Question

I'LL MARK AS BRAINLY Brenda can afford a $565-per-month automobile payment, and she's interested in either a pickup truck, which costs $28,100, or an SUV, which costs $28,500. She's being offered a 5-year loan with an APR of 7.2%, compounded monthly. Help Brenda figure out which of the vehicles she can afford.
Part I: What is the periodic interest rate of the loan that Brenda is being offered?
Part II: How many total payments does the loan that Brenda is being offered require?
Part III: What is the largest amount of money that Brenda can afford to borrow?
Part IV: Can Brenda afford the pickup truck?
Part V: Can Brenda afford the SUV?

Answer 1

Answer:

Part I:

0.6%

Part II

The total number of payment the 5 year loan Brenda is being offered requires 60 payments

Part III

The largest amount Brenda can afford to borrow is approximately $28,398.103

Part IV:

Brenda can afford the pickup truck

Part V:

Brenda cannot afford the SUV

Step-by-step explanation:

Part I:

The amount of monthly payment Brenda can afford = $565

The cost of the pickup truck = $28,100

The cost of an SUV = $28,500

The number of years loan she is offered = 5 year

The Annual Percentage Rate (APR) of the loan = 7.2%

The rate at which the interest is compounded = Monthly

Part I Given that the interest rate is compounded monthly, we have the periodic interest rate for an APR of 7.2% compounded monthly = 7.2/12 = 0.6%

∴ The periodic interest rate = 0.6%

Part II

For a 5 year loan with month payment, we have;

The number of month per year = 12 months

∴ The number of payment per year = 12

The total number of payment the 5 year loan Brenda is being offered requires 5 × 12 = 60 payments

Part III The total amount Brenda can afford to pay in 5 years, is given by the formula for monthly payment on loan as follows;

$A = P \times\dfrac{r \times \left (1 + r \right )^n}{\left (1 + r \right )^n - 1}$

Where;

A = The amount (she can) payed monthly = $565

r = The periodic interest rate (APR/12) = 7.2%/12 = 0.6%/12 = 0.006

t = The number of time period for the loan duration = 5 years

n = The number of payments made = 12 × 5 = 60 payment

P = The amount (she can) borrowed as loan

Plugging in the values gives;

$565 = P \times\dfrac{0.006 \times \left (1 + 0.006 \right )^{60}}{\left (1 + 0.006 \right )^{60} - 1} \approx P \times 1.9896 \times 10^{-2}$

$P = \dfrac{565}{1.9896 \times 10^{-2}} \approx 28398.103$

∴ The largest amount Brenda can borrow as loan = P ≈ $28,398.103

Part IV: The cost of the pickup truck = $28,100 which is less than the largest amount Brenda can borrow as loan which is approximately $28,398.103, therefore, Brenda can afford the pickup truck

Part V: The cost of the SUV = $28,500 which is more than the largest amount Brenda can borrow as loan which is approximately $28,398.103, therefore, Brenda cannot afford the SUV.