator to find the loan term that gives us a monthly payment of $400.

After performing the calculation, we find that the loan term is approximately 4.9 years (rounded to the nearest tenth of a year).

Therefore, Justin should borrow the money for approximately 4.9 years so he can afford his monthly payment of $400.

Answer 2

Explanation:

To determine how long Justin should borrow the money, we can use the formula for calculating the number of periods (or months) needed to repay a loan:

n = (log(PMT) - log(PMT - (PV × i))) / log(1 + i)

Where:

n = number of periods (in months)

PMT = monthly payment ($400)

PV = present value (loan amount, $19,864)

i = monthly interest rate (APR divided by 12 months and 100)

Let's calculate it step by step:

1. Convert the annual interest rate to a monthly rate:

i = 9% / 12 / 100 = 0.0075

2. Substitute the values into the formula:

n = (log(400) - log(400 - (19864 × 0.0075))) / log(1 + 0.0075)

3. Use a calculator to evaluate the expression inside the logarithms:

n ≈ (log(400) - log(400 - 148.98)) / log(1.0075)

4. Simplify the expression inside the logarithms:

n ≈ (2.602 - 2.575) / 0.001318

5. Evaluate the remaining expression:

n ≈ 0.027 / 0.001318

6. Calculate the final result:

n ≈ 20.5

Therefore, Justin should borrow the money for approximately 20.5 months in order to afford his monthly payment of $400.

Rounding to the nearest tenth of a year, Justin should borrow the money for approximately 20.5 / 12 = 1.7 years.