“Calculating Hazel’s Monthly Car Payment: A Step-by-Step Guide”

QUESTION

H5.

Hazel borrowed $109,720 for five years at an APR (Annual Percentage Rate) of 6.80% to buy a car. What is Hazel’s monthly car payment?(HINT: Solve the equal monthly payment. 6.80% is an annual rate, and there are 12 months in a year.)

Show proper step by step calculation

ANSWER

“Calculating Hazel’s Monthly Car Payment: A Step-by-Step Guide”

To calculate Hazel’s monthly car payment, we can use the formula for equal monthly payments on a loan, also known as an amortization formula. This formula takes into account the principal amount borrowed, the annual interest rate, and the loan term. In Hazel’s case, she borrowed $109,720 for five years at an Annual Percentage Rate (APR) of 6.80%. We’ll break down the calculation into several steps.

Step 1: Convert APR to Monthly Interest Rate

The first step is to convert the annual interest rate (APR) to a monthly interest rate. Since there are 12 months in a year, we divide the annual rate by 12 to get the monthly rate:

Monthly Interest Rate = Annual Interest Rate / 12 Monthly Interest Rate = 6.80% / 12 Monthly Interest Rate = 0.5683% (rounded to four decimal places)

Step 2: Calculate the Number of Monthly Payments

Next, we need to calculate the total number of monthly payments over the loan term. Hazel’s loan term is five years, which is equivalent to 60 months (5 years x 12 months per year).

Step 3: Use the Amortization Formula

Now, we can use the amortization formula to calculate the monthly payment (PMT). The formula for calculating equal monthly payments is as follows:

���=�⋅�⋅(1+�)�(1+�)�−1PMT=(1+r)n−1P⋅r⋅(1+r)n​

Where:

• PMT = Monthly Payment
• P = Principal Loan Amount
• r = Monthly Interest Rate (in decimal form)
• n = Total Number of Payments

In Hazel’s case:

• P = $109,720
• r = 0.5683% (0.005683 in decimal form)
• n = 60 (number of monthly payments)

Now, let’s plug these values into the formula:

���=109,720⋅0.005683⋅(1+0.005683)60(1+0.005683)60−1PMT=(1+0.005683)60−1109,720⋅0.005683⋅(1+0.005683)60​

Step 4: Calculate the Monthly Payment

Now, we can calculate the monthly payment:

���=109,720⋅0.005683⋅(1.005683)60(1.005683)60−1PMT=(1.005683)60−1109,720⋅0.005683⋅(1.005683)60​

Using a calculator or spreadsheet, we can evaluate this expression to find Hazel’s monthly car payment.

After performing the calculations, Hazel’s monthly car payment comes out to approximately $2,123.34 (rounded to two decimal places).

So, Hazel’s monthly car payment is $2,123.34, and this is the amount she will need to pay each month for the next five years to repay her car loan of $109,720 with an APR of 6.80%.